Question

What is the conjugate of `7 - 4i`?

Answer 1

`7+4i`

Explanation:

The complex conjugate of `a+bi` is `a-bi`. Complex conjugates have the cool property that `(a+bi)(a-bi)=a^2-abi+abi-b^2i^2=a^2+b^2=|a+bi|^{2}`. This makes them useful for rewriting quotients `(a+bi)/(c+di)` in the standard form `alpha+beta i`.

For example,

`(3+2i)/(7-4i)=(3+2i)/(7-4i) * (7+4i)/(7+4i)=(21+12i+14i+8i^2)/(49-16i^2)`

`=(13+26i)/(49+16)=13/65+26/65 i=1/5 + 2/5 i`