### Video Transcript

What is the modulus of the complex number three plus four π?

In this question, weβre asked to find the modulus of a given complex number, three plus four π. And to answer this question, weβre first going to need to recall what we mean by the modulus of a complex number. So we recall the modulus of a complex number is its distance from the origin on an Argand diagram. So one way of answering this question will be to plot the complex number three plus four π onto an Argand diagram and then calculate its distance from the origin.

However, we can also notice something interesting about the complex number three plus four π. This complex number is given in algebraic form. And remember, a complex number is written in algebraic form if itβs in the form π plus ππ, where π and π are real numbers. And weβve actually found a formula for finding the modulus of a complex number given in algebraic form. We know if π and π are real numbers, then the modulus of π plus ππ will be equal to the square root of π squared plus π squared. And this is just the general version of what we were talking about before.

In an algebraic form, the complex number π plus ππ is going to have coordinates π, π on an Argand diagram. And we can find the distance of the point π, π from the origin by using the Pythagorean theorem. Itβs the square root of π squared plus π squared. So we can just use this formula to find the modulus of three plus four π. Our value of π, the real part of three plus four π, is going to be equal to three. And our value of π, the imaginary part of three plus four π, is going to be equal to four.

So we get the modulus of three plus four π is equal to the square root of three squared plus four squared. And we can calculate this. Three squared plus four squared is 25, and the square root of 25 is going to be equal to five. Therefore, we were able to show the modulus of the complex number three plus four π is equal to five.