Sunday, May 22, 2016

Long Division: Why do we care?

Long division is a contentious subject in math education.  Students struggle with it, and most people fail to see its relevance in an age when you can just ask Siri.  I wanted to give some reasons why long division is worth teaching and mastering.  What you'll hopefully see is that it is absolutely crucial for us to have a general procedure which takes a division problem and gives us a quotient and remainder.   You might have seen this written as \(19÷3 = 6 \; R \;1\), but remember that \(19÷3\) is the same as \( \frac{19}{3}\) and \(6 \;R\; 1\) really means \(6 + \frac{1}{3}\) in this situation.  Another way to write it is \(19 = 3*6 + 1\), where we multiplied both sides of the equation by 3 to get rid of the fractions.

If I manage to convince you, and you're wondering how to teach it, take a look at this blog post. Hopefully, you can use what I wrote here to help motivate everyone involved in the learning process.


Algorithms

Long division is the first big example of an algorithm. An algorithm is a step-by-step description of how to do something.  You can conveniently reuse algorithms again and again in similar situations, share them with others, and teach them to a computer. Once we have an algorithm for how to complete a particular type of task, we no longer have to think about it so much.  Even if we don't always use it, it's there in our back pocket, and gives us confidence that we know how to solve any problem that comes our way.  Algorithms are fundamental to the closely related fields of math and computer science, and when kids learn some coding in elementary school (here are some resources for that), they are more prepared for understanding long division as an algorithm.


Decimal Expansions

Long division connects fractions, division, and decimals. We can use long division to write 5/7 as a decimal... we just have to keep going until it terminates or repeats.  In other words, it gives us an algorithm for finding the decimal representations of fractions.  Moreover, long division explains why all rational numbers have decimal representations that terminate or repeat.   Becoming comfortable with rational numbers and all of their forms is one of the most challenging and important tasks of middle school mathematics.


Other Bases

We write numbers in base 10.  So 1068 means \(1*10x^3 + 0*10^2 + 6*10 + 8\).  But we could also use powers of another number, like 7.  How would we write 1068 in base 7? Well, we use the division algorithm to find that \(\frac{1068}{7} = 152 + \frac{4}{7}\), or equivalently that \(1068 = 7*152 + 4\).  Next, we divide the quotient 152 by 7 to get a new quotient 21 and a new remainder 5, yielding \(152 = 7*21 + 5\).  Continuing in this way, we find that \(21 = 7*3 + 0\) and \(3 = 7*0 + 3\).  So 
$$1068 = 7*152 + 4 = 7(7*21 + 5) + 4 = 7(7(7*3) + 5) + 4 = 3*7^3 + 5*7 + 4.$$
This tells us that \((1068)_{10} = (3054)_7\), where the subscripts tell us the base the number is written in.  


Polynomial Division

The algorithm of long division can be applied to polynomials as well as positive integers.  Just as long division allows us to rewrite 17/5 as 3 + 2/5, with 3 as the quotient and 2 as the remainder, it allows us to rewrite $$\frac{x^4 - 5x^2 + 6x^2 - 18}{x^3 - 3x^2} \; \; \text{as } \; \; (x - 2) - \frac{18}{x^3 - 3x^2},$$ with \(x - 2\) as the quotient and -18 as the remainder.  This becomes useful in calculus, as its the first step to integrating a rational function where the degree of the numerator is at least as big as the degree of the denominator.


Modular Arithmetic

Modular arithmetic is born out of the idea that we can create a new number system by only caring out the remainder when we divide by n.  So, if we're looking at remainders after dividing by 7, for example, 2*4 = 1, which means that 4 = 1/2.  Crazy, right? Modular arithmetic is fundamental to many areas of mathematics and computer science, including cryptography! And it's a great topic to explore in math circles for upper elementary and above, as it's interesting, useful, and reinforces many of the main concepts of elementary and middle school mathematics.  


Continued Fractions

Expressing fractions in terms of quotient and remainder allow one to make continued fractions: $$\frac{7}{10} = \frac{1}{\frac{10}{7}} = \frac{1}{1 + \frac{3}{7}} = \frac{1}{1 + \frac{1}{\frac{7}{3}}} = \frac{1}{1 + \frac{1}{2 + \frac{1}{3}}}.$$  You can even write irrational numbers as (infinite) continued fractions.  Take, for example, the golden ratio \(\phi = \frac{1 + \sqrt{5}}{2}\).  $$\phi = \frac{1}{1 + \phi} = \frac{1}{1 + \frac{1}{1 + \phi}} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \phi}}} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}}.$$  The reason that \(\phi\) has such a nice continued fraction representation is that \(1 + \frac{1}{\phi} = \phi\).  Can you figure out why that's true? This is another great math circle topic idea, and is probably most appropriate for high school students.


Generalizations of the Integers

Long division is fundamental to number theory, the area of math that I work in. We've already seen a little of this in modular arithmetic and continued fractions, but it goes even further.  One of the main things we study in number theory is generalizations of the integers and rational numbers.  In order to do this, we must look closely at the properties that the integers and rational numbers satisfy.  One important property is unique prime factorization, and another is the existence of a division algorithm that produces a quotient and remainder.  This is what is meant when the first problem set of PROMYS and ROSS state that "long division has unexpectedly deep consequences and can be generalized in far-reaching ways".


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