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1-> ; This is a Mathomatic script that reads in all test scripts.
1->
1-> clear all
1-> ; Test simplifying trig functions:
1-> read trig.in
1-> ; Trigonometric functions as complex exponentials.
1-> ; Use m4 Mathomatic instead for easy entry of these trig functions.
1-> ; Based on Euler's identity: e^(i*x) = cos(x) + i*sin(x)
1-> ; Variable x is an angle in radians.
1->
1-> ; Unity relationship: sin(x)^2 + cos(x)^2 = 1
1->
1-> ; sin(x) (sine of x) = cos(pi/2 - x)
1-> sin=(e^(i*x)-e^(-i*x))/(2i)
((e^(i*x)) - (e^(-i*x)))
#1: sin = ------------------------
(2*i)
1->
1-> ; cos(x) (cosine of x) = sin(pi/2 - x)
1-> cos=(e^(i*x)+e^(-i*x))/2
((e^(i*x)) + (e^(-i*x)))
#2: cos = ------------------------
2
2->
2-> ; tan(x) (tangent of x) = sin(x)/cos(x) = cot(pi/2 - x)
2-> tan=(e^(i*x)-e^(-i*x))/(i*(e^(i*x)+e^(-i*x)))
((e^(i*x)) - (e^(-i*x)))
#3: tan = ----------------------------
(i*((e^(i*x)) + (e^(-i*x))))
3->
3-> ; cot(x) (cotangent of x) = cos(x)/sin(x) = tan(pi/2 - x)
3-> cot=i*(e^(i*x)+e^(-i*x))/(e^(i*x)-e^(-i*x))
i*((e^(i*x)) + (e^(-i*x)))
#4: cot = --------------------------
((e^(i*x)) - (e^(-i*x)))
4->
4-> ; sec(x) (secant of x) = 1/cos(x) = csc(pi/2 - x)
4-> sec=2/(e^(i*x)+e^(-i*x))
2
#5: sec = ------------------------
((e^(i*x)) + (e^(-i*x)))
5->
5-> ; csc(x) (cosecant of x) = 1/sin(x) = sec(pi/2 - x)
5-> csc=2i/(e^(i*x)-e^(-i*x))
2*i
#6: csc = ------------------------
((e^(i*x)) - (e^(-i*x)))
Successfully finished reading script file "trig.in".
6-> read hypertrig.in
6-> ; Definitions for hyperbolic trigonometry.
6-> ; Use m4 Mathomatic instead for easy entry of these hypertrig functions.
6-> ; Based on the identity cosh(x)^2-sinh(x)^2 = 1.
6->
6-> ; sinh(x); hyperbolic sine of x
6-> sinh=(e^x-e^-x)/2
(e^x - (e^(-x)))
#7: sinh = ----------------
2
7->
7-> ; cosh(x); hyperbolic cosine of x
7-> cosh=(e^x+e^-x)/2
(e^x + (e^(-x)))
#8: cosh = ----------------
2
8->
8-> ; tanh(x); hyperbolic tangent of x
8-> tanh=(e^x-e^-x)/(e^x+e^-x)
(e^x - (e^(-x)))
#9: tanh = ----------------
(e^x + (e^(-x)))
9->
9-> ; coth(x); hyperbolic cotangent of x
9-> coth=(e^x+e^-x)/(e^x-e^-x)
(e^x + (e^(-x)))
#10: coth = ----------------
(e^x - (e^(-x)))
10->
10-> ; sech(x); hyperbolic secant of x
10-> sech=2/(e^x+e^-x)
2
#11: sech = ----------------
(e^x + (e^(-x)))
11->
11-> ; csch(x); hyperbolic cosecant of x
11-> csch=2/(e^x-e^-x)
2
#12: csch = ----------------
(e^x - (e^(-x)))
Successfully finished reading script file "hypertrig.in".
12-> simplify all
1
i*(--------- - (e^(i*x)))
(e^(i*x))
#1: sin = -------------------------
2
1
((e^(i*x)) + ---------)
(e^(i*x))
#2: cos = -----------------------
2
2
#3: tan = i*(----------------- - 1)
((e^(2*i*x)) + 1)
2
#4: cot = i*(1 + -----------------)
((e^(2*i*x)) - 1)
2*(e^(i*x))
#5: sec = -----------------
((e^(2*i*x)) + 1)
2*i*(e^(i*x))
#6: csc = -----------------
((e^(2*i*x)) - 1)
1
(e^x - ---)
e^x
#7: sinh = -----------
2
1
(e^x + ---)
e^x
#8: cosh = -----------
2
2
#9: tanh = 1 - ---------------
((e^(2*x)) + 1)
2
#10: coth = 1 + ---------------
((e^(2*x)) - 1)
2*e^x
#11: sech = ---------------
((e^(2*x)) + 1)
2*e^x
#12: csch = ---------------
((e^(2*x)) - 1)
12-> simplify frac all
1
i*(--------- - (e^(i*x)))
(e^(i*x))
#1: sin = -------------------------
2
1
((e^(i*x)) + ---------)
(e^(i*x))
#2: cos = -----------------------
2
i*(1 - (e^(2*i*x)))
#3: tan = -------------------
((e^(2*i*x)) + 1)
i*((e^(2*i*x)) + 1)
#4: cot = -------------------
((e^(2*i*x)) - 1)
2*(e^(i*x))
#5: sec = -----------------
((e^(2*i*x)) + 1)
2*i*(e^(i*x))
#6: csc = -----------------
((e^(2*i*x)) - 1)
1
(e^x - ---)
e^x
#7: sinh = -----------
2
1
(e^x + ---)
e^x
#8: cosh = -----------
2
((e^(2*x)) - 1)
#9: tanh = ---------------
((e^(2*x)) + 1)
((e^(2*x)) + 1)
#10: coth = ---------------
((e^(2*x)) - 1)
2*e^x
#11: sech = ---------------
((e^(2*x)) + 1)
2*e^x
#12: csch = ---------------
((e^(2*x)) - 1)
12-> ; Let's simplify some trig identities without using m4:
12-> sin^2+cos^2=1
#13: sin^2 + cos^2 = 1
13-> elim all
Eliminating variable cos using solved equation #2...
Eliminating variable sin using solved equation #1...
1 1
i*(--------- - (e^(i*x))) ((e^(i*x)) + ---------)
(e^(i*x)) (e^(i*x))
#13: (-------------------------^2) + (-----------------------^2) = 1
2 2
13-> simplify
#13: 1 = 1
13-> tan=sin/cos
sin
#14: tan = ---
cos
14-> elim all
Eliminating variable tan using solved equation #3...
Eliminating variable cos using solved equation #2...
Eliminating variable sin using solved equation #1...
1
i*(--------- - (e^(i*x)))
i*(1 - (e^(2*i*x))) (e^(i*x))
#14: ------------------- = -------------------------
((e^(2*i*x)) + 1) 1
((e^(i*x)) + ---------)
(e^(i*x))
14-> csc=1/sin
1
#15: csc = ---
sin
15-> elim all
Eliminating variable csc using solved equation #6...
Eliminating variable sin using solved equation #1...
2*i*(e^(i*x)) 2
#15: ----------------- = ---------------------------
((e^(2*i*x)) - 1) 1
(i*(--------- - (e^(i*x))))
(e^(i*x))
15-> sec=1/cos
1
#16: sec = ---
cos
16-> elim all
Eliminating variable sec using solved equation #5...
Eliminating variable cos using solved equation #2...
2*(e^(i*x)) 2
#16: ----------------- = -----------------------
((e^(2*i*x)) + 1) 1
((e^(i*x)) + ---------)
(e^(i*x))
16-> cot=1/tan
1
#17: cot = ---
tan
17-> elim all
Eliminating variable cot using solved equation #4...
Eliminating variable tan using solved equation #3...
i*((e^(2*i*x)) + 1) ((e^(2*i*x)) + 1)
#17: ------------------- = ---------------------
((e^(2*i*x)) - 1) (i*(1 - (e^(2*i*x))))
17-> 1+tan^2=sec^2
#18: 1 + tan^2 = sec^2
18-> elim all
Eliminating variable sec using solved equation #5...
Eliminating variable tan using solved equation #3...
i*(1 - (e^(2*i*x))) 2*(e^(i*x))
#18: 1 + (-------------------^2) = -----------------^2
((e^(2*i*x)) + 1) ((e^(2*i*x)) + 1)
18-> cosh^2-sinh^2=1 ; The main hyperbolic trigonometry identity:
#19: cosh^2 - sinh^2 = 1
19-> elim all
Eliminating variable cosh using solved equation #8...
Eliminating variable sinh using solved equation #7...
1 1
(e^x + ---) (e^x - ---)
e^x e^x
#19: (-----------^2) - (-----------^2) = 1
2 2
19-> ; Now verify them all, to show and check the new solve command usage.
19-> solve 13-19 verifiable 0
Solving equation #13 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#13: 0 = 0
This equation is an identity.
Solving equation #14 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#14: 0 = 0
This equation is an identity.
Solving equation #15 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#15: 0 = 0
This equation is an identity.
Solving equation #16 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#16: 0 = 0
This equation is an identity.
Solving equation #17 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#17: 0 = 0
This equation is an identity.
Solving equation #18 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#18: 0 = (4*(e^(2*i*x))) - (((e^(2*i*x)) + 1)^2) + ((1 - (e^(2*i*x)))^2)
This equation is an identity.
Solving equation #19 for 0 with required identity verification...
Solve and "repeat simplify quick" successful:
#19: 0 = (4*(e^(2*x))) + (((e^(2*x)) - 1)^2) - (((e^(2*x)) + 1)^2)
This equation is an identity.
19-> pause
19-> clear all
1-> ; Next, test fixed-point mode and some financial equations:
1-> read finance
1->
1-> ; Combine 2 commonly used formulas to produce the mortgage payment formula.
1-> ; Here are 3 related financial formulas that can be "read" into Mathomatic.
1->
1-> set fixed ; Enable fixed-point money mode; rounds to the nearest cent.
Success.
1-> ; First, the variable definitions:
1-> ; pv = present value
1-> ; fv = future value (maturity value)
1-> ; interest_rate = interest rate per period (1 = 100%)
1-> ; n = number of periods
1->
1-> ; Compound Interest Future Value Formula:
1-> fv1 = pv*(1+interest_rate)^n
#1: fv1 = pv*((1.00 + interest_rate)^n)
1-> ; Future Value Annuity Formula:
1-> fv2 = payment*(((1+interest_rate)^n-1)/interest_rate)
payment*(((1.00 + interest_rate)^n) - 1.00)
#2: fv2 = -------------------------------------------
interest_rate
2-> ; Next we will combine these to produce the standard annuity formula.
2-> ; Set equal, then solve and simplify:
2-> fv1 = fv2
#3: fv1 = fv2
3-> pause
3-> eliminate all ; combine both formulas to produce the annuity formula:
Eliminating variable fv2 using solved equation #2...
Eliminating variable fv1 using solved equation #1...
payment*(((1.00 + interest_rate)^n) - 1.00)
#3: pv*((1.00 + interest_rate)^n) = -------------------------------------------
interest_rate
3-> solve verifiable pv ; solve for present value:
Solving equation #3 for pv with required verification...
Solve and "repeat simplify quick" successful:
1.00
payment*(1.00 - --------------------------)
((1.00 + interest_rate)^n)
#3: pv = -------------------------------------------
interest_rate
Solution verified.
3-> solve verifiable payment ; or solve for payment per period:
Solving equation #3 for payment with required verification...
Solve and "repeat simplify quick" successful:
pv*((1.00 + interest_rate)^n)*interest_rate
#3: payment = -------------------------------------------
(((1.00 + interest_rate)^n) - 1.00)
Solution verified.
3-> pause End of finance tutorial
3-> ; Remember we are still in fixed-point money mode,
3-> ; unless you typed "set no fixed".
Successfully finished reading script file "finance.in".
3-> a=55/-3
(-55.00)
#4: a = --------
3.00
4-> list
#4: a = (-55.00)/3.00
4-> display mixed
1.00
#4: a = -(18.00 + ----)
3.00
4-> display mixed factor
1.00
#4: a = -((2.00*3.00^2.00) + ----)
3.00
4-> display simple
(-55.00)
#4: a = --------
3.00
4-> list
#4: a = (-55.00)/3.00
4-> display
(-55.00)
#4: a = --------
3.00
4-> set no fixed_point
Success.
4-> display
-55
#4: a = ---
3
4-> clear all
1-> read quadratic
1->
1-> ; General quadratic (2nd degree polynomial) formula.
1-> ; Formula for the 2 roots (solutions for x)
1-> ; of the general quadratic equation.
1-> ;
1-> a x^2 + b x + c = 0 ; The general quadratic equation.
#1: (a*x^2) + (b*x) + c = 0
1-> copy select ; Make a copy and select it.
#2: (a*x^2) + (b*x) + c = 0
2-> solve verifiable for x ; Mathomatic can easily solve and verify that:
Solving equation #2 for x with required verification...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
Solve and "repeat simplify quick" successful:
1
((((b^2 - (4*a*c))^-)*sign) - b)
2
#2: x = --------------------------------
(2*a)
All solutions verified.
2-> ; This is the quadratic formula.
2-> ; The coefficients (a, b, and c) may be any mathematical expression not containing x.
2-> pause
2-> ; Here is the derivation and proof of the quadratic formula,
2-> ; without actually using the quadratic formula,
2-> ; because that is what we are trying to derive now, from the quadratic equation:
2-> #1:
#1: (a*x^2) + (b*x) + c = 0
1-> copy select ; make a copy of the general quadratic equation to work on and select it.
#3: (a*x^2) + (b*x) + c = 0
3-> -=c ; subtract "c" from both sides.
