Question

let u=5i and z = 1+2i what is u/z

Answer 1

\[\frac{5i}{1+2i}\times \frac{1-2i}{1-2i}\]

Answer 2

denominator is
\[1+4=5\]

Answer 3

numerator is
\[10+5i\]

Answer 4

so you get
\[\frac{10+5i}{5}=2+i\]

Answer 5

how did you get the denominator?

Answer 6

\[(a+bi)(a-bi)=a^2+b^2\]

Answer 7

once you see it it is easy

Answer 8

is that an identity

Answer 9

If you foil it out you will see why it is true. It's just how conjugate pairs work.

Answer 10

i did foil it out but i still dont get it

even though i just proved it to myself

Answer 11

Ok, well then just accept that it is true (since you proved it to yourself).

So if you want to rewrite a fraction such that it has a real denominator you just multiply top and bottom by the conjugate.

Answer 12

ok. but is it really (a +bi)(a-bi) = a^2 +bi^2 or is it (a +bi)(a-bi) = a^2 -bi^2

Answer 13

(a+bi)(a-bi)=a^2-abi+abi-b^2i^2
remember i^2=-1
so a^2-abi+abi-b^2(-1)=a^2+b^2

Answer 14

It's
\[(a +bi)(a-bi) \]\[= a^2 +(ab)i - (ab)i - b^2i^2\]\[=a^2 - b^2i^2\]\[=a^2 -b^2(-1)\]\[=a^2+b^2\]

Answer 15

ok thanks polpak
and myininaya