What is division?

What is division? Besides being one of the 4 fundamental arithmetic operations?

As we try to define it we'll use this sample equation:

X ÷ Y = Z r

These are the component parts of a division problem:

X is the dividend - the amount being divided up
Y is the divisor - the unit doing the dividing
Z is the quotient - the result of having done this division
r is the remainder if the problem does not come out evenly
Y and Z are factors of X if the division problem comes out evenly (without a remainder)
X is evenly divisible if the division problem comes out without a remainder

Here are some sample problems, before we go on with more definitions:

  9 ÷ 3 = 3 or nine divided by three equals three; three and three are factors of nine
10 ÷ 3 = 3 r1  or ten divided by three equals three with a remainder of one
10 ÷ 2 = 5 or ten divided by two equals five; two and five are factors of ten

We can call division repeated subtraction, so Z is the number of times can I remove Y from X before I get to zero.

1. 10 - 2 = 8
2.   8 - 2 = 6
3.   6 - 2 = 4
4.   4 - 2 = 2
5.   2 - 2 = 0  ZERO

The subtraction was repeated 5 times, so Z =5.

We can say division is determining how many times one quantity goes into another number, so Z is the number of times that we can put Y into X before X is "full".

1.   0 + 2 = 2
2.   2 + 2 = 4
3.   4 + 2 = 6
4.   6 + 2 = 8
5.   8 + 2 = 10 FULL

The filling was repeated 5 times, so Z =5.

We can call division sharing where Z is the number of people who can share X evenly. Each share contains Y pieces. 

In this case 5 people can share 10 evenly if each share is 2 pieces.

Here's a division problem that might amuse you - artwork was adapted from the Arif & Ali blog.

There are lots of other ways to describe division, but this is enough for now.