Question

How do you find the absolute value of the complex number z=3-4i?

Answer 1

The answer is 5.

Explanation:

One way to do this is with the general formula `|a+bi|=\sqrt{a^2+b^2}` so that `|3-4i|=\sqrt{3^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5`. This formula is essentially the distance formula from the point `(a,b)` to the origin `(0,0)` in the plane (in rectangular coordinates), which comes from the Pythagorean Theorem.

Another way to show your work if `z=a+bi` is to write the answer as `|z|=sqrt(z*\bar{z})`, where `\bar{z}=a-bi` is the complex conjugate of `z`. Since `z*\bar{z}=(a+bi)(a-bi)=a^2-abi+abi-b^2i^2=a^2+b^2`, this gives the same answer.