7.1: Division

Division can be represented with may symbols. In this section, instead of \(p \div q\) or \(p / q\), we will use \(\frac{p}{q}\). 

Definitions: Quotient, Dividend, Divisor, and Remainder

If \(p\) is any real number and \(q\) is any nonzero real number, then \(\frac{p}{q}\) is the quotient of \(p\) and \(q\). In this division equation, \(p\) is called the dividend, and \(q\) is called the divisor. If \(p\) and \(q\) are positive integers (still with \(p \neq 0\), then there exists nonnegative integers \(d\) and \(r\) such that \(0\leq r < q\) and \(\frac{p}{q} = d + \frac{r}{q}\), or equivalently, \(p = dq + r\). The nonnegative integer \(r\) is called the remainder.

As there were multiple perspectives of multiplication, there are multiple perspectives of division. Many children first learn learn division in terms of sharing, but the method most children in the U.S. use for performing division is based on subtraction. Let's consider that perspective first.

Repeated Subtraction

Definition: Division Repeated Subtraction

If \(p\), \(q\), and \(d\) are positive integers and \(r\) is a nonnegative integer less than \(q\) such that \(\frac{p}{q} = d + \frac{r}{q}\), then \(d\) is the number of times \(q\) can be subtracted from \(p\) without going below 0. This is the repeated subtraction perspective of division.

At some point, you have seen long division. Long division is based on the repeated subtraction perspective. Consider the following example.

Example \(\PageIndex{1}\)

Dee has 42 dice. She puts them into piles of 6 dice each. How many piles does she end up with?

Solution

We will solve this two ways. First, let's imagine we are directly modeling Dee's actions. Consider actually counting out 42 dice (or blocks or marbles or any other object) and then forming the piles. The first pile uses up 6 dice, leaving 36 left to make piles from. The next pile uses up 6 more dice, leaving 30 dice left. The process continues until 6 piles of 6 dice each have been created, leaving exactly 6 left, which forms the last pile. There would be 7 piles, which is the answer to the question.

The imagined procedure above involves actually subtracting 6 from the larger values repeatedly. As a second method, let's do this abstractly by simply subtracting 6 repeatedly. Below is how a young learner might do this.

The subtrahends (all sixes) are circled as the learner counts the number of times they could subtract 6. Each of those circled sixes represents a pile of 6 dice. There are 7 circled sixes, so there are 7 piles!

In the previous example, the one performing the subtraction must subtract 6 seven times. This is tedious, but not impossible. However, if we already have a good grasp of multiplication, we can do this faster. Simply knowing that \(6\times 7 = 42\) means that we could subtract all 7 sixes at once. That is precisely what long division does!

Thinking of the division \(\frac{p}{q} = d + \frac{r}{q}\) using the same letters as at the beginning of this section as repeated subtraction is equivalent to thinking that \(d\) is the number of measures of \(q\) that are in \(p\). That is why this is also called the measurement perspective of division or just measurement division. Some people also refer to this as quotitive division. Many children will abstractly model this method on a number line, which emphasizes the "measurement" concept. Try the example again by starting at 42 on a number line and making leaps of 6 units to the left until you reach 0. How many leaps did you make? It should be 7 if you did it correctly!

Before moving on, think about this: could this perspective of division be applied to non-integers? What about negative numbers? These are important questions to ask.

Sharing

Example \(\PageIndex{2}\)

Charlie has 15 balloons that he will give to three children to be shared equally. How many balloons does each child receive?

Solution

A young learner modeling this directly would probably use 15 blocks (or any other object) and "deal" them out into 3 piles until he runs out. When finished, they would count how much is in each pile (they may count each pile to make sure they are equal). The result would be that there are 5 blocks in each pile. Thus, each child receives five balloons.

Notice that we already started knowing how many piles there would be in the example above. This is the opposite of the previous example. In the example with the dice, we knew how many dice would be in each pile but did not know how many piles there would be. This leads us to another perspective of division.

