1-> ; This is a Mathomatic script that reads in all test scripts.
1->
1-> clear all
1-> read finance
1->
1-> ; Derivation of the mortgage payment formula.
1-> ; Here are 3 related financial formulas that can be "read" into Mathomatic.
1->
1-> set finance ; Enable money mode.
1->
1-> ; Compound Interest Future Value Formula:
1-> fv1 = pv*(1+interest_rate)^n
#1: fv1 = pv*((1.00 + interest_rate)^n)
1->
1-> ; 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 Amortized Loan Formula.
2-> pause
2-> ; Set equal to produce the Amortized Loan Formula (Mortgage Payment Formula):
2-> fv1 = fv2
#3: fv1 = fv2
3-> eliminate all ; combine both formulas to produce the mortgage payment formula:
Substituting the RHS of equation #2 into the current equation for variable (fv2)...
Substituting the RHS of equation #1 into the current equation for variable (fv1)...
payment*(((1.00 + interest_rate)^n) - 1.00)
#3: pv*((1.00 + interest_rate)^n) = -------------------------------------------
interest_rate
3-> solve verify payment ; solve for payment per period, verifying the result:
pv*((1.00 + interest_rate)^n)*interest_rate
#3: payment = -------------------------------------------
(((1.00 + interest_rate)^n) - 1.00)
Solution verified.
3-> ; pv = present value
3-> ; fv = future value (maturity value)
3-> ; interest_rate = interest rate per period (1 = 100%)
3-> ; n = number of periods
3-> pause End
Finished reading file "finance.in".
3-> simplify all
#1: fv1 = pv*((1.00 + interest_rate)^n)
payment*(((1.00 + interest_rate)^n) - 1.00)
#2: fv2 = -------------------------------------------
interest_rate
pv*((1.00 + interest_rate)^n)*interest_rate
#3: payment = -------------------------------------------
(((1.00 + interest_rate)^n) - 1.00)
3-> solve verify pv
payment*(((1.00 + interest_rate)^n) - 1.00)
#3: pv = -------------------------------------------
(((1.00 + interest_rate)^n)*interest_rate)
Solution verified.
3-> simplify
1.00
payment*(1.00 - --------------------------)
((1.00 + interest_rate)^n)
#3: pv = -------------------------------------------
interest_rate
3-> a=2/-3
(-2.00)
#4: a = -------
3.00
4-> list
#4: a = (-2.00)/3.00
4-> set no finance
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-> solve verify x ; Mathomatic can solve that and verify the solution:
Equation is a degree 2 polynomial equation in (x).
Equation was solved with the quadratic formula.
1
((((b^2 - (4*a*c))^-)*sign) - b)
2
#1: x = --------------------------------
(2*a)
All solutions verified.
1-> ; This is the quadratic formula.
1-> ; Use the calculate command to temporarily plug in coefficients.
1-> ; The coefficients may be any mathematical expression not containing x.
Finished reading file "quadratic.in".
1-> 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 verify r ; Solve for the resulting resistance, verifying the result.
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 verify r ; Solve for the resulting resistance, verifying the result.
r1*r2*r3
#4: r = --------------------------
(((r2 + r1)*r3) + (r1*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.
Finished reading 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
#1: 0 = ((1 - phi)*phi) + 1
1-> unfactor
#1: 0 = phi - phi^2 + 1
1-> phi ; The golden ratio will help us directly compute Fibonacci numbers.
Equation is a degree 2 polynomial equation in (phi).
Equation was solved with the quadratic formula.
1
(1 - ((5^-)*sign))
2
#1: phi = ------------------
2
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:
Substituting the RHS of equation #1 into the current equation for variable (phi)...
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.
Finished reading file "fibonacci.in".
2-> clear all
1-> read test
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)
((3*d^2) - 12)
#4: y = -------------- - 6
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)
((3*d^2) - 12)
#4: y = -------------- - 6
7
((1 + d^2)^-)
2
1 1
#5: y = (9 + (4*(2^-)))^-
2 2
5-> x^(1/99)=x
1
#6: x^-- = x
99
6-> solve verify x
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".