#3: (a*x^2) + (b*x) = -c
3-> /=a ; divide both sides by "a".
((a*x^2) + (b*x)) -c
#3: ----------------- = --
a a
3-> pause Next simplify it and turn it into a repeated factor polynomial equation
3-> simplify
x*b -c
#3: x^2 + --- = --
a a
3-> +=b^2/(4*(a^2)) ; add "b^2/(4*(a^2))" to both sides.
x*b b^2 b^2 c
#3: x^2 + --- + ------- = ------- - -
a (4*a^2) (4*a^2) a
3-> ; Now the LHS is a repeated factor polynomial, next factor it by pressing Enter to simplify.
3-> pause
3-> simplify ; Now the LHS is a factored polynomial, so solving for the single "x" is easy.
b b
(((2*x) + -)^2) (-^2)
a a c
#3: --------------- = ----- - -
4 4 a
3-> set debug 1 ; Let Mathomatic do the work and show it too.
Success.
3-> ; Show how easy it is to solve this equation now, after pressing Enter.
3-> pause
3-> x
level 1: 0.25*(((2*x) + (b/a))^2) = ((0.25*b^2) - (c*a))/a^2
Dividing both sides of the equation by "0.25":
level 1: ((2*x) + (b/a))^2 = 4*((0.25*b^2) - (c*a))/a^2
Raising both sides of the equation to the power of 0.5:
level 1: (2*x) + (b/a) = ((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0
Subtracting "b/a" from both sides of the equation:
level 1: 2*x = (((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0) - (b/a)
Dividing both sides of the equation by "2":
level 1: x = 0.5*((((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0) - (b/a))
Solve completed:
level 1: x = 0.5*((((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0) - (b/a))
Solve successful:
b^2
4*(--- - (c*a))
4 1 b
(((---------------^-)*sign0) - -)
a^2 2 a
#3: x = ---------------------------------
2
3-> ; Here is the raw solve result, press the Enter key to simplify and compare with the quadratic formula.
3-> pause
3-> set no debug
Success.
3-> repeat simplify
1
((((b^2 - (4*c*a))^-)*sign0) - b)
2
#3: x = ---------------------------------
(2*a)
3-> compare with 2
Comparing #2 with #3...
Equations are identical.
3->
Successfully finished reading script file "quadratic.in".
3-> clear all
1-> read electronics
1->
1-> ; General electrical formulas:
1->
1-> volts=amps*ohms ; Ohm's Law, commonly solved for resistance (ohms) or current (amps).
#1: volts = amps*ohms
1-> watts=volts*amps ; Power Law
#2: watts = volts*amps
2-> 1/r=1/r1+1/r2 ; Resistance (r) of 2 resistors (r1 and r2) wired in parallel.
1 1 1
#3: - = -- + --
r r1 r2
3-> solve verifiable r ; Solve for the resulting resistance.
Solving equation #3 for r with required verification...
Solve and "repeat simplify quick" successful:
r1*r2
#3: r = ---------
(r2 + r1)
Solution verified.
3-> 1/r=1/r1+1/r2+1/r3 ; Resistance (r) of 3 resistors wired in parallel.
1 1 1 1
#4: - = -- + -- + --
r r1 r2 r3
4-> solve verifiable r ; Solve for the resulting resistance.
Solving equation #4 for r with required verification...
Solve and "repeat simplify quick" successful:
r1*r2*r3
#4: r = --------------------------
((r3*r2) + (r1*(r3 + r2)))
Solution verified.
4-> frequency=1/(2*pi*(L*C)^.5) ; Resonant frequency of an LC circuit in hertz.
1
#5: frequency = ----------------
1
(2*pi*((L*C)^-))
2
5-> ; L is the inductance in henries, and C is the capacitance in farads.
Successfully finished reading script file "electronics.in".
5-> simplify all
#1: volts = amps*ohms
#2: watts = volts*amps
r1*r2
#3: r = ---------
(r2 + r1)
r1*r2*r3
#4: r = --------------------------
((r3*r2) + (r1*(r3 + r2)))
1
#5: frequency = ----------------
1
(2*pi*((L*C)^-))
2
5-> clear all
1-> read fibonacci
1->
1-> ; This Mathomatic input file contains the mathematical formula to
1-> ; directly calculate the "n"th Fibonacci number.
1-> ; The formula presented here is called Binet's formula, found at
1-> ; http://en.wikipedia.org/wiki/Fibonacci_number
1-> ;
1-> ; The Fibonacci sequence is the endless integer sequence:
1-> ; 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ...
1-> ; Any Fibonacci number is always the sum of the previous two Fibonacci numbers.
1-> ;
1-> ; Easy to understand info on the golden ratio can be found here:
1-> ; http://www.mathsisfun.com/numbers/golden-ratio.html
1->
1-> -1/phi=1-phi ; Derive the golden ratio (phi) from this quadratic polynomial.
-1
#1: --- = 1 - phi
phi
1-> 0 ; show it is quadratic
Solve successful:
#1: 0 = ((1 - phi)*phi) + 1
1-> unfactor
#1: 0 = phi - phi^2 + 1
1-> solve verifiable for phi ; The golden ratio will help us directly compute Fibonacci numbers.
Solving equation #1 for phi with required verification...
Equation is a degree 2 polynomial equation in phi.
Equation was solved with the quadratic formula.
Solve and "repeat simplify quick" successful:
1
(1 - ((5^-)*sign))
2
#1: phi = ------------------
2
All solutions verified.
1-> replace sign with -1 ; the golden ratio constant:
1
(1 + (5^-))
2
#1: phi = -----------
2
1-> fibonacci = ((phi^n) - ((1 - phi)^n))/(phi - (1 - phi)) ; Binet's Fibonacci formula.
(phi^n - ((1 - phi)^n))
#2: fibonacci = -----------------------
(phi - 1 + phi)
2-> eliminate phi ; Completed direct Fibonacci formula:
Eliminating variable phi using solved equation #1...
1 1
(1 + (5^-)) (1 + (5^-))
2 2
((-----------^n) - ((1 - -----------)^n))
2 2
#2: fibonacci = -----------------------------------------
1
(5^-)
2
2-> simplify ; Note that Mathomatic rationalizes the denominator here.
1 1 1
(5^-)*(((1 + (5^-))^n) - ((1 - (5^-))^n))
2 2 2
#2: fibonacci = -----------------------------------------
(5*2^n)
2-> for n 1 20 ; Display the first 20 Fibonacci numbers by plugging in values 1-20:
n = 1: 1
n = 2: 1
n = 3: 2
n = 4: 3
n = 5: 5
n = 6: 8
n = 7: 13
n = 8: 21
n = 9: 34
n = 10: 55
n = 11: 89
n = 12: 144
n = 13: 233
n = 14: 377
n = 15: 610
n = 16: 987
n = 17: 1597
n = 18: 2584
n = 19: 4181
n = 20: 6765
2-> ; Note that this formula should work for any positive integer value of n.
Successfully finished reading script file "fibonacci.in".
2-> clear all
1-> read test
1-> ; Read in some of the things previously fixed in Mathomatic.
1-> read fix1
1-> clear all
1-> y = (((((a+b)/(b-c))^0.25)+(((b-c)/(a+b))^0.25)+(((a-b)*i/(b-c))^0.5))*(i^0.5))^(1/n)
(a + b) 1 (b - c) 1 (a - b)*i 1 1 1
#1: y = (((-------^-) + (-------^-) + (---------^-))*(i^-))^-
(b - c) 4 (a + b) 4 (b - c) 2 2 n
1-> y = (a^a)*(1+(((a^(a^2))*(b^a))^(1/(1-a))))
1
#2: y = a^a*(1 + (((a^(a^2))*b^a)^-------))
(1 - a)
2-> y = (a^2)*(1+(((a^(2*((1.5*a)-1)))*(b^a))^(1/(1-a))))
3*a 1
#3: y = a^2*(1 + (((a^(2*(--- - 1)))*b^a)^-------))
2 (1 - a)
3-> y = (15*(d^2)/((1+(d^2))^(7/2)))-(12/((1+(d^2))^(5/2)))-6
15*d^2 12
#4: y = ------------- - ------------- - 6
7 5
((1 + d^2)^-) ((1 + d^2)^-)
2 2
4-> y = ((9 + (32^.5))^.5) ; should simplify to (1 + 2*(2^.5)) someday
1 1
#5: y = (9 + (32^-))^-
2 2
5-> simplify symbolic all
(b + a) 1 (c - b) 1 (a - b) 1 1
#1: y = ((-------^-) + (-------^-) + (-------^-))^-
(c - b) 4 (a + b) 4 (c - b) 2 n
a
#2: y = a^a + ((a*b)^-------)
(1 - a)
a
#3: y = a^2 + ((a*b)^-------)
(1 - a)
(d^2 - 4)
#4: y = 3*(------------- - 2)
7
((1 + d^2)^-)
2
1 1
#5: y = (9 + (4*(2^-)))^-
2 2
5-> simplify all
(b + a) 1 (c - b) 1 (a - b) 1 1
#1: y = ((-------^-) + (-------^-) + (-------^-))^-
(c - b) 4 (a + b) 4 (c - b) 2 n
a
#2: y = a^a + ((a*b)^-------)
(1 - a)
a
#3: y = a^2 + ((a*b)^-------)
(1 - a)
(d^2 - 4)
#4: y = 3*(------------- - 2)
7
((1 + d^2)^-)
2
1 1
#5: y = (9 + (4*(2^-)))^-
2 2
5-> x^2=|x|
1
#6: x^2 = (x^2)^-
2
6-> solve for x
Equation is a degree 2 polynomial equation in x.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 3 polynomial equation in x.
Removing possible solution: "x = 0".
Solve successful:
#6: x = sign
6-> x^(1/99)=x
1
#7: x^-- = x
99
7-> solve verifiable x
Solving equation #7 for x with required verification...
Equation is a degree 0.01010101010101 polynomial equation in x.
Raising both equation sides to the power of 99 and expanding...
Equation is a degree 99 polynomial equation in x.
Removing possible solution: "x = 0".
Solve and "repeat simplify quick" successful:
#7: x = sign0
All solutions verified.
7-> (2i)^.5+e^(pi*i)
Calculating...
answer = i
8-> (1-2i)/(3+4i)
Calculating...
answer = (-0.4*i) - 0.2, with fractions it is: (-2*i/5) - (1/5)
9-> divide (1-2i) (3+4i)
Result of complex number division:
-0.2 -0.4*i
9->
9-> y=x^3
#10: y = x^3
10-> extrema x
#11: x = 0
11-> (x+1)^4
#12: (x + 1)^4
12-> extrema x
#13: x = -1
13-> roots 4 1 0 ; The 4 roots of unity.
The polar coordinates are:
1 amplitude and
0 radians (0 degrees).
The 4 roots of (1)^(1/4) are:
1
+1*i
-1
-1*i
13-> factor number -75 100000000000000 16! 7921%14 ; should be exactly 11
-75 = 3 * 5^2 * -1
100000000000000 = 2^14 * 5^14
20922789888000 = 2^15 * 3^6 * 5^3 * 7^2 * 11 * 13
Prime number: 11 = 11
Successfully finished reading script file "fix1.in".
13-> read fix2
13-> clear all
1-> b = ((-1)^(1/((-1*n)+1)*(2+n)))*(a^(1/((-1*n)+1)))
(2 + n) 1
#1: b = ((-1)^-------)*(a^-------)
(1 - n) (1 - n)
1-> x = 1/(y^(1/(n-1)*(-2+n)))/((n^(n/(n-1)))-(n^(1/(n-1))))
1
#2: x = -----------------------------------------
(n - 2) n 1
((y^-------)*((n^-------) - (n^-------)))
(n - 1) (n - 1) (n - 1)
2-> y = (x+(((1/x)+1)*((x^m)+((a+b)/(x^n)/(c+d)))))/(x+1)
1 (a + b)
(x + ((- + 1)*(x^m + -------------)))
x (x^n*(c + d))
#3: y = -------------------------------------
(x + 1)
3-> y = 3 / (x^3+3x^2-x-3) - 2 / (x^3-x^2-3x+3) + 4 / (x^3+x^2-3x-3)
3 2 4
#4: y = ----------------------- - ----------------------- + -----------------------
(x^3 + (3*x^2) - x - 3) (x^3 - x^2 - (3*x) + 3) (x^3 + x^2 - (3*x) - 3)
4-> (x^2 - 1)^4/(x + 1)^2
((x^2 - 1)^4)
#5: -------------
((x + 1)^2)
5-> simplify all
1
#1: b = ((-1)^n*a)^-------
(1 - n)
(2 - n)
(y^-------)
(n - 1)
#2: x = ---------------------------
n 1
((n^-------) - (n^-------))
(n - 1) (n - 1)
(b + a)*(1 + x)
(x^2 + ---------------)
(x^n*(c + d))
#3: y = (x^(m - 1)) + -----------------------
(x^2 + x)
(27 - (5*x^2))
#4: y = ----------------------------------------------
((4*x^3) - x^5 + (12*x^2) - (3*(x^4 + x)) - 9)
#5: ((1 + x)^2)*((x - 1)^4)
5-> 1
1
#1: b = ((-1)^n*a)^-------
(1 - n)
1-> solve verifiable a
Solving equation #1 for a with required verification...
Solve and "repeat simplify quick" successful:
(b^(1 - n))
#1: a = -----------
(-1)^n
Solution verified.
1-> 2
(2 - n)
(y^-------)
(n - 1)
#2: x = ---------------------------
n 1
((n^-------) - (n^-------))
(n - 1) (n - 1)
2-> solve verifiable y
Solving equation #2 for y with required verification...
