Why can numbers divisible by 2 be separated in halves?

Answer 1

You asked a very good question.

If the number is divisible by 2, that means that we can continue to subtract 2 from the original number until we reach 0. That is what division means, in other words, that is by the definition of division. Put another way, if we reversed that process, we would start at 0 and continue to add 2's until we got to the orginal number (that is what multiply means). In other words, when we say a number is divisible by 2, it means the number has the same value as a certain number of 2's added together.

Now we can take each of these 2's and split them into two parts, each part containing just a 1 because 2=1+1. We put one of those 1's in pile A and another in pile B. Notice that the total in all these piles is still the original number. Since we always put the same number in each of the two piles during the process, they must each contain the same sum. (This is sometimes expressed by saying that there is a one to one correspondence between elements in the two piles.) Those two piles are the "halves" that you have separated the original number into. They are considered "halves" because there are two of them and they have the same value.

The process above shows how you can take a number divisible by 2 (sometimes called an even number) and separate it into halves. Since the procedure can be applied to any number divisible by 2, that shows why it can (always) be done.