#6: x = sign
All solutions verified.
6-> calculate
There are 2 solutions.
Solution number 1 with sign = 1:
x = 1
Solution number 2 with sign = -1:
x = -1
6-> (2i)^.5+e^(pi*i)
answer = i
7-> (1-2i)/(3+4i)
answer = (-0.4*i) - 0.2 = (-2*i/5) - (1/5)
8-> divide (1-2i) (3+4i)
Result of complex number division:
-0.2 -0.4*i
8->
8-> y=x^3
#9: y = x^3
9-> extrema x
#10: x = 0
10-> (x+1)^4
#11: (x + 1)^4
11-> extrema x
#12: x = -1
12-> 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
Inverse check: 1
+1*i
Inverse check: 1
-1
Inverse check: 1
-1*i
Inverse check: 1
12-> factor number -75 100000000000000 999999999999999
-75 = 3 * 5^2 * -1
100000000000000 = 2^14 * 5^14
999999999999999 = 3^3 * 31 * 37 * 41 * 271 * 2906161
12-> 7921%14 ; should be exactly 11
answer = 11
Finished reading 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-> 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)
4-> 1
1
#1: b = ((-1)^n*a)^-------
(1 - n)
1-> solve verify a
(b^(1 - n))
#1: a = -----------
(-1)^n
Solution verified.
1-> simplify
(b^(1 - n))
#1: a = -----------
(-1)^n
1-> 2
(2 - n)
(y^-------)
(n - 1)
#2: x = ---------------------------
n 1
((n^-------) - (n^-------))
(n - 1) (n - 1)
2-> solve verify y
x*(n - 1) 1
#2: y = ((---------^(n - 1))*n^n)^-------
n (2 - n)
Solution verified.
2-> simplify
1
#2: y = (((x*(n - 1))^(n - 1))*n)^-------
(2 - n)
2-> 1/(x+y)
1
#5: -------
(x + y)
5-> taylor x 5 0
Computing the Taylor series and simplifying...
5 derivatives applied.
1 x x^2 x^3 x^4 x^5
#6: - - --- + --- - --- + --- - ---
y y^2 y^3 y^4 y^5 y^6
6-> fraction
(y^5 + (x^2*y^3) + (x^4*y) - x^5 - (x^3*y^2) - (x*y^4))
#6: -------------------------------------------------------
y^6
6-> simplify fraction
(y^5 - (y^4*x) + (y^3*x^2) - (y^2*x^3) + (y*x^4) - x^5)
#6: -------------------------------------------------------
y^6
6-> simplify
x^5
(x^4 - ---)
y
(----------- - x^3)
y
(x^2 + -------------------)
y
(--------------------------- - x)
y
(1 + ---------------------------------)
y
#6: ---------------------------------------
y
Finished reading file "fix2.in".
6-> read fix5
6-> 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)
Finished reading 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 1
(((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)
Finished reading 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
Finished reading 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
Finished reading file "fix9.in".
Finished reading file "test.in".
8-> clear all
1-> read fraction
1->
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
Finished reading 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.
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.
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
Finished reading 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.
1
#6: x_new = y^-
n
6-> y ; But we want an algorithm to compute it without using non-integer exponentiation.
#6: y = x_new^n
6-> taylor x_new 1 x_old ; build the (nth root of y) iterative approximation formula
Computing the Taylor series of the RHS and simplifying...
1 derivative applied.
#7: y = x_old^n + (n*(x_old^(n - 1))*x_new) - (n*x_old^n)
7-> solve verify x_new ; solve for the output variable, verifying the result
(y - x_old^n)
#7: x_new = (------------- + 1)*x_old
(x_old^n*n)
Solution verified.
7-> simplify ; convergent nth root approximation formula:
y
(------- - 1)
x_old^n
#7: x_new = x_old*(------------- + 1)
n
7-> copy
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".
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
Computing the Taylor series and simplifying...
10 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.
Finished reading file "demo.in".
12-> ; read limits
12-> ; read how_limit_works
12-> read test3
12-> ; Test solving linear equations with Mathomatic.