Solve and "repeat simplify quick" successful:
1
#2: y = (((x*(n - 1))^(n - 1))*n)^-------
(2 - n)
Solution verified.
2-> 1/(x+y)
1
#6: -------
(x + y)
6-> taylor x, 5, 0
Taylor series with respect to x, simplified...
5 non-zero derivatives applied.
1 x x^2 x^3 x^4 x^5
#7: - - --- + --- - --- + --- - ---
y y^2 y^3 y^4 y^5 y^6
7-> fraction
(y^5 + (x^2*y^3) + (x^4*y) - x^5 - (x^3*y^2) - (x*y^4))
#7: -------------------------------------------------------
y^6
7-> simplify fraction
(y^5 - (y^4*x) + (y^3*x^2) - (y^2*x^3) + (y*x^4) - x^5)
#7: -------------------------------------------------------
y^6
7-> simplify
x^5
(x^4 - ---)
y
(----------- - x^3)
y
(x^2 + -------------------)
y
(--------------------------- - x)
y
(1 + ---------------------------------)
y
#7: ---------------------------------------
y
Successfully finished reading script file "fix2.in".
7-> read fix5
7-> clear all
1-> a = (x+1/2^.5)^3
1
#1: a = (x + -----)^3
1
(2^-)
2
1-> a = (b+((c+1)^0.5))^3
1
#2: a = (b + ((c + 1)^-))^3
2
2-> a = b*c*x*((((x^2)*c)+(b^4))^3)*(x+c)
#3: a = b*c*x*(((x^2*c) + b^4)^3)*(x + c)
3-> a = (((b^2)+x)^3)*((1/x)+x)*b
1
#4: a = ((b^2 + x)^3)*(- + x)*b
x
4-> a = b*(((1/b)+(1/c))^3)
1 1
#5: a = b*((- + -)^3)
b c
5-> a = (b^2)*(((1/b)+(1/c))^3)
1 1
#6: a = b^2*((- + -)^3)
b c
6-> a = (b^2)*((b-c)^3)
#7: a = b^2*((b - c)^3)
7-> simplify all
1
(((2*x) + (2^-))^3)
2
#1: a = -------------------
8
1
#2: a = (b + ((c + 1)^-))^3
2
#3: a = ((b^4 + (c*x^2))^3)*b*((c*x^2) + (c^2*x))
1
#4: a = ((b^2 + x)^3)*b*(- + x)
x
1 1
#5: a = ((- + -)^3)*b
c b
b
((1 + -)^3)
c
#6: a = -----------
b
#7: a = b^2*((b - c)^3)
Successfully finished reading script file "fix5.in".
7-> read fix7
7-> ; Algebraic fractions test
7-> clear all
1-> (c+a-b)/(b-a)
(c + a - b)
#1: -----------
(b - a)
1-> ((d*(b+c))+(a*(e1+f)))/(e1+f)/(b+c)
((d*(b + c)) + (a*(e1 + f)))
#2: ----------------------------
((e1 + f)*(b + c))
2-> ((((e1^2)+d)*b*((b^2)+2))-e1-f)/b/((b^2)+2)/(e1+f)
(((e1^2 + d)*b*(b^2 + 2)) - e1 - f)
#3: -----------------------------------
(b*(b^2 + 2)*(e1 + f))
3-> ((b*((((e1^2)+d)*((b^2)+2))+(b*(e1+f))))+e1+f)/(e1+f)/b/((b^2)+2)
((b*(((e1^2 + d)*(b^2 + 2)) + (b*(e1 + f)))) + e1 + f)
#4: ------------------------------------------------------
((e1 + f)*b*(b^2 + 2))
4-> ((1/(x^(1+n)))+(1/(x^n))+(x^(m-1))+(x^m)+x)/(x+1)
1 1
(----------- + --- + (x^(m - 1)) + x^m + x)
(x^(1 + n)) x^n
#5: -------------------------------------------
(x + 1)
5-> (1/(a + b)) + (1/(b + c))
1 1
#6: ------- + -------
(a + b) (b + c)
6-> ((x - 1)^2)*(2 + x)/((1 + x)*((x - 3)^2))
((x - 1)^2)*(2 + x)
#7: ---------------------
((1 + x)*((x - 3)^2))
7-> simplify all
c
#1: ------- - 1
(b - a)
d a
#2: -------- + -------
(e1 + f) (b + c)
(d + e1^2) 1
#3: ---------- - -------------
(e1 + f) (b^3 + (2*b))
(d + e1^2) (1 + b^2)
#4: ---------- + -------------
(e1 + f) (b^3 + (2*b))
1
(x^m + ---)
x^n x
#5: ----------- + -------
x (x + 1)
1 1
#6: ------- + -------
(b + c) (b + a)
(1 - x)
(-------^2)*(2 + x)
(3 - x)
#7: -------------------
(1 + x)
7-> fraction all
(c - b + a)
#1: -----------
(b - a)
((d*(b + c)) + (a*(e1 + f)))
#2: ----------------------------
((c + b)*(e1 + f))
(((d + e1^2)*(b^3 + (2*b))) - e1 - f)
#3: -------------------------------------
(((2*b) + b^3)*(e1 + f))
(((d + e1^2)*(b^3 + (2*b))) + e1 + f + (b^2*(e1 + f)))
#4: ------------------------------------------------------
(((2*b) + b^3)*(e1 + f))
((x^n*((x^m*(x + 1)) + x^2)) + x + 1)
#5: -------------------------------------
(x^n*(x^2 + x))
((2*b) + a + c)
#6: -----------------
((b + c)*(b + a))
(2 + x)*((1 - x)^2)
#7: ---------------------
((1 + x)*((3 - x)^2))
7-> simplify fraction all
(c - b + a)
#1: -----------
(b - a)
((d*(b + c)) + (a*(e1 + f)))
#2: ----------------------------
((e1 + f)*(c + b))
(((d + e1^2)*(b^3 + (2*b))) - e1 - f)
#3: -------------------------------------
((e1 + f)*((2*b) + b^3))
(((d + e1^2)*(b^3 + (2*b))) + (b^2*(e1 + f)) + e1 + f)
#4: ------------------------------------------------------
((e1 + f)*((2*b) + b^3))
1
(x^m + ---)
x^n 1
(x^m + ----------- + --- + x)
x x^n
#5: -----------------------------
(x + 1)
((2*b) + a + c)
#6: -----------------
((b + c)*(b + a))
((1 - x)^2)*(2 + x)
#7: ---------------------
(((3 - x)^2)*(1 + x))
7-> simplify all
c
#1: ------- - 1
(b - a)
d a
#2: -------- + -------
(e1 + f) (c + b)
(d + e1^2) 1
#3: ---------- - -------------
(e1 + f) ((2*b) + b^3)
(d + e1^2) (1 + b^2)
#4: ---------- + -------------
(e1 + f) ((2*b) + b^3)
1
(x^m + ---)
x^n x
#5: ----------- + -------
x (x + 1)
1 1
#6: ------- + -------
(b + c) (b + a)
(1 - x)
(-------^2)*(2 + x)
(3 - x)
#7: -------------------
(1 + x)
Successfully finished reading script file "fix7.in".
7-> read fix8
7-> clear all
1-> a = (((b^2)*(x^2))+(4*(b^2)*x)+(b^2)+(2*(b^3)*x)+(2*(b^3))+(b^4)+(2*b*(x^2))+(2*b*x)+(x^2))/(((b^3)*(x^2))+(2*(b^4)*x)+(b^5))
((b^2*x^2) + (4*b^2*x) + b^2 + (2*b^3*x) + (2*b^3) + b^4 + (2*b*x^2) + (2*b*x) + x^2)
#1: a = -------------------------------------------------------------------------------------
((b^3*x^2) + (2*b^4*x) + b^5)
1-> y = (((b+1)^0.5)*((b^2.5)+c))+((((b^2)+b)^0.5)*a)
1 5 1
#2: y = (((b + 1)^-)*((b^-) + c)) + (((b^2 + b)^-)*a)
2 2 2
2-> a = (b^(1-n))/(1+(b^(m-n)))
(b^(1 - n))
#3: a = -----------------
(1 + (b^(m - n)))
3-> a = (((b^2)+(b*(c^(1-n)))+(b^0.5))/(b^n)/(1+(b^(m-n))))^0.5
1
(b^2 + (b*(c^(1 - n))) + (b^-))
2 1
#4: a = -------------------------------^-
(b^n*(1 + (b^(m - n)))) 2
4-> simplify all
1
((- + 1)^2)
b
#1: a = -----------
b
1 5 1
#2: y = ((b + 1)^-)*((b^-) + ((b^-)*a) + c)
2 2 2
b
#3: a = -----------
(b^n + b^m)
1
(b^2 + (b*(c^(1 - n))) + (b^-))
2 1
#4: a = -------------------------------^-
(b^n + b^m) 2
Successfully finished reading script file "fix8.in".
4-> read fix9
4-> clear all
1-> ((1/b) + (1/c) + (1/d))^3
1 1 1
#1: (- + - + -)^3
b c d
1-> ((+/-1000*(b!^4)+/-x)^2)*((1/x)+x)*b
1
#2: (((1000*sign*((b!)^4)) + (sign0*x))^2)*(- + x)*b
x
2-> ((b+(2*i))^5)
#3: (b + (2*i))^5
3-> (((1/(b^2))+c)^2)*((1/b)+(c*b))
1 1
#4: ((--- + c)^2)*(- + (c*b))
b^2 b
4-> (6*(b^0.5)-3)^3
1
#5: ((6*(b^-)) - 3)^3
2
5-> (2-(4/(c-b)))^3
4
#6: (2 - -------)^3
(c - b)
6-> (((e*((2*(x^3)) + 24 + (x!) - zy)) - pi)/e)^2
((e*((2*x^3) + 24 + x! - zy)) - pi)
#7: -----------------------------------^2
e
7-> (2+3x)^3
#8: (2 + (3*x))^3
8-> simplify all
1 1 1
#1: (- + - + -)^3
b d c
1
#2: ((x + (1000*sign*sign0*((b!)^4)))^2)*b*(- + x)
x
#3: (b + (2*i))^5
1 1
#4: ((--- + c)^2)*(- + (c*b))
b^2 b
1
#5: -27*((1 - (2*(b^-)))^3)
2
2
#6: 8*((1 - -------)^3)
(c - b)
pi
#7: (zy - 24 + -- - x! - (2*x^3))^2
e
#8: (2 + (3*x))^3
8-> display factor all
1 1 1
#1: (- + - + -)^3
b d c
1
#2: ((x + ((2^3*5^3)*sign*sign0*((b!)^(2^2))))^2)*b*(- + x)
x
#3: (b + (2*i))^5
1 1
#4: ((--- + c)^2)*(- + (c*b))
b^2 b
1
#5: (3^3*-1)*((1 - (2*(b^-)))^3)
2
2
#6: 2^3*((1 - -------)^3)
(c - b)
pi
#7: (zy - (2^3*3) + -- - x! - (2*x^3))^2
e
#8: (2 + (3*x))^3
Successfully finished reading script file "fix9.in".
Successfully finished reading script file "test.in".
8-> clear all
1-> read fraction
1->
1-> ; Algebraic fractions tutorial.
1-> ; This Mathomatic input shows how "simplify fraction" and "unfactor fraction" work.
1-> 1/x+1/y+1/z
1 1 1
#1: - + - + -
x y z
1-> fraction ; Convert expressions with algebraic fractions into a single fraction.
((y*z) + (x*(z + y)))
#1: ---------------------
(x*y*z)
1-> simplify
1 1 1
#1: - + - + -
x y z
1-> simplify fraction ; does the same as the above fraction command, but simplifies more.
((z*y) + (x*(z + y)))
#1: ---------------------
(x*y*z)
1-> unfactor ; Expand the products of sums.
((z*y) + (x*z) + (x*y))
#1: -----------------------
(x*y*z)
1-> unfactor fraction ; Fully expand algebraic fractions by also expanding division of sums.
1 1 1
#1: - + - + -
x y z
Successfully finished reading script file "fraction.in".
1-> clear all
1-> read pie
1->
1-> ; This is the famous Bailey-Borwein-Plouffe (BBP) formula.
1-> ; Sum this n = 0 to infinity to compute pi.
1-> ; This is especially useful for calculating pi in hexadecimal.
1-> ; One hexadecimal digit of pi is generated with each iteration.
1-> ((4/((8*n)+1))-(2/((8*n)+4))-(1/((8*n)+5))-(1/((8*n)+6)))/(16^n)
4 2 1 1
(----------- - ----------- - ----------- - -----------)
((8*n) + 1) ((8*n) + 4) ((8*n) + 5) ((8*n) + 6)
#1: -------------------------------------------------------
16^n
1-> simplify ; BBP simplifies to the ratio of two polynomials.
((120*n^2) + (151*n) + 47)
#1: ----------------------------------------------------------
(16^n*((512*n^4) + (1024*n^3) + (712*n^2) + (194*n) + 15))
1-> sum n=0 to 10 ; Numerically sum BBP from n = 0 to 10 in steps of 1.
#2: 3.1415926535898
1-> pi ; The digits should be the same.
Calculating...
answer = 3.1415926535898
3-> repeat echo *
*******************************************************************************
3-> x^n/n! ; Sum this n = 0 to infinity to compute (e^x).
x^n
#4: ---
n!
4-> replace x with 1 ; Sum this n = 0 to infinity to compute e:
1
#4: --
n!
4-> sum n=0 to 20 ; Numerically sum from n = 0 to 20 in steps of 1.
#5: 2.718281828459
4-> e ; The digits should be the same.
