## anonymous one year ago A simple * question

1. anonymous

whats the question

2. anonymous

Develop a proof to show that $a ° b =\frac{a+b}{ab}$Has no identity (i.e. $$a ° b = a$$).

3. anonymous

Any clue?

4. anonymous

Here is what I got

5. wolf1728

Is that ever true? Let's use some real numbers: 7 * 3 = (7 + 3) / 7*3 7*3 =? 10/21 21 =? 10/21

6. anonymous

Assume that $a*b=a$ then, $\frac{ab}{a+b}=a$$\frac{1}{a}+\frac{1}{b}=a$$\frac{1}{b}=a-\frac{1}{a}$$\frac{1}{b}=\frac{a^2-1}{a}$$b=\frac{a}{a^2-1}$ Here b would be the identity element, for which all a*b=a however this is not true as at a=1 b is not defined, therefore there is no identity element

7. wolf1728

Why do we assume that a*b = a?

8. anonymous

If our assumption is correct, then we'd get an expression for b which is defined for all a, but since b is not defined at a=1, our assumption is incorrect and there's no identity element

9. anonymous

for all a belonging to the required set

10. anonymous

Ahh, that make's sense @Nishant_Garg. Thank you! :D. Also, from my textbook, it says that there can only be one identity element. Since in this case, the identity element would change (the squared would affect it), is that another reason aswell?

11. anonymous

@Nishant_Garg, the $$ab$$ should be at the bottom, not the top

12. anonymous

13. anonymous

Yeah, that changes a bit ;)

14. anonymous

but it should be something like that if u just solve it

15. anonymous

I don't know how to rearrange to get something similar, can you help @Nishant_Garg?

16. anonymous

ill try

17. anonymous

Thanks so much @Nishant_Garg :D

18. anonymous

ok there's only 1 mistake, i've written $\frac{ab}{a+b}$ instead of $\frac{a+b}{ab}$ but the rest of the calculation is ok As for the identity element, yes there's only one identity element...so about that I'm also confused O.o

19. anonymous

Maybe, what we are meant to do is to sub in some value for $$a$$, and find the corresponding $$b$$ vaule. Then, substitute anoter $$a$$ value. If the $$b$$ values don't match, then there is no identity element, @Nishant_Garg?

20. anonymous

yep, and clearly they won't match! and it's not even defined for some numbers like +-1, so there r definitely many factors to it...

21. anonymous

Yeah. Thanks so much @Nishant_Garg :D