12->
12-> read linear
12->
12-> ; Combine 3 simultaneous linear equations with 3 unknowns (x, y, z).
12-> ; Solve for all 3 unknowns using the eliminate, solve, and simplify commands.
12->
12-> 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
Substituting the RHS of equation #1 into the current equation for variable (x)...
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-> z ; solve and find z
((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
Substituting the RHS of equation #3 into the current equation for variable (z)...
((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
Substituting the RHS of equation #3 into the current equation for variable (z)...
Substituting the RHS of equation #2 into the current equation for variable (y)...
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))))
Finished reading 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
((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
((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
((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
((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
((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
((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
((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
((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
((((b1*c3) - (c1*b3))*((a2*x) - d2)) - (((b2*c3) - (b3*c2))*((x*a1) - d1)))
#1: d3 = --------------------------------------------------------------------------- + (x*a3)
((c1*b2) - (b1*c2))
1-> a3
-((((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
((((c1*b2) - (b1*c2))*((a3*x) - d3)) + (((b1*c3) - (c1*b3))*((a2*x) - d2)))
#1: d1 = -(--------------------------------------------------------------------------- - (x*a1))
((b2*c3) - (b3*c2))
1-> a1
((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
((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...
Completely simplifying both equations...
((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))))
((d1*((b2*c3) - (b3*c2))) + (b1*((d3*c2) - (d2*c3))) + (c1*((d2*b3) - (d3*b2))))
#1: x = --------------------------------------------------------------------------------
((a1*((b2*c3) - (b3*c2))) + (b1*((a3*c2) - (a2*c3))) + (c1*((a2*b3) - (a3*b2))))
Equations are identical.
Finished reading 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
Substituting the RHS of equation #1 into the current equation for variable (a)...
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-> c ; solve and find (c)
((y2*(x1 - x3)) + (y1*(x3 - x2)) - (y3*(x1 - x2)))
#3: c = -------------------------------------------------------------
((x1*(x2^2 + (x1*(x3 - x2)))) - (x3*(x2^2 + (x3*(x1 - x2)))))
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
Substituting the RHS of equation #3 into the current equation for variable (c)...
(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)
Substituting the RHS of equation #3 into the current equation for variable (c)...
Substituting the RHS of equation #2 into the current equation for variable (b)...
(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))
Finished reading file "poly.in".
1-> clear all
1-> read examples
1->
1-> ; This is a line comment. This file shows some simple examples of Mathomatic usage.
1->
1-> ; Equations are entered by just typing 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".
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-> b ; Solve for variable "b".
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.
#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-> b ; Solve for variable "b".
Equation is a degree 2 polynomial equation in (b).
Equation was solved with the quadratic formula.
1
((((a^2 - 4)^-)*sign) + a)
2
#2: b = --------------------------
2
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...
(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
answer = 5
3-> 495/44 ; Fractions are always reduced to their simplest form:
answer = 11.25 = 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:
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 verify y ; Solve for y, verifying the result.
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 verify x ; Solve for x, plugging the results into the original equation to verify.
Equation is a degree 2 polynomial equation in (x).
Equation was solved with the quadratic formula.
(3 - (13*sign))
#7: x = ---------------
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 = -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
Finished reading file "examples.in".
7-> read test1
7-> clear all
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
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.
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*pi^2*r^3 1
4*((------------- - d^2)^-)*sign
(g*m*epsilon) 2
#1: y = 0.6666 - --------------------------------
(15*d)
Finished reading 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
Substituting the RHS of equation #2 into the current equation for variable (l)...
Substituting the RHS of equation #3 into the current equation for variable (m)...
Substituting the RHS of equation #4 into the current equation for variable (n)...
Substituting the RHS of equation #1 into the current equation for variable (y)...
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
((f*((64*b^3) - (4*b*a^2))) + (64*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
((f*((64*b^3) - (4*b*a^2))) + (64*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.
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
((f*((64*b^3) - (4*b*a^2))) + (64*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.
