Question about unique real number

[citea]

hello guys I am new in this forum
i start to self learn calculus tom apostol (in my college we got another book)
i start from the very beginning of the chapter. so this not really a homework. if i post thread in wrong sub-forum please remind me :)

Homework Statement

prove : If a+b = a+c, then b = c.

Homework Equations

this also prove zero number is unique.

The Attempt at a Solution

the prove was easy

using AXIOM EXISTENCE OF IDENTITY ELEMENTS & ASSOCIATIVE

there is a number y such that y + a = O
then
y+(a+b) = y+(a+c)

(y+a)+b=(y+a)+c

0+b=0+c

b=c

but why in the book say this also prove number 0 is unique? in other words only one real number have the property zero?!

thank u :)

 

[R136a1]

How did you define the zero number?

 

[citea]

How did you define the zero number?

axiom says : there exist real number denote by 0, such that for every real x, we have x+0=x

see the underlined, it doesn't say : there exist ONE real number.
so we have to prove only exactly one real number denote by 0. but i still don't get it how to prove this.. :)

 

[PeroK]

Can you see how to use the result you have just proved:

a + b = a + c => b = c

To prove that there is only one 0?

 

[R136a1]

Assume there are two real numbers 0 and 0'. Both satisfy x+0 = x and x+0' = 0' for all x. Can you deduce 0 = 0'?

 

[citea]

ok guys thanks for the help..
i don't have problem anymore.. its clear now