Calculating...
answer = 2.718281828459
6-> repeat echo *
*******************************************************************************
6-> ; Euler's identity is made of these most important universal constants:
6-> e^(pi*i)+1=0
#7: (e^(pi*i)) + 1 = 0
7-> simplify ; An identity is when the LHS is identical to the RHS:
#7: 0 = 0
Successfully finished reading script file "pie.in".
7-> 1
((120*n^2) + (151*n) + 47)
#1: ----------------------------------------------------------
(16^n*((512*n^4) + (1024*n^3) + (712*n^2) + (194*n) + 15))
1-> fraction
((120*n^2) + (151*n) + 47)
#1: ----------------------------------------------------------
(16^n*((512*n^4) + (1024*n^3) + (712*n^2) + (194*n) + 15))
1-> read demo
1-> clear all
1-> ; Some symbolic differentiation examples follow.
1->
1-> ; Take the derivative of the absolute value function:
1-> |x|
1
#1: (x^2)^-
2
1-> derivative ; The result is the sign function sgn(x), which gives the sign of x.
Differentiating with respect to x and simplifying...
x
#2: ---------
1
((x^2)^-)
2
2-> repeat echo *
*******************************************************************************
2-> ; Mathomatic can differentiate anything that doesn't require symbolic logarithms.
2-> y=e^(1+1/x)
1
#3: y = e^(1 + -)
x
3-> derivative ; The first order derivative is:
Differentiating the RHS with respect to x and simplifying...
1
-(e^(1 + -))
x
#4: y' = ------------
x^2
4-> derivative ; The second order derivative is:
Differentiating the RHS with respect to x and simplifying...
1 1
(e^(- + 1))*(- + 2)
x x
#5: y'' = -------------------
x^3
5-> expand fraction ; Perhaps easier to read:
1 1
(e^(- + 1)) 2*(e^(- + 1))
x x
#5: y'' = ----------- + -------------
x^4 x^3
5-> repeat echo *
*******************************************************************************
5-> ; A Taylor series demonstration:
5-> y=x_new^n ; x_new is what we want, without using the root operator.
#6: y = x_new^n
6-> x_new ; It is easily solved for in Mathomatic.
Solve successful:
1
#6: x_new = y^-
n
6-> y ; But we want an algorithm to compute it without using non-integer exponentiation.
Solve successful:
#6: y = x_new^n
6-> taylor x_new, 1, x_old ; build the (nth root of y) iterative approximation formula
Taylor series of the RHS with respect to x_new, simplified...
1 non-zero derivative applied.
#7: y = x_old^n + (n*(x_old^(n - 1))*x_new) - (n*x_old^n)
7-> solve verifiable x_new ; solve for the output variable
Solving equation #7 for x_new with required verification...
Solve and "repeat simplify quick" successful:
y
(------- - 1)
x_old^n
#7: x_new = x_old*(------------- + 1)
n
Solution verified.
7-> ; That is the convergent nth root approximation formula.
7-> copy ; "calculate x_old 10000" tests this formula, if you would like to see for yourself.
y
(------- - 1)
x_old^n
#8: x_new = x_old*(------------- + 1)
n
7-> replace x_old x_new with x ; make x_old (input) and x_new (output) the same
y
(--- - 1)
x^n
#7: x = x*(--------- + 1)
n
7-> x ; make sure the formula was correct by solving for x
Removing possible solution: "x = 0".
Solve successful:
1
#7: x = y^-
n
7-> repeat echo *
*******************************************************************************
7-> ; Another Taylor series demo:
7-> e^x ; enter the exponential function
#9: e^x
9-> taylor x, 10, 0 ; generate a 10th order taylor series of the exponential function
Taylor series with respect to x, simplified...
10 non-zero derivatives applied.
x^2 x^3 x^4 x^5 x^6 x^7 x^8 x^9 x^10
#10: 1 + x + --- + --- + --- + --- + --- + ---- + ----- + ------ + -------
2 6 24 120 720 5040 40320 362880 3628800
10-> laplace x ; do a Laplace transform on it
1 1 1 1 1 1 1 1 1 1 1
#11: - + --- + --- + --- + --- + --- + --- + --- + --- + ---- + ----
x x^2 x^3 x^4 x^5 x^6 x^7 x^8 x^9 x^10 x^11
11-> simplify ; show the structure of the result
1
(1 + -)
x
(1 + -------)
x
(1 + -------------)
x
(1 + -------------------)
x
(1 + -------------------------)
x
(1 + -------------------------------)
x
(1 + -------------------------------------)
x
(1 + -------------------------------------------)
x
(1 + -------------------------------------------------)
x
(1 + -------------------------------------------------------)
x
#11: -------------------------------------------------------------
x
11-> laplace inverse x ; undo the Laplace transform
x^2 x^3 x^4 x^5 x^6 x^7 x^8 x^9 x^10
#12: 1 + x + --- + --- + --- + --- + --- + ---- + ----- + ------ + -------
2 6 24 120 720 5040 40320 362880 3628800
12-> compare with 10 ; check the result
Comparing #10 with #12...
Expressions are identical.
Successfully finished reading script file "demo.in".
12-> read limits
12->
12-> ; Tests for the experimental limit command.
12->
12-> clear all
1-> ; find the derivative of:
1-> y = 1/(x^.5)
1
#1: y = -----
1
(x^-)
2
1-> ; using the difference quotient:
1-> y' = (1/(x+delta_x)^.5-1/x^.5)/delta_x
1 1
(----------------- - -----)
1 1
((x + delta_x)^-) (x^-)
2 2
#2: y' = ---------------------------
delta_x
2-> limit delta_x 0 ; take the limit as delta_x (change in x) goes to 0
Taking the limit as delta_x goes to 0
Solving...
Equation is a degree 0.5 polynomial equation in delta_x.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 3 polynomial equation in delta_x.
Removing possible solution: "delta_x = 0".
Equation is a degree 2 polynomial equation in delta_x.
Equation was solved with the quadratic formula.
Equation is a degree 0.5 polynomial equation in y'.
Raising both equation sides to the power of 2 and expanding...
-1
#3: y' = ---------
3
(2*(x^-))
2
2-> 3
-1
#3: y' = ---------
3
(2*(x^-))
2
3-> integrate x ; take the antiderivative to see if it's right
Only the RHS will be transformed.
Integrating the RHS with respect to x and simplifying...
1
#4: y = -----
1
(x^-)
2
4-> compare 1
Comparing #1 with #4...
Equations are identical.
4->
4-> ; test infinity limits:
4-> 2x/(x+1)
2*x
#5: -------
(x + 1)
5-> limit x inf ; answer should be 2
Taking the limit as x goes to inf
Solving...
#6: limit = 2
5->
5-> (3x+100-a)/(x-b)
((3*x) + 100 - a)
#7: -----------------
(x - b)
7-> limit x inf ; answer should be 3
Taking the limit as x goes to inf
Solving...
#8: limit = 3
7->
7-> (((x^2) - (5*x) + 6)^(1/2)) - x
1
#9: ((x^2 - (5*x) + 6)^-) - x
2
9-> limit x inf ; answer should be -5/2
Taking the limit as x goes to inf
Solving...
Equation is a degree 0.5 polynomial equation in x.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
-5
#10: limit = --
2
9->
9-> x*((x^2+1)^.5-x)
1
#11: x*(((x^2 + 1)^-) - x)
2
11-> limit x inf ; answer should be 1/2
Taking the limit as x goes to inf
Solving...
Equation is a degree 1.5 polynomial equation in x.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 3 polynomial equation in x.
Removing possible solution: "x = 0".
1
#12: limit = -
2
11->
11-> 1/x^2+1/x
1 1
#13: --- + -
x^2 x
13-> limit y inf ; result should be original expression with a warning.
Warning: Limit variable not found; answer is original expression.
1 1
#13: limit = --- + -
x^2 x
13-> limit x inf ; result should be 0
Taking the limit as x goes to inf
Solving...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
#14: limit = 0
13->
13-> ((2*(x^2)) - x - 6)/((x^2) + (2*x) - 8)
((2*x^2) - x - 6)
#15: -----------------
(x^2 + (2*x) - 8)
15-> limit x inf ; result should be 2
Taking the limit as x goes to inf
Solving...
#16: limit = 2
15->
15-> x^2+x
#17: x^2 + x
17-> limit x 0 ; result should be 0
Taking the limit as x goes to 0
Solving...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
#18: limit = 0
17-> limit x 2 ; result should be 6
Taking the limit as x goes to 2
Solving...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
#19: limit = 6
17-> display
#17: limit = x^2 + x
17-> ; The following currently gives the wrong answer:
17-> limit x inf ; result should be inf
Taking the limit as x goes to inf
Solving...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
#20: limit = 0
17-> ; The following currently gives errors:
17-> y=x+1/x
1
#21: y = x + -
x
21-> :limit x 0 ; result should be inf
Taking the limit as x goes to 0
Solving...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
Equation is a degree 0.5 polynomial equation in y.
Raising both equation sides to the power of 2 and expanding...
There are no possible values for the solve variable.
Can't take the limit because solve failed.
21-> :limit x inf; result should be inf
Taking the limit as x goes to inf
Solving...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
Equation is a degree 0.5 polynomial equation in y.
Raising both equation sides to the power of 2 and expanding...
There are no possible values for the solve variable.
Can't take the limit because solve failed.
Successfully finished reading script file "limits.in".
21-> ; read how_limit_works
21-> read test3
21-> ; Test solving linear equations with Mathomatic.
21->
21-> read linear
21->
21-> ; Combine 3 simultaneous linear equations with 3 unknowns (x, y, z).
21-> ; Solve for all 3 unknowns using the eliminate, solve, and simplify commands.
21->
21-> clear all ; restart Mathomatic
1-> ; enter all 3 equations:
1-> d1=a1*x+b1*y+c1*z
#1: d1 = (a1*x) + (b1*y) + (c1*z)
1-> d2=a2*x+b2*y+c2*z
#2: d2 = (a2*x) + (b2*y) + (c2*z)
2-> d3=a3*x+b3*y+c3*z
#3: d3 = (a3*x) + (b3*y) + (c3*z)
3-> 2 ; select equation number 2 as the current equation
#2: d2 = (a2*x) + (b2*y) + (c2*z)
2-> eliminate x ; eliminate variable x from the current equation
Solving equation #1 for x and substituting into the current equation...
a2*((b1*y) + (c1*z) - d1)
#2: d2 = (b2*y) - ------------------------- + (c2*z)
a1
2-> 3 ; select equation number 3
#3: d3 = (a3*x) + (b3*y) + (c3*z)
3-> eliminate x y ; eliminate variables x and then y from the current equation
Eliminating variable x using solved equation #1...
Solving equation #2 for y and substituting into the current equation...
b1*((z*((c2*a1) - (a2*c1))) + (a2*d1) - (d2*a1))
a3*(------------------------------------------------ + (c1*z) - d1)
b3*((z*((c2*a1) - (a2*c1))) + (a2*d1) - (d2*a1)) ((a2*b1) - (b2*a1))
#3: d3 = ------------------------------------------------ - ------------------------------------------------------------------- + (c3*z)
((a2*b1) - (b2*a1)) a1
3-> solve for z
Solve successful:
((d3*((a2*b1) - (a1*b2))) + (b3*((d2*a1) - (a2*d1))) + (a3*((b2*d1) - (b1*d2))))
#3: z = --------------------------------------------------------------------------------
((b3*((c2*a1) - (a2*c1))) + (a3*((b2*c1) - (b1*c2))) + (c3*((a2*b1) - (a1*b2))))
3-> 2 ; select equation number 2
((z*((c2*a1) - (a2*c1))) + (a2*d1) - (d2*a1))
#2: y = ---------------------------------------------
((a2*b1) - (b2*a1))
2-> eliminate z using 3 ; find y by combining equation numbers 2 and 3
Eliminating variable z using solved equation #3...
((d3*((a2*b1) - (a1*b2))) + (b3*((d2*a1) - (a2*d1))) + (a3*((b2*d1) - (b1*d2))))*((c2*a1) - (a2*c1))
(---------------------------------------------------------------------------------------------------- + (a2*d1) - (d2*a1))
((b3*((c2*a1) - (a2*c1))) + (a3*((b2*c1) - (b1*c2))) + (c3*((a2*b1) - (a1*b2))))
#2: y = --------------------------------------------------------------------------------------------------------------------------
((a2*b1) - (b2*a1))
2-> simplify
((a1*((d3*c2) - (d2*c3))) + (d1*((a2*c3) - (c2*a3))) + (c1*((d2*a3) - (d3*a2))))
#2: y = --------------------------------------------------------------------------------
((a1*((b3*c2) - (c3*b2))) + (c1*((a3*b2) - (b3*a2))) + (b1*((c3*a2) - (a3*c2))))
2-> 1 ; select equation number 1
-((b1*y) + (c1*z) - d1)
#1: x = -----------------------
a1
1-> eliminate z using 3, y using 2; find x
Eliminating variable z using solved equation #3...
