Question

How do you simplify `1/(2+5i)`?

Answer 1

To simplify a complex fraction, multiply top and bottom (denominator and numerator) by the complex conjugate of the bottom. This expression simplifies to `(2-5i)/29`.

Explanation:

The 'complex conjugate' of a complex number `(a+bi)` is `(a-bi)`, and multiplying a complex number by its complex conjugate will yield a real number.

`1/(2+5i)*(2-5i)/(2-5i)`

Note that

`(2-5i)/(2-5i) = 1` so we are not changing the original number by simplifying it.

`1/(2+5i)*(2-5i)/(2-5i) = (2-5i)/(4+10i-10i-25i^2) = (2-5i)/(4-25i^2)`

Note that `i^2=-1` because `i=sqrt(-1)`

So the simplified version of this expression is

`(2-5i)/(4-(-25)) = (2-5i)/29`