### Video Transcript

Is it true that the modulus of π§ is equal to the modulus of π§ star for all π§?

Letβs begin by defining each part of our question. π§ is a complex number, and we generally say that itβs of the form π plus ππ, where π and π are real constants. We say π is the real part of π§, whereas the imaginary part of π§ is the coefficient of π, so itβs π. The modulus of π§ is the square root of the sum of the squares of the real and imaginary parts. So for a complex number of the form π plus ππ, the modulus of π§ is the square root of π squared plus π squared.

And what about π§ star? Well, π§ star is the conjugate or the complex conjugate of π§. We find the complex conjugate by changing the sign of the imaginary part. So for a complex number π§ of the form π plus ππ, its conjugate π§ star is π minus ππ. We can see that the real part of π§ star here is still π. But this time its imaginary part, the coefficient of π, is negative π.

Now, we said that these bars mean the modulus of π§ star. And the modulus of a complex number is the square root of the sum of the squares of its real and imaginary parts. So the modulus of π§ star is the square root of π squared plus negative π squared. But negative π times negative π is positive π squared. So we find π§ star is equal to the square root of π squared plus π squared. And weβve proven algebraically that the modulus of π§ is always equal to the modulus of π§ star.

And so, we can say that yes, it is indeed true that the modulus of π§ is equal to the modulus of π§ star for all π§.