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

I only brought out apples and kids as a demonstration of how we were thought, and how wrong that could be." I recall you telling me English was your 2nd language, so can I assume you mean "taught?" There are several other grammatical errors, which I do not hold against you. However I am forced to make reasonable assumptions as to what you have meant. "My point is, can division give the result of anything but, of how many times did you divide? Can division.... with the same rules." Maybe I missed the point of this debate and your argument then? I tried to show you the flaw of your arguments, point by point, as you made them. Also, I went chronologically, point by point, to maintain a coherent flow to the discussion. "If by having 4/2 I count 2 of the result as being a quantity of an object, then if I .... adding won't work." Two examples: I have two groups of apples (group A and B). In each group, there is half an apple. I want to figure out how many apples from group A to send to group B. In other words, I want to "divide" my group A members evenly among my group B members. My answer will be: 1/2 apple / 1/2 apple = 1 apple/apple. I have a rate. For every apple in group B, I send an apple from group A to it. The rate would be the same with group A and B each having 1/4 an apple in the group: 1/4 apple / 1/4 apple = 1 apple/apple. Again, for every apple in group B, I send an apple from group A. Since I have 1/2 an apple in group B, and I must maintain that rate, I send 1/2 an apple from group A. So, in summary, the 1 object (in my example, apple) you get is the number of apples in group A you send per apple in Group B. "I notice your use of; "pro advances arguments that do not use math properly to show what is happening", and other places you also call my methods to not use mathematics properly, even know I active the same result by my method. *remember I'm accusing math to be flawed, ....This is also the reason of why I said you must show flaws in reason, and in reason only, and not in mathematical statement since if I'm right that there is a flaw in mathematics .... Therefore, the rules of mathematics is into question." I had to point this out because the math was not flawed; rather it was your interpretation and application of it" the "reasoning" of your approach. This is also why I went point by point. I had to show the exact flaws you were making when setting up your equations, which then resulted in a flaw in reason" It was your premise that was incorrect, leading to an incorrect conclusion based on a faulty premise. This is why I believe I have showed your flaw in reasoning. You have set up incorrect mathematical statements to frame your argument. The reasoning behind the mathematical statements you set up was wrong. I then set up the proper mathematical statements to show you the proper reasoning which led to the proper result. "Do you mean, one 1/2 of an apple divided by one 1/2 of an apple? It feels very confusing." Yes. I pasted my argument from Microsoft Word directly into DDO and it resulted in formatting errors. My apologies. Here is the correction: "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, 1/2 of an apple to 1/2 of an apple. This equation also works: "1/2 apple / 1/2 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 + 1/2 apple = 1 apple"" "1/2 is a unit in itself, just like 1 is just 1 unit, but 1/2 is not the same as 1." You use addition to obtain 1 apple here, but yet with whole numbers you never use any addition to demonstrate the same thing. Can you show me this same example using 1/4 / 1/4 and 4/2?"" You can set up the equation both ways and it will still work because 1, 1/2 of an apple is the same as 1/2 of an apple. With the 1/4 apples, the RATE will be the same: 1/4 apple / 1/4 apple = 1 apple/apple When you apply that rate to your 1/4 apples, you send and equal amount of apples for every apple in group b. A 1 to 1 ratio. The process is the same for the 4/2. "Why did you feel my statement was incorrect? What is the math problem that you see? You feel.... Now you feel my pain." Hopefully my past examples have shown how your statements were incorrect. The fact that 1/2 / 1/2 is the same as 1/2 + 1/2 is a coincidence. "I do not get your use of grouping, by saying group of two you count 2 as being just 1 set." Hopefully, using group A and B clarifies this. "I see the end result being how many times you divide, so I see 1 as being, 1 (whole) time. I cannot see how you can get 1 whole apple out of this. So all you got to do, is show me, how exactly do you get half of an apple to be divided by half of an apple and get the result of 1 in apples? If you want to continue to show me this by adding, then also do it using 1/4 / 1/4." Again, the math here gives a rate. Watch the units and observe my previous examples. "Therefore, the only time I see 1 not equal to 1 is when you have an equation using the operation of division that equals to 1 in one side, and on the other side any of the other basic operations to also equal 1. I feel that both 1's from both sides cannot be equal since they represent different things. So basically division is not comparable with the other operations since they cannot active the same purposes. So it goes beyond 1=|=1, of why I told you, I need you to prove me wrong." Instead of going beyond, simplify. The quantities are the same and the units are what change. Since the quantities remain the same, 1 still equals 1. An apple is not the same as a knife, how many I have can be the same. "By going point by point, my opponent missed my reasoning, and have of his round was filled with misinterpretation of what I was trying to convey, and assume something that in the begin of this round I state to be completely untrue." By going point by point, I was pointing out the flaws of your reasoning in setting up your equations. This was necessary to show the flaw in your reasoning. "By going point by point, my opponent missed that half of my previous round, as I was just trying to demonstrate that, by going by what such of teacher would ... Therefore, the quotient and the divisor cannot represent the same thing." Hopefully, the introduction of units in my argument has answered my opponents objections. The objects become irrelevant. The math remains true and correct as long as the problems are set up properly. The numbers represent quantities of objects, not the objects themselves. When the problem is calculated properly with the units, the quotient and divisor describe the result. "By going point by point, and missing what I did in the beginning, my opponent just use that same argument as I previously show not to work." When the units and problems are set up correctly, they do work. Hopefully I have addressed any misunderstandings from my opponent. Allow me to reiterate as I close this round: The flaws in: my opponents reasoning, setting up of mathematical equations to depict an event, and misunderstanding of units, have left my opponent with an erroneous conclusion. Perhaps I misunderstood his original argument, or perhaps he misrepresented it. That is at this point irrelevant. At the heart of this debate is "A number not always represent the same thing, therefore 1 not always equal 1." Regardless of division, addition, multiplication or addition, using units clarifies the problem as the numbers themselves only represent quantities. 1 will always equal 1.