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Reliance on Technology in Mathematics

I read a very interesting article today from the Notices of the American Mathematical Society (AMS) about the intelligent use of technology in mathematics.  This article, titled “The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?” describes the experiences of these three researchers in using the mathematical software packages Mathematica and Maple.  The details of their research are interesting but the important point for students of mathematics is to be aware of the limitations of the technology you use.  A key quote from the article is:

…even more dramatically, his algorithm yielded different outputs given the same inputs.

A more detailed explanation of what was going wrong:

…given the same matrix, the determinant function can give different values!

The authors do give credit to technology as a groundbreaking aid in modern mathematical research, but as is true in other research disciplines, they recommend using multiple sources.  In this case, checking the results of one mathematical software package against another software package to compare the results:

Having made this criticism, let us stress that software systems have proved very useful to research mathematicians.  Some well-known instances are the proof of the four-color problem by Kenneth Appel and Wolfgang Haken and the Kepler conjecture by Thomas Hales….Software bugs should not prevent us from continuing this mutually beneficial relationship in the future.  However, for the time being, when dealing with a problem whose answer cannot be easily verified without a computer, it is highly advisable to perform the computations with at least two computer algebra systems.

And, for students of mathematics, I would add,

– When dealing with a problem whose answer CAN be easily verified without a computer, do so!

Synthetic Division

We just finished talking about synthetic division in the College Algebra course today and got into a discussion of how to represent the remainder.  For example, given the problem:

$\frac{x^4-2x^3-x+10}{x-2}$

The answer turns out to be: $x^3-1 R: 8$ or you can say that the answer is: $x^3-1+\frac{8}{x-2}$.  This all goes back to the division algorithm which says that given two numbers $a$ and $b$, then solving the problem $\frac{a}{b}$ means finding $q$, the quotient and $r$, the remainder such that $a=b*q+r$ (with $r) .

If we take the expression $a=b*q+r$ and divide on both sides by $b$, then we’ll have $\frac{a}{b}=\frac{b*q}{b}+\frac{r}{b}$ or $\frac{a}{b}=q+\frac{r}{b}$.

Which of these forms we prefer depends on whether we want to say that:

$x^4-2x^3-x+10=(x-2)(x^3-1)+8$

or

$\frac{x^4-2x^3-x+10}{x-2}=x^3-1+\frac{8}{x-2}$