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

I will start with the definition of what a number is: A number is a mathematical object used to count, label, and measure. This is from Wikipedia, but I am sure pro will agree with this definition of number. Pro has made a good point in showing that an apple, a child and anything else a number represents is not the same thing. This would be correct. However, a key distinction is being missed. If pro picks up an apple from his kitchen table, how many "1"s" is he holding? He is not holding any "1"s," he is holding 1 apple. The "1" is only representing the quantity of apples he is holding in his hand. If he picks up a knife with his other hand, and then grabs a napkin with the same hand, how many 2"s is he holding in his other hand? He"s not holding any "2"s." He is holding 2 objects. He is also holding 1 knife and 1 napkin. Again, the numbers are not physical things he is holding, but rather mathematical objects used to represent the quantity of the things he is holding. Is 1 apple equal to 1 knife? No, of course not. Is the quantity of apples equal to the quantity of knives? Yes. There is one of each. The quantity is the same" 1 = 1. Pro makes a valid mathematical example in his argument: "1/2 / 1/2= 1 1+0= 1" etc. But next, pro advances arguments that do not use math properly to show what is happening. As a matter of fact, once units are introduced to the math, all of pros examples can be shown to make physical sense. However, his basic argument will still be shown to be wrong. Using units 1 = 1. 1 apple =/= 1 knife. But pro has not made the argument that "1 apple =/= 1 knife." There"s no disagreement there at all. Pro has made the argument that since "1 apple =/= 1 knife," then "1 =/= 1" But the units have changed with the new declaration. A more accurate statement from pro"s example would be "the quantity (1) of apples is equal to the quantity (1) of knives." This is a true statement. There is one of each thing. They are not the same thing. But their quantities are equal to each other" 1 = 1. Pro"s next argument: "Imagine 4 apples, and another 2 apples in different places, you like to join those 4 apples to the 2 separated apples equally, you send two apples to each apple. To have the right result, you will have to only consider the 2 apples that you sent to each apple, as if you add the 2 apples to that 1 apple that you had as a separation of the 2 apples, you would count 3 apples, and that wouldn't match the result. Just like that, you will notice adding won't work in this example, so in the example 1/2 / 1/2, adding should not be consider. But wait, what did I just do? I show that the result of 2 wasn't a result of how many times it was divided, but actual apples. OH no, is this debate over?" In this example, you have incorrectly used math to describe what has happened: "Imagine 4 apples, and another 2 apples in different places, you like to join those 4 apples to the 2 separated apples equally, you send two apples to each apple." There is one math problem there, and it has the correct result also. Why did you send 2 apples to each apple? Because: 4 / 2 = 2. When you add units it makes more sense: 4 apples / 2 apples = 2 apples/apple. Here is how this equation would be read: "4 apples, divided by/amongst 2 apples equals 2 apples per apple." Each apple in the group of two, receives 2 apples from the group of 4. This equation has not addressed adding the apples or joining them. It has only solved how many from the group of 4 you send to each apple in the group of 2. "To have the right result, you will have to only consider the 2 apples that you sent to each apple, as if you add the 2 apples to that 1 apple that you had as a separation of the 2 apples, you would count 3 apples, and that wouldn't match the result." The 3 apples you have mentioned here are a new mathematical equation: "2 apples + 1 apple = 3 apples." This new equation represents "adding" the 2 apples from the group of 4 with the 1 apple from the group of 2. The two equations represent the two things you have done in your example: Dividing up the 4 apples equally amongst two apples, and then adding those 2 apples to the 1 apple. Moving to pro"s next argument: "Imagine 1/2 of an apple. Another1/2 of an apple awaits to be equally dived. I send 1/2 of an apple to the other 1/2 of an apple. That half apple now has half of an apple, the result is 1, the only way I can get 1 apple is if I join them (which is the mistake of my last opponents), but wait, when I did 4/2 I show that adding cannot work. Oh no, we have a contradiction, ok I guess this debate is pointless, and it should end in a tie." By "sending" the half of the apple to the other half of an apple you get the correct mathematical result when the units are set up properly: "1 (1/2 of an apple) / 1 (1/2 of an apple) = 1 (1/2 of an apple)/(1/2 of an apple)" It is this equation that represent how to divide, or divvy up evenly, " of an apple to " of an apple. This equation also works: "" apple / " apple = 1 apple/apple" This is a rate which also applies. For every 1 apple I have, I send 1 apple to it. Next, you join them: "1/2 apple + " apple = 1 apple" You would agree this is correct right? "That division gives you the result of how many times did you divide whatever you had with whatever it is?" What the math problem represents all depends on the units and how the math problem is set up. "4/2=2 The two in this equation is a container, the 4 is whatever you are dividing into the container." I have 4 apples, I have a container that holds 2 apples, how many times can this container be filled to the limit? 2 times correct?" So, to properly set up this equation using units (we will use apples going into the container): "4 apples / 2 containers = 2 apples / container" Or "4 apples divided into two containers equals 2 apples per container" The equation you used is correct, I have merely added units to clarify. "I have 4 apples, I have 2 container, each of which holds 1 apple (since I split the containers into two), how many times will I fill each container if I dived the apples equally? 2 correct?" Again, this is two problems: "4 apples / 2 containers = 2 apples / container" Remember, all this is saying is that to evenly divide the 4 apples into the 2 containers, you take 2 of the four apples to each container. It says nothing about putting the apples into the container, nor the quantity of apples the container can hold. And, for one container, since it can only hold 1 apple, but I have 2 apples to go into it: "1 / 2 = 1 remainder 1." (not "one half" but one divided by 2) You can put in one apple and you have one apple left (the remainder) you cannot put in. Even equating the problem to one half, you can put half of your 2 apples into the container. "The numeric 1 from 1 apple does not equal the numeric 1 that you get by how many times 1/2 apple can be inserted in a container that holds 1/2." Again, the numeric 1 does because the numeric 1 is only representing a quantity of things, not the actual things. I hope I have explained things thoroughly enough for my opponent! I look forward to your response!