## idku one year ago Maybe it is too early for me to ask this question, but why .... ?

1. idku

$$\large\color{black}{ \displaystyle \left|1+i\right|=\sqrt{2} }$$

2. idku

(and so $$\large\color{black}{ \displaystyle \left|1-i\right|=\sqrt{2} }$$ )

3. idku

can someone explain this?

4. anonymous

both $$1+i$$ and$$1-i$$ are $$\sqrt2$$ units away from $$0$$

5. idku

Is that because i is a rotation?

6. idku

A 45 degree rotation?

7. anonymous

|dw:1442713983303:dw|

8. anonymous

just like both $$(1,1)$$ and $$(1,-1)$$ have distance $$\sqrt2$$ from $$(0,0)$$

9. anonymous

or if you prefer, just like the diagonal of any square with side 1 has length $$\sqrt2$$

10. idku

Wait, so i is a diagonal?

11. anonymous

|dw:1442714099587:dw|

12. idku

|dw:1442714123626:dw|

13. anonymous

ok i didn't mean to confuse you $$i$$ is not the diagonal the absolute value of a complex number is its distance from 0

14. idku

and that would mean that $$\large\color{black}{ \displaystyle \left|a+i\right|=a\sqrt{2} }$$ $$\forall a\in\mathbb{R}$$

15. idku

wait, sorry for a>0 too

16. anonymous

oh no

17. idku

oh, that is not right I guess. (I just heard someone said that i is sort of a rotation)

18. anonymous

$|a+i|=\sqrt{a^2+1}$

19. anonymous

what is true is that $|a+bi|=|a-bi|=\sqrt{a^2+b^2}$

20. anonymous

it has no bearing on the sign of $$a$$ or $$b$$ since you are going to square them in any case

21. idku

oh, ok.... I guess I will explore that a little more. For now I will just have the rules at least;)

22. idku

yes as long as a, b are Real...

23. anonymous

in $$a+bi$$ $$a$$ and $$b$$ are always real

24. idku

ok...