A number not always represent the same thing, therefore 1 not always equal 1.

If you have two groups, and each has half an apple, and you send half an apple to one group, you are left with an empty group (in this case group A). How is this dividing? " It"s not. The division told you how much to send from group a to group b. When you take the half apple from group a, you have subtracted half an apple from half an apple and end up with zero. " If you do the same thing with whole numbers, it wouldn't work to gain the result desired. Example: Both groups A and B has 1 apple. I send one apple to group B, group B has now 2 apples. " It does work actually. You have subtracted 1 apple from group a leaving it with no apples. 1 " 1 = 0. And you have added 1 apple to group b which has an apple already. 1 + 1 = 2. " When you divide, you don't start with groups, you have one unit, and you divide that unit. After you divide, you have more than one unit since you split that unit. " What you start with depends on how the problem is framed. You didn"t split the unit, you merely ended with a new unit to describe the result. Ex: apple / apple = "apples per apple" In this case, the two units have given you a rate. " Let's say I have a bar of chocolate, at this point I have one unit. If I brake this unit in half, now I have 2 half pieces. This can be shown as 1/ 1/2 or 1 / 2 (one divided by two). Both times in the left (represented by the number 1), it will represent the bar of chocolate I had before splitting into 2 pieces. The first equation, the number in the right, represents which way I cut it, which was in half giving me the result of 2 pieces I acquire from that 1 unit. Second equation, the number on the right represents what would happen if I divide this into two groups evenly. Each group would both get half piece. Or you could have the answer to; after dividing the chocolate in two equal parts (groups if you prefer), what size is each part? " This is an excellent example that demonstrates the equality of the objects based on the unit. One, one half of something is equal to one half of that same something. Two halves are equal to one whole. The amounts are the same, but since you changed the unit, you had to change the quantity to represent the amount. "Anyway, both times, the number in the left is the only number that represent the unit, the number in the right simply tells you what you are doing with that unit. Both numbers never represent the same role. " The units have modified the number representing the object. Ex: 2*Pie Radians equal 360 degrees. The numbers are different, as are the units. The amount is the same though, hence the equality. " This works great if I have a whole number, but if I have fraction, I cannot do the same, unless the other function on the right is not a fraction. Note* The result 2, even know are chocolates, should be seen as parts because, if you see as chocolates it results in 1=2. I don't know, maybe that is how my opponent arrived on 1 apple from 1/2. " Again, the units describe this. " Try to follow what I just wrote, but this time use half of a bar of chocolate, and see what happens. If you feel what I say is wrong, please tell me which equation I should have used instead, and use both types of equation to solve both problems, 1 bar of chocolate, and a 1/2 of a bar. If you feel I was right, follow my example using half of bar instead. " Does 1 = 2? No, however: 1 bar of chocolate = 2 half bars of chocolate. Hopefully, I have fully explained how units are the bridge between math and the real world. 1 = 1. Always. Forever. Thank you for a wonderful debate!