## The Wonders of WolframAlpha

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 Normal form 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 chisquarecdf normalcdf poissoncdf 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 Unit conversions projectile 23 degrees 42 m/sec kinetic energy 47 kg 125 miles per hour

### 18 Responses to “The Wonders of WolframAlpha”

1. Carol Cronin Says:

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

2. Stephen Hutter Says:

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5. Using Wolfram Alpha | Memoirs of a Super Scripter Says:

[...] 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 [...]

6. Ferdinand Reuer Says:

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7. Lampe Says:

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…

8. Annuity Rates UK Says:

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.

9. Brianna Jones Says:

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.

10. Taylor Maya Says:

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.

11. Vicenta Croutch Says:

12. Emma kart Says:

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.

13. nathaniel Méthode Says:

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.

14. Luminaire Anna Says:

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.

15. Brown Says:

@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.

16. Ariana normes Says:

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.

17. Joshua Parker Says:

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.

18. Alexis Hall Says:

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!