Eliminating variable y using solved equation #2...
b1*((a1*((d3*c2) - (d2*c3))) + (d1*((a2*c3) - (c2*a3))) + (c1*((d2*a3) - (d3*a2)))) c1*((d3*((a2*b1) - (a1*b2))) + (b3*((d2*a1) - (a2*d1))) + (a3*((b2*d1) - (b1*d2))))
-(----------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- - d1)
((a1*((b3*c2) - (c3*b2))) + (c1*((a3*b2) - (b3*a2))) + (b1*((c3*a2) - (a3*c2)))) ((b3*((c2*a1) - (a2*c1))) + (a3*((b2*c1) - (b1*c2))) + (c3*((a2*b1) - (a1*b2))))
#1: x = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
a1
1->
1-> simplify all ; simplify and display all solutions
((c1*((b2*d3) - (b3*d2))) + (b1*((c3*d2) - (c2*d3))) + (d1*((b3*c2) - (c3*b2))))
#1: x = --------------------------------------------------------------------------------
((a1*((b3*c2) - (c3*b2))) + (c1*((a3*b2) - (b3*a2))) + (b1*((c3*a2) - (a3*c2))))
((a1*((d3*c2) - (d2*c3))) + (d1*((a2*c3) - (c2*a3))) + (c1*((d2*a3) - (d3*a2))))
#2: y = --------------------------------------------------------------------------------
((a1*((b3*c2) - (c3*b2))) + (c1*((a3*b2) - (b3*a2))) + (b1*((c3*a2) - (a3*c2))))
((b1*((d3*a2) - (a3*d2))) + (a1*((b3*d2) - (d3*b2))) + (d1*((a3*b2) - (b3*a2))))
#3: z = --------------------------------------------------------------------------------
((a1*((b3*c2) - (c3*b2))) + (c1*((a3*b2) - (b3*a2))) + (b1*((c3*a2) - (a3*c2))))
Successfully finished reading script file "linear.in".
1-> copy
((c1*((b2*d3) - (b3*d2))) + (b1*((c3*d2) - (c2*d3))) + (d1*((b3*c2) - (c3*b2))))
#4: x = --------------------------------------------------------------------------------
((a1*((b3*c2) - (c3*b2))) + (c1*((a3*b2) - (b3*a2))) + (b1*((c3*a2) - (a3*c2))))
1-> b2
Solve successful:
((x*((b3*((a1*c2) - (c1*a2))) + (b1*((c3*a2) - (a3*c2))))) - (d2*((b1*c3) - (c1*b3))) - (c2*((d1*b3) - (b1*d3))))
#1: b2 = -----------------------------------------------------------------------------------------------------------------
((c1*(d3 - (x*a3))) + (c3*((x*a1) - d1)))
1-> c2
Solve successful:
((b2*((c1*(d3 - (x*a3))) + (c3*((x*a1) - d1)))) + (((b1*c3) - (c1*b3))*(d2 - (x*a2))))
#1: c2 = --------------------------------------------------------------------------------------
((x*((b3*a1) - (b1*a3))) - (d1*b3) + (b1*d3))
1-> b3
Solve successful:
((b1*((c2*(d3 - (x*a3))) + (c3*((x*a2) - d2)))) - (b2*((c1*(d3 - (x*a3))) + (c3*((x*a1) - d1)))))
#1: b3 = -------------------------------------------------------------------------------------------------
((c1*((x*a2) - d2)) + (c2*(d1 - (x*a1))))
1-> c3
Solve successful:
((b3*((c1*((x*a2) - d2)) + (c2*(d1 - (x*a1))))) + (((x*a3) - d3)*((b1*c2) - (b2*c1))))
#1: c3 = --------------------------------------------------------------------------------------
((b1*((x*a2) - d2)) + (b2*(d1 - (x*a1))))
1-> b1
Solve successful:
((b2*((c3*(d1 - (x*a1))) + (c1*((x*a3) - d3)))) - (b3*((c1*((x*a2) - d2)) + (c2*(d1 - (x*a1))))))
#1: b1 = -------------------------------------------------------------------------------------------------
((c2*((x*a3) - d3)) + (c3*(d2 - (x*a2))))
1-> c1
Solve successful:
((b1*((c2*((x*a3) - d3)) + (c3*(d2 - (x*a2))))) + (((x*a1) - d1)*((b2*c3) - (b3*c2))))
#1: c1 = --------------------------------------------------------------------------------------
((b2*((x*a3) - d3)) + (b3*(d2 - (x*a2))))
1-> d2
Solve successful:
((c1*((b2*((x*a3) - d3)) - (b3*x*a2))) - (((b2*c3) - (b3*c2))*((x*a1) - d1)) + (b1*((c2*(d3 - (x*a3))) + (c3*x*a2))))
#1: d2 = ---------------------------------------------------------------------------------------------------------------------
((b1*c3) - (c1*b3))
1-> a2
Solve successful:
((d2*((b1*c3) - (c1*b3))) + (((b2*c3) - (b3*c2))*((x*a1) - d1)) + ((d3 - (x*a3))*((c1*b2) - (b1*c2))))
#1: a2 = ------------------------------------------------------------------------------------------------------
(x*((b1*c3) - (c1*b3)))
1-> d3
Solve successful:
((((b1*c3) - (c1*b3))*((a2*x) - d2)) - (((b2*c3) - (b3*c2))*((x*a1) - d1)))
#1: d3 = --------------------------------------------------------------------------- + (x*a3)
((c1*b2) - (b1*c2))
1-> a3
Solve successful:
-((((b1*c3) - (c1*b3))*((a2*x) - d2)) - (((b2*c3) - (b3*c2))*((x*a1) - d1)) - (d3*((c1*b2) - (b1*c2))))
#1: a3 = -------------------------------------------------------------------------------------------------------
(((c1*b2) - (b1*c2))*x)
1-> d1
Solve successful:
((((c1*b2) - (b1*c2))*((a3*x) - d3)) + (((b1*c3) - (c1*b3))*((a2*x) - d2)))
#1: d1 = -(--------------------------------------------------------------------------- - (x*a1))
((b2*c3) - (b3*c2))
1-> a1
Solve successful:
((d1*((b2*c3) - (b3*c2))) + (((c1*b2) - (b1*c2))*((a3*x) - d3)) + (((b1*c3) - (c1*b3))*((a2*x) - d2)))
#1: a1 = ------------------------------------------------------------------------------------------------------
(((b2*c3) - (b3*c2))*x)
1-> x
Solve successful:
((d1*((b2*c3) - (b3*c2))) + (d3*((b1*c2) - (c1*b2))) + (d2*((c1*b3) - (b1*c3))))
#1: x = --------------------------------------------------------------------------------
((a1*((b2*c3) - (b3*c2))) + (a3*((b1*c2) - (c1*b2))) + (a2*((c1*b3) - (b1*c3))))
1-> compare with 4
Comparing #4 with #1...
Simplifying both equations...
Equations are identical.
Successfully finished reading script file "test3.in".
1-> read poly
1->
1-> ; Combine 3 quadratic polynomial equations with 3 unknown coefficients (a, b, c).
1-> ; Solve for variables (a), (b), and (c).
1->
1-> clear all ; restart Mathomatic
1-> ; enter all 3 equations:
1-> y1=a+b*x1+c*x1^2
#1: y1 = a + (b*x1) + (c*x1^2)
1-> y2=a+b*x2+c*x2^2
#2: y2 = a + (b*x2) + (c*x2^2)
2-> y3=a+b*x3+c*x3^2
#3: y3 = a + (b*x3) + (c*x3^2)
3-> 2 ; select equation number 2 as the current equation
#2: y2 = a + (b*x2) + (c*x2^2)
2-> eliminate a ; eliminate variable (a) from the current equation
Solving equation #1 for a and substituting into the current equation...
#2: y2 = (b*x2) - (x1*(b + (c*x1))) + y1 + (c*x2^2)
2-> 3 ; select equation number 3
#3: y3 = a + (b*x3) + (c*x3^2)
3-> eliminate a b ; eliminate variables (a) and then (b) from the current equation
Eliminating variable a using solved equation #1...
Solving equation #2 for b and substituting into the current equation...
(y1 - y2 + (c*(x2^2 - x1^2)))*x3 (y1 - y2 + (c*(x2^2 - x1^2)))
#3: y3 = -------------------------------- - (x1*(----------------------------- + (c*x1))) + y1 + (c*x3^2)
(x1 - x2) (x1 - x2)
3-> solve verifiable c
Solving equation #3 for c with required verification...
Solve and "repeat simplify quick" successful:
((x1*(y2 - y3)) + (x3*(y1 - y2)) + (x2*(y3 - y1)))
#3: c = --------------------------------------------------
((x2 - x1)*(x3 - x1)*(x3 - x2))
Solution verified.
3-> simplify
(y1 - y2) (y3 - y2)
(--------- + ---------)
(x2 - x1) (x3 - x2)
#3: c = -----------------------
(x3 - x1)
3-> 2 ; select equation number 2 again
(y1 - y2 + (c*(x2^2 - x1^2)))
#2: b = -----------------------------
(x1 - x2)
2-> eliminate c using 3 ; find (b) by combining equation numbers 2 and 3
Eliminating variable c using solved equation #3...
(y1 - y2) (y3 - y2)
(--------- + ---------)*(x2^2 - x1^2)
(x2 - x1) (x3 - x2)
(y1 - y2 + -------------------------------------)
(x3 - x1)
#2: b = -------------------------------------------------
(x1 - x2)
2-> simplify
((x1^2*(y2 - y3)) + (x3^2*(y1 - y2)) + (x2^2*(y3 - y1)))
#2: b = --------------------------------------------------------
((x2 - x1)*(x3 - x1)*(x2 - x3))
2-> 1 ; select equation number 1
#1: a = -((x1*(b + (c*x1))) - y1)
1-> eliminate c using 3, b using 2 ; find (a)
Eliminating variable c using solved equation #3...
Eliminating variable b using solved equation #2...
(y1 - y2) (y3 - y2)
(--------- + ---------)*x1
((x1^2*(y2 - y3)) + (x3^2*(y1 - y2)) + (x2^2*(y3 - y1))) (x2 - x1) (x3 - x2)
#1: a = -((x1*(-------------------------------------------------------- + --------------------------)) - y1)
((x2 - x1)*(x3 - x1)*(x2 - x3)) (x3 - x1)
1->
1-> simplify fraction all ; display all solutions, converting to simple fractions first
((x1^2*((y2*x3) - (y3*x2))) + (x1*((x2^2*y3) - (x3^2*y2))) + (y1*((x3^2*x2) - (x3*x2^2))))
#1: a = ------------------------------------------------------------------------------------------
((x2 - x1)*(x3 - x1)*(x3 - x2))
((x1^2*(y2 - y3)) + (x3^2*(y1 - y2)) + (x2^2*(y3 - y1)))
#2: b = --------------------------------------------------------
((x2 - x1)*(x3 - x1)*(x2 - x3))
((x3*(y1 - y2)) + (x2*(y3 - y1)) + (x1*(y2 - y3)))
#3: c = --------------------------------------------------
((x2 - x1)*(x3 - x1)*(x3 - x2))
Successfully finished reading script file "poly.in".
1-> clear all
1-> read examples
1->
1-> ; This is a line comment. This script shows some simple examples of Mathomatic usage.
1-> ; Mathomatic input files are scripts that may be read in with the "read" command.
1->
1-> ; Equations are entered by just typing or pasting them in:
1-> c^2=a^2+b^2 ; The Pythagorean theorem, "c" squared equals "a" squared plus "b" squared.
#1: c^2 = a^2 + b^2
1-> ; The entered equation becomes the current equation and is displayed.
1-> ; The current equation can be solved by simply typing in a variable name:
1-> c ; which is shorthand for the solve command. Solve for variable "c".
Solve successful:
1
#1: c = ((a^2 + b^2)^-)*sign
2
1-> ; "sign" variables are special two-valued variables that may only be +1 or -1.
1-> solve for b ; Another way to solve for a variable, using English.
Solve successful:
1
#1: b = ((c^2 - a^2)^-)*sign0
2
1-> ; To output programming language code, use the code command:
1-> code ; C language code is the default.
b = (pow(((c*c) - (a*a)), (1.0/2.0))*sign0);
1->
1-> code java ; Mathomatic can also generate Java
b = (Math.pow(((c*c) - (a*a)), (1.0/2.0))*sign0);
1->
1-> code python ; and Python code.
b = ((((c*c) - (a*a))**(1.0/2.0))*sign0)
1->
1-> repeat echo *
*******************************************************************************
1-> a=b+1/b ; Enter another equation; this is actually a quadratic equation.
1
#2: a = b + -
b
2-> 0 ; Solve for zero.
Solve successful:
#2: 0 = (b*(b - a)) + 1
2-> unfactor ; Expand, showing that this is a quadratic polynomial equation in "b".
#2: 0 = b^2 - (b*a) + 1
2-> solve verifiable b ; Require solution verification with the "verifiable" option.
Solving equation #2 for b with required verification...
Equation is a degree 2 polynomial equation in b.
Equation was solved with the quadratic formula.
Solve and "repeat simplify quick" successful:
1
((((a^2 - 4)^-)*sign) + a)
2
#2: b = --------------------------
2
All solutions verified.
2-> a ; Solve back for "a" and we should get the original equation.
Equation is a degree 0.5 polynomial equation in a.
Raising both equation sides to the power of 2 and expanding...
Solve successful:
(b^2 + 1)
#2: a = ---------
b
2-> simplify ; The simplify command makes expressions simpler and prettier.
1
#2: a = b + -
b
2-> repeat echo *
*******************************************************************************
2-> ; Mathomatic is also handy as an advanced calculator.
2-> ; Expressions without variables entered at the main prompt are instantly evaluated:
2-> 2+3
Calculating...
answer = 5
3-> 495/44 ; Fractions are always reduced to their simplest form:
Calculating...
answer = 11.25, with fractions it is: 45/4
4-> ; Fractions greater than 1 can easily be displayed as mixed fractions.
4-> display mixed ; Display above fraction as a mixed fraction:
1
#4: answer = 11 + -
4
4-> display factor ; Integers and fractions are easily factored:
(3^2*5)
#4: answer = -------
2^2
4-> 2^.5 ; The square root of 2, rounded to the default 14 digits:
Calculating...
answer = 1.4142135623731
5->
5-> repeat echo *
*******************************************************************************
5-> ; Symbolic logarithms like log(x) are not implemented, yet.
5-> 27^y=9 ; An example that uses numeric logarithms.
