It is unworthy of excellent men to lose hours like slaves in the labor of calculation.
Gottfried Wilhelm Leibniz
I have often told students that becoming good at math is a lot like becoming good at a sport or at playing an instrument – practice is extremely important. Hours and hours of practice.
However, there is a lot that can be learned from having computer software do math for you, especially if the math is algebra and calculus. The kind of programs that can do this are called Computer Algebra Systems (CAS), and they can actually do much more than just algebra. The most famous CAS is Mathematica from Wolfram Research, and another one called Maple is widely used. The mathematical manipulations these systems can do are remarkable, and the term artificial intelligence seems appropriate to describe them.
Unfortunately, both Mathematica and Maple cost hundreds of dollars, and most students could not justify that much expense. There are a few CAS programs that are free, and for a year or so, I experimented with one called Maxima. However, when Wolfram Research introduced WolframAlpha (W|A) in 2009, it seemed clear it was much superior. As many people know, W|A is web based and free, and it can do much more than just mathematics. My focus here is just on its math capability. W|A gives us access to a substantial subset of Mathematica, and those with an appreciation for math will often be amazed by what it can do.
Of special interest to the student, the page showing the problem solution often has a link called Show Steps. Clicking that will bring up a step-by-step explanation of how the solution was obtained.
An oddity of W|A is that there appears to be no formal documentation – just a series of sample commands that actually don’t cover nearly all of its capabilities. I have found a good many of the “hidden” commands by just trying things, or by looking at the Mathematica documentation for hints about things to try. The table below summarizes my exploration so far, and is more comprehensive than the W|A web site.
I welcome corrections and additions, and may post a revised version of the table later.
| Algebra, Trigonometry |
|
Equations |
|
|
solve ax^3 + bx + c = 0 for x |
|
|
|
solve sin(x)/x = .5 |
|
|
|
solve x^3-3x^2 = x^6-2 |
|
|
Linear Algebra |
|
|
{{2,0,5},{1,3,-6},{3,4,7}} |
all properties of the 3×3 matrix |
|
|
rotate 60 degrees |
generate a rotation matrix |
|
|
{1/4, -1/2, 1} cross {1/3, 1, -2/3} |
Vector cross product |
|
|
{1/4,-1/2,1} dot {1/3,1,-2/3} |
Vector dot product |
|
|
linearProgramming ({1, 1}, {{1, 2}},
{{3,-1}}) |
minimize x+y, subject to the constraint x+2y<=3.
For >=, make the last number 1. For =, make it 0 |
|
|
eliminate ({a x + y = 0, 2 x + (1 – a) y = 1}, y) |
derive one equation with y eliminated |
|
|
Simultaneous linear equations: |
|
|
solve {{2,5},{1,3}},{x,y}={3,5} |
Matrix form |
|
|
or solve 2x+5y=3, x+3y=5 |
Normal form |
|
|
or
2x+5y=3, x+3y=5 |
without “solve”, it also plots the equations |
|
Miscellaneous |
|
|
partial fractions (x^2-4)/(x^4-x) |
|
|
|
expand (2x-3y)^12 |
|
|
|
factor 6x^3 – 65x^2 + 13x + 84 |
|
|
|
complete the square 3x^2-4x+13 |
|
|
|
x^2 – 2x + 4 – y^2 + 5y + 1 = 0 |
Analyze and plot the conic |
|
|
2.467401100272339654708622 |
recognizing numbers |
|
|
(53+60i)/29 |
computes rectangular & polar form plus gaussian prime factors |
|
|
((3-2i)(4+i)/(2-5i) +1+2i)e^(2i) |
simplify complex expression |
|
|
factor(4829490) |
|
| Graphing |
|
|
plot x^3-3x^2, x^4-2 |
|
|
|
logplot e^(3x) |
y axis is logarithmic |
|
|
line through (2,3) and (6,4) |
|
|
|
plot 3d (z=3x^2-4y^2) |
3 dimensional plot |
|
|