Definition: Division as Sharing Equally

Suppose \(p\), \(q\), and \(d\) are positive integers and \(r\) is a nonnegative integer less than \(q\) such that \(\frac{p}{q} = d + \frac{r}{q}\). This means that if \(p\) is split evenly into \(q\) groups (with \(r\) left over), there will be \(d\) in each group. This is the sharing equally perspective of division.

This is also sometimes called partitive division. Also, sometimes the word "distribute" is used instead of "share". This is commonly how children first get exposed to division because it pairs nicely with lessons on sharing. In this way, children see a mathematical operation linked to a social activity! One might say that division is a friendly operation. Unfortunately, sometimes the fact that this division is mathematically equivalent to the quotitive division discussed previously is not always made clear. These are two very different actions when modeled directly. Thus, to most children, they feel like completely different operations. Children must be exposed to both viewpoints early on in order to make the connections between them.

Factors

After children have learned about quotitive and partitive division, but before they learn long division, it is important that they connect division to multiplication. 

Definition: Division as Finding a Factor

Suppose \(p\), \(q\), and \(d\) are positive integers and \(r\) is a nonnegative integer less than \(q\) such that \(\frac{p}{q} = d + \frac{r}{q}\). This is equivalently written as \(p = dq+r\). Because \(d\) is the unknown factor in the product \(dq\), this is the unknown or missing factor perspective of division.

In addition to this perspective being foundational to using long division instead of the long-hand repeated subtraction shown in the first example, thinking of division in terms of multiplication makes it much clearer as to how division can work with non-integer and negative numbers. It also allows us to succinctly model what happens if we try to let \(q\) equal 0. The following example is more advanced than what children first learning division would learn, but it is important to see where this perspective leads to later in their education.

Example \(\PageIndex{1}\)

A. Use multiplication to divide \(10\) by \(\frac{1}{2}\).

B. Use multiplication to show that it is impossible to divide \(10\) by \(0\).

Solution

A. We know that the quotient of \(10\) and \(\frac{1}{2}\) is the value of \(d\) such that \(d\cdot \frac{1}{2} = 10\). If we were to use a guess-and-check strategy, we might start with \(2\cdot \frac{1}{2} = 1\). We either know or would quickly find out that we need to only bother trying even numbers, so the next try might be 4: \(4\cdot \frac{1}{2} = 2\). Seeing that we need something bigger, we might try 8: \(8\cdot \frac{1}{2} = 4\). This is still too small and our product is not increasing fast enough, so we might make a bigger leap, like 30: \(30\cdot \frac{1}{2} = 15\). This is too big, so we try something smaller, such as 20: \(20\cdot \frac{1}{2} = 10\). This is exactly what we want! So, since \(20\cdot \frac{1}{2} = 10\), we know that \(\frac{10}{1/2} = 20\).

Of course, many people do not need to guess-and-check. They might notice that \(d\cdot \frac{1}{2} = 10\) is the same as \(\frac{d}{2} = 10\), which means \(d = 20\). In this way, we can see that sometimes thinking about division as multiplication leads to an easier division problem!

B. We will show that it is impossible to divide \(10\) by \(0\) by pretending it is and then showing why this does not work. Let's suppose there is some real number, say \(x\) that is the quotient of 10 and 0: \(\frac{10}{0} = x\). If we think of this as multiplication, we would have that \(x\cdot 0 = 10\). But we know that \(x \cdot 0 = 0\). So, this only way this could be true is if \(0 = 10\). Since that is not true, our original supposition was incorrect. There is no quotient of 10 and 0.

Practice Problems

1. Sophie needs to pass out papers to her coworkers. She has 28 papers and each coworker needs four papers. How many coworkers does she have? What type of division is this?
2. Later that day, Sophie speaks to ten other coworkers who need different papers. She runs off 30 copies. How many papers did each coworker receive? What type of division is this?