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Division Explained in 5 Minutes

Division Explained in 5 Minutes

Mathematicians call a number even if it's divisible by 2. But wait, what does divisible mean? What's division?

Let's suppose you have 10 apples. You can break your group of 10 apples in all types of ways. For example, you could break it up into three chunks, one containing 2 apples, one with 3 apples and one with 5 apples. Of course there are lots of other ways you can break the group up. Mathematician call a set of chunks a "partition". Of course, the more apples in your collection the more partitions you could make.

10=2+3+5 or 10={2,3,5}
One of the partitions of 10. There are more.

But now suppose I told you that I was only interested in breaking the group of apples into chunks that are all the same size. Now there are far fewer ways you can do it. For example, you could break it into two chunks, each with 5 apples. Or you could break it into 5 chunks each 2 with apples.

10=5+5 or 10={5,5}
10=2++2+2+2+2 or 10={2,2,2,2,2}
Two partitions of 10 that have equal size "chunks".

When you break a group of n objects into chunks that are all the same size it's called division. We say that the original number can be divided by the number of chunks. It can also be divided by the size of each chunk. So in the above example 10 is divisible by 2 and 5.

Get it?

10 can be broken into 2 chunks that are all the same size, so it's divisible by 2.
10 can also be broken into 5 chunks that are all the same size, so it's divisible by 5.

Oh, there's also another way to define division. It's abstract and not as visual as the above definition. Take an integer n. If you can find two other integers, say a and b, such that n=a*b then we say that n is divisible by a. We also say n is divisible by b. Note that by definition divisors come in pairs! Note also that this defines division in terms of multiplication. So, division is not really a new operation, we still have only two operations: addition and multiplication.

But let's go back to the nice visual definition of division. Can every number be broken up into chunks of equal size? Sounds like the answer is an easy yes, but the answer is no! There are some numbers that cannot be broken into chunks of equal size. These numbers have created endless problems for mathematicians and they still do! They are called prime numbers.

Content written and posted by Ken Abbott