Division
Division is just repeatedly subtracting until you have zero left, and seeing how many times subtracting it took.
Eight divided by two. Eight minus two minus two minus two minus two - zero. So, you can make four groups of two with eight objects.
For nine, the story becomes slightly different. Nine minus two minus two minus two minus two... equals one.
So, we must now see how many times a tenth of two fits in what we have left. One minus 0.2 minus 0.2... we can do this five times, and as such this means that Two fits in four point five times. The answer is 4.5.
Division by two and five will always result in a clear decimal result because each cycle through the digits takes 10 in base 10, and 10 divides evenly by two and five. You can play around with this more to verify.
But, what about the others...
The Rational Numbers
Any number that you can write in the form of division is considered a rational number. Division, or, writing things as fractions, are a way to represent all the infinitely continuing numbers such as 10/3.
You will notice that 10 minus three minus three minus three equals one. Then, one minus 0.3 minus... equals 0.1. Then, 0.1 minus 0.03 minus 0.03... this is a rational number, but not an integer.
You can write it also as 3.33333... or just 3.(3), the parenthesis symbolizing that the three never really, as long as you keep looking, stops.
This is the problem we discussed in an earlier page about dividing evenly. You can keep dividing a one into three and getting something left. There is always a remainder.
This is, here, however, an artifact of our base. Watch this cool trick:
In base 60, '10' would probably mean "60". So, in base 60, 10 divided by three is 20. Or, if we use letters to represent the new numbers in base 60, onsidering that J is the 10th letter of the alphabet, "10 divided by three is J". Here, you might see the problem of not having letters, of course, and we will look more at that little issue (which actually makes base 60 seem like hell) when we get to Babylonian Math.