#6: 27^y = 9
6-> solve verifiable y ; Require solution verification with the "verifiable" option.
Solving equation #6 for y with required verification...
Solve and "repeat simplify quick" successful:
2
#6: y = -
3
Solution verified.
6->
6-> repeat echo *
*******************************************************************************
6-> 0=2x^2-3x-20 ; A simple quadratic equation, to show how the calculate command works.
#7: 0 = (2*x^2) - (3*x) - 20
7-> solve verifiable x ; Solve for x, plugging the results into the original equation to verify.
Solving equation #7 for x with required verification...
Equation is a degree 2 polynomial equation in x.
Equation was solved with the quadratic formula.
Solve and "repeat simplify quick" successful:
3 13*sign
#7: x = - - -------
4 4
All solutions verified.
7-> calculate ; Expand "sign" variables and approximate the RHS (Right-Hand Side).
There are 2 solutions.
Solution number 1 with sign = 1:
x = -2.5, with fractions it is: -5/2
Solution number 2 with sign = -1:
x = 4
7-> ; The calculate command also lets you plug values into a formula with variables, if any.
7-> display; Display the current equation, showing that it was not modified by calculate.
3 13*sign
#7: x = - - -------
4 4
Successfully finished reading script file "examples.in".
7-> clear all
1-> read test1
1-> y = .6666 - (4*(((10*(pi^2)*(r^3)/((d^2)*g*m*epsilon)) - 1)^(1/2))/15)
10*pi^2*r^3 1
4*((----------------- - 1)^-)
(d^2*g*m*epsilon) 2
#1: y = 0.6666 - -----------------------------
15
1-> simplify
pi
10*(--^2)*r^3
d 1
4*((------------- - 1)^-)
(g*m*epsilon) 2
#1: y = 0.6666 - -------------------------
15
1-> simplify symbolic
10*pi^2*r^3 1
4*((------------- - d^2)^-)
(g*m*epsilon) 2
#1: y = 0.6666 - ---------------------------
(15*d)
1-> r
Solve successful:
15*(0.6666 - y)*d
g*m*epsilon*((-----------------^2) + d^2)
4 1
#1: r = -----------------------------------------^-
(10*pi^2) 3
1-> repeat simplify
d 2 45*y^2 1
#1: r = (--^-)*((g*m*epsilon*(0.72487500625 + ------ - (1.8748125*y)))^-)
pi 3 32 3
1-> y
Equation is a degree 2 polynomial equation in y.
Equation was solved with the quadratic formula.
Solve successful:
1
((0.6666*d^2*g*m*epsilon) - (0.25773333555556*((((2.5863941835972*d^2*g*m*epsilon)^2) + (d^2*g*m*epsilon*((10.705236737008*r^3*pi^2) - (7.7599585466463*d^2*g*m*epsilon))))^-)*sign))
2
#1: y = -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(d^2*g*m*epsilon)
1-> repeat simplify symbolic
10*r^3*pi^2 1
4*((------------- - d^2)^-)*sign
(g*m*epsilon) 2
#1: y = 0.6666 - --------------------------------
(15*d)
Successfully finished reading script file "test1.in".
1-> read test2
1-> clear all
1-> y=(a/2)^2/b/4
a
(-^2)
2
#1: y = -----
(4*b)
1-> l=f*(b-y)+z*(a-f)
#2: l = (f*(b - y)) + (z*(a - f))
2-> m=2*(b-y)-a+f
#3: m = (2*(b - y)) - a + f
3-> n=2*(b-y)+a-f
#4: n = (2*(b - y)) + a - f
4-> o=l*(1/m-1/n)/2
1 1
l*(- - -)
m n
#5: o = ---------
2
5-> eliminate l m n y
Eliminating variable l using solved equation #2...
Eliminating variable m using solved equation #3...
Eliminating variable n using solved equation #4...
Eliminating variable y using solved equation #1...
a
(-^2)
2 1 1
((f*(b - -----)) + (z*(a - f)))*(------------------------- - -------------------------)
(4*b) a a
(-^2) (-^2)
2 2
((2*(b - -----)) - a + f) ((2*(b - -----)) + a - f)
(4*b) (4*b)
#5: o = ---------------------------------------------------------------------------------------
2
5-> simplify
4*((f*((16*b^3) - (b*a^2))) + (16*b^2*z*(a - f)))*(a - f)
#5: o = -----------------------------------------------------------
((256*b^4) + (b^2*((128*a*f) - (96*a^2) - (64*f^2))) + a^4)
5-> copy
4*((f*((16*b^3) - (b*a^2))) + (16*b^2*z*(a - f)))*(a - f)
#6: o = -----------------------------------------------------------
((256*b^4) + (b^2*((128*a*f) - (96*a^2) - (64*f^2))) + a^4)
5-> f
Equation is a degree 2 polynomial equation in f.
Equation was solved with the quadratic formula.
Solve successful:
1
((((4*b*a*((16*b*((2*(z + o)) - b)) + a^2))^2) + (16*b*((16*b*((b*((16*b*((b*((a^2*((4*z) + (7*o))) + (16*b*o*(z - b + o)))) - (2*a^2*((z*((2*z) + (5*o))) + (3*o^2))))) - (a^4*((7*o) + (4*z))))) + (o*a^4*(z + o)))) + (a^6*o))))^-)*sign
2
(------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + (2*b*a*((16*b*(b - (2*(z + o)))) - a^2)))
2
#5: f = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(4*b*((16*b*(b - z - o)) - a^2))
5-> simplify symbolic
1 a^2
((sign*(((b^2*(a^2 + (16*(o^2 + (o*z))))) + (o*((a^2*b) - (16*b^3))))^-)*((8*b^2) - ---)) - (8*b^2*a*(z + o)))
a 2 2
#5: f = - + --------------------------------------------------------------------------------------------------------------
2 ((16*(b^3 - (b^2*(z + o)))) - (b*a^2))
5-> 6
4*((f*((16*b^3) - (b*a^2))) + (16*b^2*z*(a - f)))*(a - f)
#6: o = -----------------------------------------------------------
((256*b^4) + (b^2*((128*a*f) - (96*a^2) - (64*f^2))) + a^4)
6-> derivative z
Differentiating the RHS with respect to z and simplifying...
64*((b*(a - f))^2)
#7: o' = -----------------------------------------------------------
((256*b^4) + (b^2*((128*a*f) - (96*a^2) - (64*f^2))) + a^4)
7-> f
Equation is a degree 2 polynomial equation in f.
Equation was solved with the quadratic formula.
Solve successful:
1
((((128*b^2*a*(1 + o'))^2) + (256*b^2*((o'*((32*b^2*((8*b^2*(1 + o')) - (a^2*(5 + (3*o'))))) + (a^4*(1 + o')))) - (64*((b*a)^2)))))^-)*sign0
2
-(-------------------------------------------------------------------------------------------------------------------------------------------- - (64*b^2*a*(1 + o')))
2
#7: f = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------
(64*b^2*(1 + o'))
7-> repeat simplify symbolic
o' 1 a^2
#7: f = a + (sign0*(--------^-)*(----- - (2*b)))
(1 + o') 2 (8*b)
Successfully finished reading script file "test2.in".
7-> read test6
7-> ; Combine the equations for conservation of momentum and kinetic energy
7-> ; to solve for the resulting velocity of two objects colliding head on.
7-> clear all
1-> ; equations for energy:
1-> e1=1/2*mass1*velocity1_old^2
mass1*velocity1_old^2
#1: e1 = ---------------------
2
1-> e2=1/2*mass2*velocity2_old^2
mass2*velocity2_old^2
#2: e2 = ---------------------
2
2-> e3=1/2*mass1*velocity1_new^2
mass1*velocity1_new^2
#3: e3 = ---------------------
2
3-> e4=1/2*mass2*velocity2_new^2
mass2*velocity2_new^2
#4: e4 = ---------------------
2
4-> e1+e2=e3+e4
#5: e1 + e2 = e3 + e4
5-> eliminate all
Eliminating variable e4 using solved equation #4...
Eliminating variable e3 using solved equation #3...
Eliminating variable e2 using solved equation #2...
Eliminating variable e1 using solved equation #1...
mass1*velocity1_old^2 mass2*velocity2_old^2 mass1*velocity1_new^2 mass2*velocity2_new^2
#5: --------------------- + --------------------- = --------------------- + ---------------------
2 2 2 2
5-> ; equations for momentum:
5-> #1: u1=mass1*velocity1_old
#1: u1 = mass1*velocity1_old
1-> #2: u2=mass2*velocity2_old
#2: u2 = mass2*velocity2_old
2-> #3: u3=mass1*velocity1_new
#3: u3 = mass1*velocity1_new
3-> #4: u4=mass2*velocity2_new
#4: u4 = mass2*velocity2_new
4-> u1+u2=u3+u4
#6: u1 + u2 = u3 + u4
6-> eliminate all
Eliminating variable u4 using solved equation #4...
Eliminating variable u3 using solved equation #3...
Eliminating variable u2 using solved equation #2...
Eliminating variable u1 using solved equation #1...
#6: (mass1*velocity1_old) + (mass2*velocity2_old) = (mass1*velocity1_new) + (mass2*velocity2_new)
6-> clear 1-4
6-> eliminate velocity1_new
Solving equation #5 for velocity1_new and substituting into the current equation...
mass2*(velocity2_old^2 - velocity2_new^2) 1
#6: (mass1*velocity1_old) + (mass2*velocity2_old) = (mass1*((----------------------------------------- + velocity1_old^2)^-)*sign) + (mass2*velocity2_new)
mass1 2
6-> velocity2_new
Equation is a degree 0.5 polynomial equation in velocity2_new.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 2 polynomial equation in velocity2_new.
Equation was solved with the quadratic formula.
Solve successful:
1
((((2*((mass1*velocity1_old) + (mass2*velocity2_old)))^2) + (4*velocity2_old*((mass1*((mass1*(velocity2_old - (2*velocity1_old))) - (2*mass2*velocity1_old))) - (mass2^2*velocity2_old))))^-)*sign0
2
(--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + (mass1*velocity1_old) + (mass2*velocity2_old))
2
#6: velocity2_new = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(mass1 + mass2)
6-> simplify
1
(((((mass1*(velocity1_old - velocity2_old))^2)^-)*sign0) + (mass1*velocity1_old) + (mass2*velocity2_old))
2
#6: velocity2_new = ---------------------------------------------------------------------------------------------------------
(mass1 + mass2)
6-> velocity2_new = ((sign*((mass1*(velocity1_old-velocity2_old))^2)^.5)+(mass1*velocity1_old)+(mass2*velocity2_old))/(mass1+mass2)
1
((sign*(((mass1*(velocity1_old - velocity2_old))^2)^-)) + (mass1*velocity1_old) + (mass2*velocity2_old))
2
#7: velocity2_new = --------------------------------------------------------------------------------------------------------
(mass1 + mass2)
7-> compare 6
Comparing #6 with #7...
Equations are identical.
Successfully finished reading script file "test6.in".
7-> clear all
1-> read simplify
1->
1-> ; Some complete simplifications Mathomatic has always been able to do.
1-> ; The result is the smallest expression that gives exactly the same results.
1->
1-> 2*(x^2-y^2)^16-(x^2-y^2)^15*(2x^2-3)
#1: (2*((x^2 - y^2)^16)) - (((x^2 - y^2)^15)*((2*x^2) - 3))
1-> simplify ; Simplify the previously entered expression above.
#1: ((x^2 - y^2)^15)*(3 - (2*y^2))
1-> repeat echo *
*******************************************************************************
1-> a^3/((a-b)*(a-c))+b^3/((b-c)*(b-a))+c^3/((c-a)*(c-b))
a^3 b^3 c^3
#2: ----------------- + ----------------- + -----------------
((a - b)*(a - c)) ((b - c)*(b - a)) ((c - a)*(c - b))
2-> simplify ; Simplify algebraic fractions.
#2: a + b + c
2-> repeat echo *
*******************************************************************************
2-> (x^6+a^6)*(x+1)/((x^6+a^6)*(x^2-a^2)+a^2*x^2*(x^4-a^4))+a^2*x^2*(x+1)/(x^6-a^6-a^2*x^2*(x^2-a^2))
(x^6 + a^6)*(x + 1) a^2*x^2*(x + 1)
#3: --------------------------------------------------- + -----------------------------------
(((x^6 + a^6)*(x^2 - a^2)) + (a^2*x^2*(x^4 - a^4))) (x^6 - a^6 - (a^2*x^2*(x^2 - a^2)))
3-> simplify
(x + 1)
#3: -----------
(x^2 - a^2)
3-> repeat echo *
*******************************************************************************
3-> (1-(1-(y+1)/(x+y+1))/(1-x/(x+y+1)))/((y+1)^2-x/(1+x/(y-x+1))*(x*(y+1)/(y-x+1)-x))
(y + 1)
(1 - -----------)
(x + y + 1)
(1 - -----------------)
x
(1 - -----------)
(x + y + 1)
#4: -----------------------------------
x*(y + 1)
x*(----------- - x)
(y - x + 1)
(((y + 1)^2) - -------------------)
x
(1 + -----------)
(y - x + 1)
4-> simplify fraction ; Any complex fraction can be reduced to a simple fraction with this command.
1
#4: -------------------------------------
(1 + y^2 + (2*y) + (x*(y + 1)) + x^2)
4-> repeat echo *
*******************************************************************************
4-> ((2*((x*(x+(((x^2)-1)^(1/2))))-1))+1)/((2*x*((x^2)-1))+((((x^2)-1)^(1/2))*((2*(x^2))-1)))
1
((2*((x*(x + ((x^2 - 1)^-))) - 1)) + 1)
2
#5: -------------------------------------------------
1
((2*x*(x^2 - 1)) + (((x^2 - 1)^-)*((2*x^2) - 1)))
2
5-> simplify ; Simplify an expression containing radicals (roots).
