# Question Video: Comparing the Moduli of a Complex Number and Its Conjugate Mathematics

Is it true that |π§| = |π§^(*)| for all π§?

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### Video Transcript

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

Letβs begin by defining each part of our question. π§ is a complex number, and we generally say that itβs of the form π plus ππ, 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 itβs 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 ππ, 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 ππ, its conjugate π§ star is π minus ππ. 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 weβve 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§.