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

Is it true that |𝑧| = |𝑧^(*)| for all 𝑧?

01:39

Video Transcript

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

Let’s begin by defining each part of our question. 𝑧 is a complex number, and we generally say that it’s of the form π‘Ž plus 𝑏𝑖, where π‘Ž and 𝑏 are real constants. We say π‘Ž is the real part of 𝑧, whereas the imaginary part of 𝑧 is the coefficient of 𝑖, so it’s 𝑏. The modulus of 𝑧 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‘Ž plus 𝑏𝑖, the modulus of 𝑧 is the square root of π‘Ž squared plus 𝑏 squared.

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

Now, we said that these bars mean the modulus of 𝑧 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 𝑧 star is the square root of π‘Ž squared plus negative 𝑏 squared. But negative 𝑏 times negative 𝑏 is positive 𝑏 squared. So we find 𝑧 star is equal to the square root of π‘Ž squared plus 𝑏 squared. And we’ve proven algebraically that the modulus of 𝑧 is always equal to the modulus of 𝑧 star.

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

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