1
#5: -------------
1
((x^2 - 1)^-)
2
5-> ; Rationalizing the denominator was required to simplify the above expression.
5-> repeat echo *
*******************************************************************************
5-> ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) / (4*y^2 + x^2)
((x - (2*y))^4)
(------------------- + 1)*(y + a)*((2*y) + x)
((x^2 - (4*y^2))^2)
#6: ---------------------------------------------
((4*y^2) + x^2)
6-> repeat simplify
2*(y + a)
#6: -----------
((2*y) + x)
Successfully finished reading script file "simplify.in".
6-> read heron
6-> clear all
1-> ; This Mathomatic script shows two reverse derivations of Heron's formula.
1-> ; This is Heron's formula for the area of any triangle,
1-> ; given side lengths "a", "b", and "c".
1->
1-> 2s = a+b+c
#1: 2*s = a + b + c
1-> triangle_area = (s*(s-a)*(s-b)*(s-c))^.5
1
#2: triangle_area = (s*(s - a)*(s - b)*(s - c))^-
2
2-> eliminate s ; Heron's formula:
Solving equation #1 for s and substituting into the current equation...
(a + b + c) (a + b + c) (a + b + c)
(a + b + c)*(----------- - a)*(----------- - b)*(----------- - c)
2 2 2 1
#2: triangle_area = -----------------------------------------------------------------^-
2 2
2-> simplify ; Heron's formula simplified by Mathomatic:
1
(((2*((a^2*(b^2 + c^2)) + ((b*c)^2))) - a^4 - b^4 - c^4)^-)
2
#2: triangle_area = -----------------------------------------------------------
4
2-> pause
2-> ; This is how we arrive at Heron's formula for the area
2-> ; of any triangle, given side lengths a, b, and c, using the formula
2-> ; for the area of a trapezoid with side lengths a, b, c, and d,
2-> ; where a and c are the parallel sides (a is the longer parallel side).
2->
2-> ; A trapezoid is a quadrilateral with
2-> ; two sides that are parallel to each other.
2->
2-> ; Formula for the area of a trapezoid that is not a parallelogram:
2-> trapezoid_area=(a+c)/(4*(a-c))*((a+b-c+d)*(a-b-c+d)*(a+b-c-d)*(-a+b+c+d))^.5
1
(a + c)*(((a + b - c + d)*(a - b - c + d)*(a + b - c - d)*(b - a + c + d))^-)
2
#3: trapezoid_area = -----------------------------------------------------------------------------
(4*(a - c))
3-> pause
3-> copy
1
(a + c)*(((a + b - c + d)*(a - b - c + d)*(a + b - c - d)*(b - a + c + d))^-)
2
#4: trapezoid_area = -----------------------------------------------------------------------------
(4*(a - c))
3-> replace c with 0 ; make the shorter parallel side length = 0
1
(((a + b + d)*(a - b + d)*(a + b - d)*(b - a + d))^-)
2
#3: trapezoid_area = -----------------------------------------------------
4
3-> replace d with c ; Heron's formula in its simplest form:
1
(((a + b + c)*(a - b + c)*(a + b - c)*(b - a + c))^-)
2
#3: trapezoid_area = -----------------------------------------------------
4
3-> replace trapezoid_area with triangle_area
1
(((a + b + c)*(a - b + c)*(a + b - c)*(b - a + c))^-)
2
#3: triangle_area = -----------------------------------------------------
4
3-> pause Please press the Enter key to verify the result.
3-> copy
1
(((a + b + c)*(a - b + c)*(a + b - c)*(b - a + c))^-)
2
#5: triangle_area = -----------------------------------------------------
4
3-> display 2
1
(((2*((a^2*(b^2 + c^2)) + ((b*c)^2))) - a^4 - b^4 - c^4)^-)
2
#2: triangle_area = -----------------------------------------------------------
4
3-> compare 5 with 2 ; simplify and compare the result with Heron's formula:
Comparing #5 with #2...
Simplifying both equations...
Equations are identical.
3-> clear 5
3-> pause
3->
3-> ; This is how we arrive at Heron's formula for the area
3-> ; of any triangle, given side lengths a, b, and c, using
3-> ; Brahmagupta's formula for the area of a cyclic quadrilateral,
3-> ; making one side length equal zero, to make a cyclic triangle.
3-> ; Since all triangles are cyclic (can be circumscribed by a circle),
3-> ; this gives the area for any triangle.
3->
3-> 2s=a+b+c+d ; cyclic quadrilateral side lengths are a, b, c, and d
#5: 2*s = a + b + c + d
5-> cyclic_area = ((s-a)*(s-b)*(s-c)*(s-d))^.5
1
#6: cyclic_area = ((s - a)*(s - b)*(s - c)*(s - d))^-
2
6-> eliminate s ; Brahmagupta's formula:
Solving equation #5 for s and substituting into the current equation...
(a + b + c + d) (a + b + c + d) (a + b + c + d) (a + b + c + d) 1
#6: cyclic_area = ((--------------- - a)*(--------------- - b)*(--------------- - c)*(--------------- - d))^-
2 2 2 2 2
6-> pause
6-> copy
(a + b + c + d) (a + b + c + d) (a + b + c + d) (a + b + c + d) 1
#7: cyclic_area = ((--------------- - a)*(--------------- - b)*(--------------- - c)*(--------------- - d))^-
2 2 2 2 2
6-> replace d with 0 ; make one side length zero to get Heron's formula:
(a + b + c) (a + b + c) (a + b + c)
(----------- - a)*(----------- - b)*(----------- - c)*(a + b + c)
2 2 2 1
#6: cyclic_area = -----------------------------------------------------------------^-
2 2
6-> pause Please press the Enter key to verify the result.
6-> compare 2 ; simplify and compare the result with Heron's formula:
Comparing #2 with #6...
Simplifying both equations...
Variable triangle_area in the first equation
is equal to cyclic_area in the second equation.
6-> clear
6-> clear 1 5
Successfully finished reading script file "heron.in".
6-> clear all
1-> read radius
1->
1-> ; Some more fun formulas. These are very similar to Heron's formula
1-> ; for the area of a triangle (see "heron.in"). a, b, and c are the
1-> ; lengths of the sides of the triangle.
1->
1-> s=(a+b+c)/2 ; semiperimeter
(a + b + c)
#1: s = -----------
2
1-> ; radius of a circle inscribed in a triangle, called an incircle:
1-> inradius=(s*(s-a)*(s-b)*(s-c))^.5/s
1
((s*(s - a)*(s - b)*(s - c))^-)
2
#2: inradius = -------------------------------
s
2-> eliminate s
Eliminating variable s using solved equation #1...
(a + b + c) (a + b + c) (a + b + c)
(a + b + c)*(----------- - a)*(----------- - b)*(----------- - c)
2 2 2 1
2*(-----------------------------------------------------------------^-)
2 2
#2: inradius = -----------------------------------------------------------------------
(a + b + c)
2-> simplify
8*((b^2*c) + (b*c^2)) 1
(((2*((b*(a - (3*c))) + (a*c))) - a^2 + --------------------- - b^2 - c^2)^-)
(a + b + c) 2
#2: inradius = -----------------------------------------------------------------------------
2
2->
2-> ; The following is the equation for the radius of a circle circumscribing
2-> ; a triangle, called a circumcircle, which is a circle that passes through
2-> ; all the vertices of a polygon.
2-> radius=a*b*c/(4*(s*(s-a)*(s-b)*(s-c))^.5)
a*b*c
#3: radius = -----------------------------------
1
(4*((s*(s - a)*(s - b)*(s - c))^-))
2
3-> eliminate s
Eliminating variable s using solved equation #1...
a*b*c
#3: radius = -------------------------------------------------------------------------
(a + b + c) (a + b + c) (a + b + c)
(a + b + c)*(----------- - a)*(----------- - b)*(----------- - c)
2 2 2 1
(4*(-----------------------------------------------------------------^-))
2 2
3-> simplify
a*b*c
#3: radius = -----------------------------------------------------------
1
(((2*((a^2*(b^2 + c^2)) + ((b*c)^2))) - a^4 - b^4 - c^4)^-)
2
3-> #s ; Search backwards for the s variable; "/" searches forwards.
Searching backwards for variable.
(a + b + c)
#1: s = -----------
2
1-> clear ; No longer needed.
1-> display all
8*((b^2*c) + (b*c^2)) 1
(((2*((b*(a - (3*c))) + (a*c))) - a^2 + --------------------- - b^2 - c^2)^-)
(a + b + c) 2
#2: inradius = -----------------------------------------------------------------------------
2
a*b*c
#3: radius = -----------------------------------------------------------
1
(((2*((a^2*(b^2 + c^2)) + ((b*c)^2))) - a^4 - b^4 - c^4)^-)
2
Successfully finished reading script file "radius.in".
1-> clear all
1-> read pyth3d
1->
1-> ; This arrives at the distance between two points in 3D space from the
1-> ; Pythagorean theorem (distance between two points on a 2D plane).
1-> ; The coordinate of point 1, 2D: (x1, y1), 3D: (x1, y1, z1).
1-> ; The coordinate of point 2, 2D: (x2, y2), 3D: (x2, y2, z2).
1->
1-> distance2D^2=(x1-x2)^2+(y1-y2)^2 ; Distance formula for a 2D Cartesian plane.
#1: distance2D^2 = ((x1 - x2)^2) + ((y1 - y2)^2)
1-> distance3D^2=distance2D^2+(z1-z2)^2 ; Add another leg.
#2: distance3D^2 = distance2D^2 + ((z1 - z2)^2)
2-> eliminate distance2D ; Combine the two equations.
Solving equation #1 for distance2D and substituting into the current equation...
#2: distance3D^2 = ((x1 - x2)^2) + ((y1 - y2)^2) + ((z1 - z2)^2)
2-> distance3D ; Solve to get the distance in 3D Cartesian space.
Solve successful:
1
#2: distance3D = ((((x1 - x2)^2) + ((y1 - y2)^2) + ((z1 - z2)^2))^-)*sign0
2
Successfully finished reading script file "pyth3d.in".
2-> clear all
1-> read distance
1->
1-> ; This input arrives at the shortest distance between a point and a line
1-> ; in 2 dimensions. The point is at (x0, y0) in cartesian coordinates.
1-> ; (x, y) are the points on the line.
1->
1-> a*x+b*y+c=0 ; equation of the line
#1: (a*x) + (b*y) + c = 0
1-> y ; solve for y
Solve successful:
-((a*x) + c)
#1: y = ------------
b
1-> unfactor fraction ; equation of the line in slope-intercept form:
-c a*x
#1: y = -- - ---
b b
1-> distance=|a*(x-x0)+b*(y-y0)|/(a^2+b^2)^.5
1
((((a*(x - x0)) + (b*(y - y0)))^2)^-)
2
#2: distance = -------------------------------------
1
((a^2 + b^2)^-)
2
2-> eliminate y ; Combine the above two equations to eliminate x and y.
Eliminating variable y using solved equation #1...
-c a*x 1
((((a*(x - x0)) + (b*(-- - --- - y0)))^2)^-)
b b 2
#2: distance = --------------------------------------------
1
((a^2 + b^2)^-)
2
2-> simplify ; The beautiful answer is:
(((a*x0) + c + (b*y0))^2) 1
#2: distance = -------------------------^-
(a^2 + b^2) 2
2-> ; Replacing a with -m, b with 1, and c with -b results in the shortest distance from the line y=m*x+b.
Successfully finished reading script file "distance.in".
2-> clear all
1-> read circles
1->
1-> ; This is a simple example of eliminate command usage.
1-> ; Combine the equations for 2 circles of radius "r" on a 2D Cartesian plane
1-> ; to find the points of intersection (x, y).
1->
1-> (x-x1)^2+(y-y1)^2=r^2 ; circle of radius "r" with center at (x1, y1)
#1: ((x - x1)^2) + ((y - y1)^2) = r^2
1-> (x-x2)^2+(y-y2)^2=r^2 ; circle of radius "r" with center at (x2, y2)
#2: ((x - x2)^2) + ((y - y2)^2) = r^2
2-> eliminate x ; combine the two equations, removing the x variable from the result
Solving equation #1 for x and substituting into the current equation...
1
#2: (((((r^2 - ((y - y1)^2))^-)*sign) + x1 - x2)^2) + ((y - y2)^2) = r^2
2
2-> solve for y
Equation is a degree 0.5 polynomial equation in y.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 2 polynomial equation in y.
Equation was solved with the quadratic formula.
