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.
September 23rd, 2010 at 7:02 am
Thank you for providing this great blog. Check out mine!
October 12th, 2010 at 3:41 pm
Hi this blog entry was very interesting and funny for me. But it was really hard to find it with ask. Maybe you should improve it with seo plugins for wordpress like headspace. Just a tip
October 15th, 2010 at 3:05 pm
[...] 6. Calculator. where can I start? Solving the travelling salesman problem “FindShortestTour[{{0, 0}, {5, 8}, {10, 36}, {12, 75}, {4, 33},{3,3},{7,7}}]” or “Minimize[{x^2 + y^2 + z^2, x >= 2 && y >= 2 && z >= 2}, {x, y, z}]” and see Maths Examples [...]
October 11th, 2011 at 1:08 pm
You must take part in a contest for top-of-the-line blogs on the web. I will recommend this web site!
March 30th, 2012 at 3:12 pm
Thanks a lot for this great post. I am a beginner with W|A and had problems with the short list of commands given. Your post is a very useful compendum of formulas. This post sould be featured on the offical website, it would definitely provide better information than what is already on the website…
May 17th, 2012 at 6:27 pm
Math and I were never really friends when I was in school. Its a good thing we have applications and software that makes life easy for us when it comes to math.
June 30th, 2012 at 5:48 am
I liked maths in my early schooling but when I was in High School maths was my biggest enemy. I never got good grades in maths. Now again maths is attracting me as I found it can be more interesting and i can do fun with maths. Thanks for writing this article.
September 3rd, 2012 at 3:42 pm
Yes i agree , practice is extremely important,atleat this statement should be known by every math’s student.Macsyma is the same as open-source Maxima system.TI-Nspire is good CAS,but its not free.Java Algebra System,Mathomatic are free CAS.I used Casio Algebra FX 2.0 is good Hand-held calculator CAS.This post is very informative.
September 13th, 2012 at 10:37 pm
I was recommended this blog by my cousin. I’m not sure whether this article is written by him as nobody else know such detailed about my difficulty. You are remarkable! Many thanks!
September 18th, 2012 at 10:14 pm
I appreciate you work.I like math but always below average in Graphing. Show Steps, you have given are very useful to understand the things. I also handy with Computer Algebra Systems (CAS), its an intersting tool to work.
Thank you for sharing the post.
September 19th, 2012 at 5:03 am
The toughest part in high school math is calculus. I hope most of you agree with me. With a software to solve it, I think I can practice more calculus. Thanks for giving us the shortcuts also. I also had problems with graphing data, which has also become easy now. Once again thanks for this blog.
October 5th, 2012 at 8:59 am
Thanks for this wonderful article. I always find statistics very interesting, but sometimes the calculations are too tedious. Iam happy to find there is a software to calculate the required values. Though I have always used excel,SPSS,Minitab and the like, each and every software has its own limitations. I hope this software helps me in the required areas. As this software also solves calculus, it will be very helpful for me.
October 20th, 2012 at 5:33 am
@Anna – Absolutely, this software really helps to calculate complex requirements.
I am not very good in this area, but with the help of this software, now I can say that I am genius to this area.
Thanks for sharing such a good post.
October 21st, 2012 at 12:14 pm
I think maths is all about practicing, more you practice maths more you will be proficient in maths. These types of software will definitely help you to check the results and these are very efficient and fast but if you are learning maths then you should practice it manual not by any software.
November 8th, 2012 at 6:54 am
Great job! when I was student I never prctice math so, I never get good marks on Math. Thanks for sharing such a nice information about the software to help those like me never love to do math. This software help me to solve calculus problem.
November 26th, 2012 at 11:26 pm
Remarkable post! Math is very interesting as well as toughest subject. Who finds math very difficult, your software will give them incredible assistance . But only with the practice one can overcome the fear and score good in math. Thanks for sharing such a nice post!