Division

Background

We've talked about how we can use multiplication to do many additions at the same time. Is there something that works the same way for subtraction? Well... kind of...

Question

What's 6 ÷ 3? 

Answer

2

Analysis

There are many many ways to think about division. It's used widely throughout mathematics and can be used in a variety of ways (which we'll explore in further entries). One way to view division, and the way I want to present it here, is to think of it as a way to take a starting number of things and divide those things into a number of groups.

Let's look at the question we have. We have 6 things - let's say we have cookies. Now, if it was just you, all alone, with those 6 cookies, they'd all be yours. But what if you have two friends over, so you want to divide the cookies evenly with your friends. How many cookies do you and your friends get?

Division is the opposite, or inverse, of multiplication, so one way we can find 6 ÷ 3 is to remember that 3 x 2 = 6, and so when we go the other way, 6 ÷ 3 = 2. Or in other words, you and your friends get 2 cookies each. That's 3 groups of 2 cookies each, or 6 cookies in total.

We can do the same type of calculation if you have one friend over and so there are 2 of you in total. That's 6 cookies being divided into 2 groups, and so each of you gets 3 cookies each. So 6 ÷ 2 = 3.

Let's do one more: 6 ÷ 4. What happens?

We can give each person 1 cookie. That's 4 cookies handed out, which means there's 2 left over - let's call that the remainder.

As we make our way through more entries, we'll find different things we can do with that remainder. For right now, let's put the extra 2 cookies back in the cookie jar.

One thing we can't do that all the other operators have been able to do is to divide by 0. We can add by 0 (as much as we want, with 0 being the additive identity), we can subtract by 0 (again, as much as you like), we can multiply by 0 (and end up with 0), but we can't divide by 0. Let's talk about why.

Let's talk about 6 ÷ 0 and what that means. We have 6 cookies and we're going to divide it into... 0 categories. We don't eat them (that's 1 group), we don't brush them off the table (also just 1 group), or throw them in the trash (again, 1 group). We can smash the cookies into a billion little pieces and flick the crumbs everywhere (that's 6 cookies divided into a billion little crumb groups). While we can divide things like cookies into groups, we can't divide into 0 groups. And so when faced with 6 ÷ 0, we say the result Does Not Exist.

Of course, we need special division terminology for the different numbers! The first number is the dividend and the number we are dividing by is the divisor. The result of the dividend being divided by the divisor is the quotient. And so in 6 ÷ 3 = 2, 6 is the dividend, 3 is the divisor, and 2 is the quotient.

Division can be shown in the following ways:

6 ÷ 3
6 / 3


Vocabulary used:

For more information check out these links (comment to add your favourite link):

http://studyjams.scholastic.com/studyjams/jams/math/multiplication-division/relate-mult-div.htm

Where might you have come from?

Fact-orials Index
Numbers:

• The Number 0
• Negative Numbers

Operations:

• Multiplication

Where might we go?

Numbers:

• Fractions and Decimals
• Expressing Values, and Bases

Operations:

• Order of Operations (PEMDAS)

Operations with different kinds of numbers:

• Multiplication, Division, and Whole Numbers
• Multiplication, Division, and Integers
• Multiplication, Division, and Rational Numbers
• Addition, Subtraction, and Variables

Associated Operations:

• Dividing Evenly
• Remainders and Modulo
• Long Division
• Factors and Factoring

Properties:

• Commutative Property
• Associative Property

Relations:

• Inverse Relations