Solve successful:
1
((((4*((y1*((x2*((2*x1) - x2)) - x1^2 + (y1*(y2 - y1)) + y2^2)) + (y2*((x1*((2*x2) - x1)) - x2^2 - y2^2))))^2) + (16*((y2*((y2*((y1*((y1*((2*((x2*((2*x1) - x2)) - x1^2)) + (y2*(y2 - (4*y1))) + y1^2)) + (2*y2*((2*((x1*(x1 - (2*x2))) + x2^2)) + y2^2)))) + (x1*((x1*((3*((x1*((4*x2) - x1)) - y2^2 - (6*x2^2))) + (4*r^2))) + (2*x2*((6*x2^2) + (3*y2^2) - (4*r^2))))) + (x2^2*((4*r^2) - (3*(x2^2 + y2^2)))) - y2^4)) + (2*y1*((y1^2*((2*((x2*(x2 - (2*x1))) + x1^2)) + y1^2)) + (x1*((x1*((x1*(x1 - (4*x2))) + (6*x2^2) - (4*r^2))) + (4*x2*((2*r^2) - x2^2)))) + (x2^2*(x2^2 - (4*r^2))))))) + (y1^2*((y1^2*((3*((x2*((2*x1) - x2)) - x1^2)) - y1^2)) + (x1*((x1*((3*((x1*((4*x2) - x1)) - (6*x2^2))) + (4*r^2))) + (4*x2*((3*x2^2) - (2*r^2))))) + (x2^2*((4*r^2) - (3*x2^2))))) + (x2*((x2*((x1*((x1*((5*((x1*((4*x2) - (3*x1))) - (3*x2^2))) + (24*r^2))) + (2*x2*((3*x2^2) - (8*r^2))))) + (x2^2*((4*r^2) - x2^2)))) + (2*x1^3*((3*x1^2) - (8*r^2))))) + (x1^4*((4*r^2) - x1^2)))))^-)*sign0
2
(------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + (2*((y1*((x2*(x2 - (2*x1))) + x1^2 + (y1*(y1 - y2)) - y2^2)) + (y2*((x1*(x1 - (2*x2))) + x2^2 + y2^2)))))
2
#2: y = -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(4*((y2*(y2 - (2*y1))) + y1^2 + (x2*(x2 - (2*x1))) + x1^2))
2-> repeat simplify
4*r^2 1
(y2 + y1 + (((((x1 - x2)^2)*(----------------------------------------------------- - 1))^-)*sign0))
(y2^2 - (2*((y2*y1) + (x2*x1))) + x2^2 + y1^2 + x1^2) 2
#2: y = ---------------------------------------------------------------------------------------------------
2
Successfully finished reading script file "circles.in".
2-> clear all
1-> read ellipse
1->
1-> ; This is an equation for an ellipse that was created using the rule
1-> ; that the sum of the distances from any point on the perimeter (x, y)
1-> ; to the two foci: (x1, y1) and (x2, y2), is a constant k. This can
1-> ; represent any ellipse of any orientation on the Cartesian plane.
1->
1-> k = ((x1-x)^2+(y1-y)^2)^0.5 + ((x2-x)^2+(y2-y)^2)^0.5
1 1
#1: k = ((((x1 - x)^2) + ((y1 - y)^2))^-) + ((((x2 - x)^2) + ((y2 - y)^2))^-)
2 2
1->
1-> ; A simplified equation for a right ellipse centered at the origin (0, 0)
1-> ; of the Cartesian plane:
1->
1-> 1 = x^2/radius1^2 + y^2/radius2^2
x^2 y^2
#2: 1 = --------- + ---------
radius1^2 radius2^2
2-> ; The x-intercepts are radius1 and -radius1 because y=0 there.
2-> ; The y-intercepts are radius2 and -radius2 because x=0 there.
Successfully finished reading script file "ellipse.in".
2-> solve all y
Equation is a degree 0.5 polynomial equation in y.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 0.5 polynomial equation in y.
Raising both equation sides to the power of 2 and expanding...
Equation is a degree 2 polynomial equation in y.
Equation was solved with the quadratic formula.
Solve successful:
1
((((4*((2*x*(y1 - y2)*(x1 - x2)) + (y2*((y2*(y1 - y2)) - x2^2 + x1^2 + y1^2 + k^2)) + (y1*(x2^2 - x1^2 - y1^2 + k^2))))^2) + (64*((y1*((y1*((x2*((x2*((x*(x2 - x1 - x)) - (y1*y2))) - (x*(x1^2 + y1^2)))) + (x*((x1*((x1*(x1 - x)) + y1^2)) + (x*k^2))) + (x1^2*(k^2 + (y1*y2))) - (y1*y2*(y2^2 + k^2)))) + (y2*((x2^2*(y2^2 - x1^2 - k^2)) - (y2^2*(k^2 + x1^2)) - ((x1*k)^2))))) + (k^2*((x2*((x2*((x*(x + x1 - x2)) + y2^2)) + (x*(k^2 + x1^2)))) + (x*((x1*((x1*(x - x1)) + k^2)) + (x*(y2^2 - k^2)))))) + (y2^2*x*((x2*((x2*(x2 - x1 - x)) + y2^2 - x1^2)) + (x1*((x1*(x1 - x)) - y2^2)))))) + (32*((y1*((y1*((x2^2*(x1^2 + y1^2)) + (y1^2*((y1*y2) - x1^2)) + ((y2*k)^2))) + (y2*(x2^4 + y2^4 + x1^4 + k^4)))) - (k^2*((x2^2*(x1^2 + k^2)) + ((k*x1)^2))) + (y2^2*((x2^2*(x1^2 - y2^2)) + ((y2*x1)^2))))) + (16*((y1*((y1*((x2*((8*x*((x*x1) + (y1*y2))) - x2^3)) - (x1*((8*x*(k^2 + (y1*y2))) + x1^3)) + (y2^2*(y2^2 + y1^2)) + (y1^2*((3*k^2) - y1^2)) - (3*k^4))) + (8*y2*x*((x2*((x2*(x + x1 - x2)) + k^2 + (x1*(x1 - (2*x))) - y2^2)) + (x1*(y2^2 + (x1*(x - x1)) + k^2)) - (x*k^2))))) + (k^2*((x2*(x2^3 - (8*x*((x*x1) + y2^2)))) + (3*y2^2*(y2^2 - k^2)) + x1^4 + k^4)) + (y2^2*((x2*((8*x^2*x1) - x2^3)) - y2^4 - x1^4)))))^-)*sign
2
(-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + (4*x*(y2 - y1)*(x1 - x2)) + (2*((y2*((y2*(y2 - y1)) + x2^2 - x1^2 - y1^2 - k^2)) + (y1*(x1^2 - x2^2 + y1^2 - k^2)))))
2
#1: y = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(4*((y1*(y1 - (2*y2))) - k^2 + y2^2))
Solve successful:
x 1
#2: y = ((-((-------^2) - 1))^-)*sign0*radius2
radius1 2
2-> simplify all
1
((((k^2*(x1^2 + (4*(x^2 - (x*x2))) + y1^2 + x2^2 + y2^2 + (2*((x1*(x2 - (2*x))) - (y1*y2))))) - k^4)*(y1^2 - (2*((y1*y2) + (x2*x1))) + y2^2 + x2^2 + x1^2 - k^2))^-)*sign
2 (x1 + x2)
(------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ((y1 - y2)*(x - ---------)*(x2 - x1)))
(y1 + y2) 2 2
#1: y = --------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 (y1^2 - (2*y1*y2) - k^2 + y2^2)
x 1
#2: y = ((1 - (-------^2))^-)*sign0*radius2
radius1 2
2-> pause
2-> clear all
1-> help examples
*******************************************************************************
1-> ; Example 1:
1-> ; Here the derivative of the absolute value function is computed.
1-> ; Expressions are entered by just typing them in:
1-> |x| ; The absolute value of x
1
#1: (x^2)^-
2
1-> derivative ; The result gives the sign of x:
Differentiating with respect to x and simplifying...
x
#2: ---------
1
((x^2)^-)
2
2-> pause
2-> repeat echo -
-------------------------------------------------------------------------------
2-> ; Example 2:
2-> ; Here the calculate command is used to plug values into a solved formula.
2-> ; A common temperature conversion formula (from "help conversions"):
2-> fahrenheit = (9*celsius/5) + 32
9*celsius
#3: fahrenheit = --------- + 32
5
3-> repeat calculate ; plug in values until an empty line is entered
fahrenheit = (1.8*celsius) + 32, with fractions it is: (9*celsius/5) + 32
3->
3-> ; Solve for the other variable and simplify the result:
3-> solve for celsius
Solve successful:
-5*(32 - fahrenheit)
#3: celsius = --------------------
9
3-> simplify
5*(fahrenheit - 32)
#3: celsius = -------------------
9
3-> repeat calculate ; plug in values until an empty line is entered
celsius = (0.55555555555556*fahrenheit) - 17.777777777778, with fractions it is: (5*fahrenheit/9) - (160/9)
3->
3-> variables count; count all variables that occur in expressions
fahrenheit /* count = 1 */
celsius /* count = 1 */
3-> pause
3-> repeat echo -
-------------------------------------------------------------------------------
3-> ; Example 3:
3-> ; Expand the following to polynomial form, then refactor and differentiate:
3-> (x+y+z)^3
#4: (x + y + z)^3
4-> expand count ; Expand and count the resulting number of terms:
#4: x^3 + (3*x^2*y) + (3*x^2*z) + (3*x*y^2) + (6*x*y*z) + (3*x*z^2) + y^3 + (3*y^2*z) + (3*y*z^2) + z^3
#4: Expression consists of a total of 10 terms.
4-> pause
4-> simplify ; refactor:
#4: (x + y + z)^3
4-> derivative x ; here is the derivative, with respect to x:
Differentiating with respect to x and simplifying...
#5: 3*((x + y + z)^2)
5-> expand count ; and its term count, when expanded:
#5: (3*x^2) + (6*x*y) + (6*x*z) + (3*y^2) + (6*y*z) + (3*z^2)
#5: Expression consists of a total of 6 terms.
5-> clear all
1-> help conversions
*******************************************************************************
Help conversions:
-----------------
Commonly used metric/English conversions.
Select the equation you want (for example, with "1" or "/celsius")
and type the unit name you want, to solve for it (like "celsius").
Then type "repeat calculate" for units conversion and trying different values.
These values are correct for the US and UK.
-------------------------------------------
1-> ; Temperature
1-> fahrenheit = (9*celsius/5) + 32
9*celsius
#1: fahrenheit = --------- + 32
5
1-> kelvin = celsius + 273.15
5463
#2: kelvin = celsius + ----
20
2-> ; Distance
2-> inches = centimeters/2.54
50*centimeters
#3: inches = --------------
127
3-> miles = kilometers/1.609344
#4: miles = 0.62137119223733*kilometers
4-> ; Weight
4-> pounds = kilograms/0.45359237
#5: pounds = 2.2046226218488*kilograms
5-> simplify all
9*celsius
#1: fahrenheit = --------- + 32
5
5463
#2: kelvin = celsius + ----
20
50*centimeters
#3: inches = --------------
127
#4: miles = 0.62137119223733*kilometers
#5: pounds = 2.2046226218488*kilograms
5-> clear all
1-> help geometry
*******************************************************************************
Help geometry:
--------------
Commonly used standard (Euclidean) geometric formulas
-----------------------------------------------------
1-> ; Triangle area, "b" is the "base" side:
1-> triangle_area = b*height/2
b*height
#1: triangle_area = --------
2
1-> ; Here is Heron's formula for the area of any triangle
1-> ; given all three side lengths ("a", "b", and "c"):
1-> triangle_area = (((a + b + c)*(a - b + c)*(a + b - c)*(b - a + c))^(1/2))/4
1
(((a + b + c)*(a - b + c)*(a + b - c)*(b - a + c))^-)
2
#2: triangle_area = -----------------------------------------------------
4
2->
2-> ; Rectangle of length "l" and width "w":
2-> rectangle_area = l*w
#3: rectangle_area = l*w
3-> rectangle_perimeter = 2*l + 2*w
#4: rectangle_perimeter = (2*l) + (2*w)
4->
4-> ; Trapezoid of parallel sides "a" and "b",
4-> ; and the "distance" between them:
4-> trapezoid_area = distance*(a + b)/2
distance*(a + b)
#5: trapezoid_area = ----------------
2
5->
5-> ; Circle of radius "r":
5-> circle_area = pi*r^2
#6: circle_area = pi*r^2
6-> circle_perimeter = 2*pi*r
#7: circle_perimeter = 2*pi*r
7->
7-> ; 3D rectangular solid of length "l", width "w", and height "h":
7-> brick_volume = l*w*h
#8: brick_volume = l*w*h
8-> brick_surface_area = 2*l*w + 2*l*h + 2*w*h
#9: brick_surface_area = (2*l*w) + (2*l*h) + (2*w*h)
9->
9-> ; 3D sphere of radius "r":
9-> sphere_volume = 4/3*pi*r^3
4*pi*r^3
#10: sphere_volume = --------
3
10-> sphere_surface_area = 4*pi*r^2
#11: sphere_surface_area = 4*pi*r^2
11->
11-> ; Convex 2D polygon with straight sides,
11-> ; sum of all interior angles formula in degree, radian, and gradian units:
11-> sum_degrees = (sides - 2)*180
#12: sum_degrees = 180*(sides - 2)
12-> sum_radians = (sides - 2)*pi
#13: sum_radians = (sides - 2)*pi
13-> sum_grads = (sides - 2)*180*10/9 ; Rarely used gradian formula.
#14: sum_grads = 200*(sides - 2)
14-> ; "sides" is the number of sides of any convex 2D polygon.
14-> ; Convex means that all interior angles are less than 180 degrees.
14-> ; Type "elim sides" to get the radians/degrees/grads conversion formulas.
14-> simplify all
b*height
#1: triangle_area = --------
2
1
(((2*((b^2*(a^2 + c^2)) + ((a*c)^2))) - a^4 - b^4 - c^4)^-)
2
#2: triangle_area = -----------------------------------------------------------
4
#3: rectangle_area = l*w
#4: rectangle_perimeter = 2*(l + w)
distance*(a + b)
#5: trapezoid_area = ----------------
2
#6: circle_area = pi*r^2
#7: circle_perimeter = 2*pi*r
#8: brick_volume = l*w*h
#9: brick_surface_area = 2*((l*(w + h)) + (w*h))
4*pi*r^3
#10: sphere_volume = --------
3
#11: sphere_surface_area = 4*pi*r^2
#12: sum_degrees = 180*(sides - 2)
#13: sum_radians = pi*(sides - 2)
#14: sum_grads = 200*(sides - 2)
14-> quit
ByeBye!! from Mathomatic.
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