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.

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