"he does not get it and starts guessing:
nothing? x? zero? one? "
One could proceed by elimination from the answers the student proposes.
(1) Is x/x equal to " nothing"?
Here the student probably means : " when x is divided by x, nothing remains". He is confusing the quotient with the remainder.
(2) Is it equal to x itself?
If it were equal to x, then, (x/x) - x would be 0. Is it the case?
(3) Is it equal to zero?
In order x/x to be equal to 0, x should be equal to zero, since only 0/x can be equal to 0. But that's impossible, since that would also mean " dividing by 0"!
(4) Is it 1 ?
He, one could use the fact that division "undoes" multiplication.
A little dialogue.
Teacher. Let's begin with the " multiply by x game". Which number should you begin with on the left to get the result x?
? ----- multiply by x ----------> x
Student. I should begin by 1.
Teacher. Good answer : 1 is the identity element for multiplication. Now, let's play the " division game". You are on the right side and you "have" x. Wich division should you perform to " return" to 1? How could you "undo" what you've done?
1 <--------- divide by ? -------- x
Student. I can't "divide by nothing" since nothing is not a number. I can't divide by 0. Should I divide by 1 ?
Teacher.- It would work in only one case, that is if x=1. Suppose x is 1000, do you hope to obtain a result of 1 when dividing 1000 by 1? As you now, in general, x/1 is not equal to 1, but to x itself!
Student. I know, I should divide by x! If I multipliy 1 by x I get x, and if I divide x by x, I find back 1, my original number!
Teacher.- Right. And you can express this truth either in the " operation language" or in the " fraction language". In the first case, you'll say that "dividing x by x gives you as result the number 1". In the other case, you'll say that, provided x is not equal to 0,
the number x/x is the same number as 1",
in symbols ,
for all number x ( not equal to 0) : x/x = 1.
Student. Isn't it strange, however, that x divided by itself gives as result precisely that special number 1 which is the identity element?
Teacher. It will look less strange to you if you notice that :
x/x = x * (1/x)
in other words, that x/x is x multiplied by it's multiplicative inverse! And as you know - this is the definition of a multiplicative inverse - if I multiply a number A by it's inverse 1/A, the result is ( by definition I repeat) ....
Student .- .... the identity element ( for multiplication) : 1 !