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Long Division

February 28, 2011

It was with great regret that last year I learnt that long division was no longer an oft-taught technique in Western Australian schools. Why didn’t someone tell me! I found myself teaching polynomial rings in a second year undergraduate course, but had to back-peddle a tad to show the students how it was possible to find the remainder when you divide 7 out of 10334 (the answer is 2, by the way).

Technology might well have made this calculation redundant, but it doesn’t take long to think up some good reasons why long division should go back into the curriculum. First of all, it’s an algorithm. The notion of an algorithm is a very important one in mathematics, and indeed, science. It is difficult to come up with examples of genuine sophisticated algorithms for secondary school students, beyond the wholly sequential.

Secondly, what makes long division work is the “Division Theorem”, and it is an interesting question as to what things a Division Theorem applies. The Division Theorem doesn’t work for the real numbers, but it does for integers (including the negative ones!). (We refer the interested reader to the theory of Euclidean domains).

Thirdly, we need some idea of the division algorithm to get a grip on what a real number is. Think about it, what is a real number? The idea that an infinite decimal expansion can define something uniquely, hinges on the uniqueness of the remainder in the Division theorem. Moreover, any decimal expression which repeats represents a rational number. After this is understood a student can proceed to understand what an irrational number is.

Fourthly, division of polynomials is important in tertiary level mathematics. For example, I saw just today an exercise given in class where it was asked to find the eigenvalues of the triangle (3-cycle) graph. The characteristic polynomial of the adjacency matrix of this graph is X^3-3X-2 and we can see that 2 is a root of this polynomial. So we  use long division to find the quadratic factor, which turns out to be (X+1)^2.

We conclude with a link to an article on the role of long division in school.

Can anyone think of another reason why long division is a necessary skill?

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One Comment leave one →
  1. May 25, 2011 7:43 am

    The dreaded X^3-3X-2 reared its head again today; this time, in a first-year tutorial. None of the students could find its roots. Some students were surprised to learn that once I know that X+1 divides this polynomial, then the quotient must be a quadratic! For the record people, X^3-3X-2 = (X+1)^2(X-2).

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