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-> simplify symbolic
o' 1 a^2
#7: f = a + (sign0*(--------^-)*(----- - (2*b)))
(1 + o') 2 (8*b)
Finished reading 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
Substituting the RHS of equation #4 into the current equation for variable (e4)...
Substituting the RHS of equation #3 into the current equation for variable (e3)...
Substituting the RHS of equation #2 into the current equation for variable (e2)...
Substituting the RHS of equation #1 into the current equation for variable (e1)...
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
Substituting the RHS of equation #4 into the current equation for variable (u4)...
Substituting the RHS of equation #3 into the current equation for variable (u3)...
Substituting the RHS of equation #2 into the current equation for variable (u2)...
Substituting the RHS of equation #1 into the current equation for variable (u1)...
#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.
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.
Finished reading 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)
Finished reading file "simplify.in".
6-> read heron
6-> clear all
1-> ; This Mathomatic script shows two 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-> pause Please press the Enter key to verify the result.
3-> compare 2 ; simplify and compare the result with Heron's formula:
Comparing #2 with #3...
Completely simplifying both equations...
1
(((2*((a^2*(b^2 + c^2)) + (b^2*c^2))) - a^4 - b^4 - c^4)^-)
2
#2: triangle_area = -----------------------------------------------------------
4
1
(((2*((a^2*(b^2 + c^2)) + (c^2*b^2))) - a^4 - b^4 - c^4)^-)
2
#3: trapezoid_area = -----------------------------------------------------------
4
Variable triangle_area in the first equation
is equal to trapezoid_area in the second equation.
3-> clear
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
#3: 2*s = a + b + c + d
3-> cyclic_area = ((s-a)*(s-b)*(s-c)*(s-d))^.5
1
#5: cyclic_area = ((s - a)*(s - b)*(s - c)*(s - d))^-
2
5-> eliminate s ; Brahmagupta's formula:
Solving equation #3 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
#5: cyclic_area = ((--------------- - a)*(--------------- - b)*(--------------- - c)*(--------------- - d))^-
2 2 2 2 2
5-> pause
5-> copy
(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
5-> 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
#5: cyclic_area = -----------------------------------------------------------------^-
2 2
5-> pause Please press the Enter key to verify the result.
5-> compare 2 ; simplify and compare the result with Heron's formula:
Comparing #2 with #5...
Completely simplifying both equations...
1
(((2*((a^2*(b^2 + c^2)) + (b^2*c^2))) - a^4 - b^4 - c^4)^-)
2
#2: triangle_area = -----------------------------------------------------------
4
1
(((2*((a^2*(b^2 + c^2)) + (c^2*b^2))) - a^4 - b^4 - c^4)^-)
2
#5: cyclic_area = -----------------------------------------------------------
4
Variable triangle_area in the first equation
is equal to cyclic_area in the second equation.
5-> clear
5-> clear 1 3
Finished reading file "heron.in".
5-> 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
Substituting the RHS of equation #1 into the current equation for variable (s)...
(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
1
(((2*((a^2*(b^2 + c^2)) + ((b*c)^2))) - a^4 - b^4 - c^4)^-)
2
#2: inradius = -----------------------------------------------------------
(2*(a + b + c))
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
Substituting the RHS of equation #1 into the current equation for variable (s)...
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
Finished reading file "radius.in".
3-> 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.
1
#2: distance3D = ((((x1 - x2)^2) + ((y1 - y2)^2) + ((z1 - z2)^2))^-)*sign0
2
Finished reading 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
-((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.
Substituting the RHS of equation #1 into the current equation for variable (y)...
-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
Finished reading 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-> y ; 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.
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) + (x1*x2))) + x1^2 + y1^2 + x2^2) 2
#2: y = ---------------------------------------------------------------------------------------------------
2
Finished reading 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.
Finished reading 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.
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))
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 + (x1*((2*x2) - (4*x))) - (2*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-> quit
ByeBye!! from Mathomatic.
distrib
>
Mandriva
>
2010.2
>
i586
>
media
>
contrib-backports
>
by-pkgid
>
b8ec8bfa2b3d0923a5da0c96615f305f
>
files
>
33