plot3D (sin(x + y^2), {x, -3, 3}, {y, -2, 2}) |
specify a range for x and y |
|
|
polar plot r=theta, theta=0 to 8 pi |
|
|
|
parametric plot (t-sin(t)),2(1-cos(t)) for t=0 to 4*pi |
|
| Financial |
In each case, entering the command results in a form for specifying the input variables |
|
|
compound interest |
|
|
|
amortization calculator |
|
|
|
future value |
|
|
|
annuity |
|
| Statistics and Probability |
|
|
{25, 35, 10, 17, 29, 14, 21, 31} |
Descriptive statistics of the array |
|
|
linear fit {{1.3, 2.2}, {2.1, 5.8}, {3.7, 10.2}, {4.2, 11.8}} |
Linear regression. There’s also quadratic fit, cubit fit, log fit, and exponential fit |
|
Distributions |
In each case, enter the distribution name – a form will then request the input variables |
|
|
binomialcdf, binomialpdf |
|
|
|
chisquarecdf, chisquarepdf |
|
|
|
normalcdf, normalpdf |
|
|
|
poissoncdf, poissonpdf |
|
|
Probability |
|
|
combination 10,2 |
|
|
|
permutation 10,2 |
|
|
|
2 heads in 10 tosses |
|
|
|
poker royal flush |
|
|
|
3 dice |
|
|
|
5 10 sided dice |
|
| Calculus |
|
Differential calculus |
|
|
derivative ( x^2 sin(x)^3+x^3sin(x)^2) |
d/dx can be used in place of “derivative” |
|
|
critical points x^4-5x^2+4x |
|
|
|
d/dy (3ax^2y^3 + xy*sin(y)^2) |
partial derivative |
|
|
maximize -x^3+4x^2 on [-1, 5] |
|
|
|
grad (3x^3y, xy^2, xyz) |
vector calculus (also div, curl operators) |
|
Integral calculus |
|
|
integral Log[x] |
antiderivitative |
|
|
integrate e^(-x^2) from 0 to +infinity |
definite integrals |
|
|
integrate e^(-x)sin(x) from 0 to pi |
|
|
|
integrate e^(-x) from 0 to infinity
|
improper integral |
|
Limits and Series |
|
|
limit sin[x]/x x->0 |
|
|
|
taylor series of 1/ln(x) |
|
|
|
sum n^4 from 1 to k |
Get the formula for the sum |
|
|
sum of [(ln(2)^n)+(1/(ln(4)^n))] from 0 to infinity |
|
|
|
6,15,30,51,78,… |
compute a possible formula for the series |
|
Differential equations |
|
|
solve y”(x)+x=0 |
|
|
|
solve y”(x) – y = 3x^2, y’(0)=1, y(0)=10 |
|
| Functions |
|
|
table [sin(x)/x^2,{x,1,pi,.2}] |
Table of function values from 1 to pi, in increments of 0.2 |
|
|
j1() from -10 to 10 |
Plot Bessel function of first kind |
|
|
zeta(.5+2i) |
Evaluate function at a point |
|
|
log (e^3) |
Natural logarithm |
|
|
log (12,144) |
Logarithm to any base |
|
|
laplaceTransform (e^(2x) sin(x)) |
|
|
|
inverseLaplace (1/(s^2+5s-16)) |
|
|
|
fourierTransform (e^(2x) sin(x)) |
|
|
|
19! |
factorial |
| Physics |
|
|
.007 joules in ergs |
Unit conversions |
|
|
projectile 23 degrees 42 m/sec |
|
|
|
kinetic energy 47 kg 125 miles per hour |
|
This entry was posted
on Friday, January 8th, 2010 at 2:27 am and is filed under Computers and Mathematics, Learning Math, Thrilling Math, Trig/PreCalculus.
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I'm Larry Phillips, a former engineer, programmer, math teacher, math /physics tutor, and currently owner of a tutoring company. I'm on a mission to show that math is more interesting than the schools made you think it was.
January 11th, 2010 at 1:48 pm
Dear Larry,
I did want to mention that the price of Mathematica for Students is $139.95 and many students may access it for much less or even for free, if their school has a site license agreement.
I’m glad you mentioned the documentation. We have a Wolfram|Alpha community site where users exchange ideas and suggestions, and we’d be thrilled to have you join.
http://community.wolframalpha.com/
Kindest regards,
Carol
March 3rd, 2010 at 10:14 am
Wow. What a great reference. Thanks for writing this up! Am also enjoying your articles on Euler and primes.
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