Question Video: Multiplying a Complex Number by Its Conjugate | Nagwa Question Video: Multiplying a Complex Number by Its Conjugate | Nagwa

Question Video: Multiplying a Complex Number by Its Conjugate Mathematics • Third Year of Secondary School

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Given that |𝑍| = 3, determine the value of 𝑍 times the conjugate of 𝑍.

03:14

Video Transcript

Given that the modulus of 𝑍 is equal to three, determine the value of 𝑍 times the conjugate of 𝑍.

In this question, we’re given a complex number 𝑍 and we’re told the modulus of this complex number 𝑍 is equal to three. We need to determine the value of 𝑍 multiplied by its complex conjugate. And before we answer this question, it’s worth noting there’re several different notations for the complex conjugate for 𝑍. For example, you may have seen this written with a star notation instead of an over bar. Both of these mean the same thing. The complex conjugate of a complex number just means we flip the sign of the imaginary part of that complex number. If 𝑍 is equal to π‘Ž plus 𝑏𝑖, where π‘Ž and 𝑏 are real numbers, then the complex conjugate of 𝑍 is equal to π‘Ž minus 𝑏𝑖.

There’s a few different ways of answering this question. We can do this directly from the following property: 𝑍 multiplied by its complex conjugate is equal to the modulus of 𝑍 squared. This is true for any complex number. Then, we just substitute the modulus of 𝑍 is equal to three into this equation, 𝑍 multiplied by its complex conjugate is equal to three squared, which is, of course, just equal to nine. And this is enough to answer our question. However, it can be useful to recall exactly where this property comes from.

One way of seeing this is to use the algebraic form of a complex number. Let’s set 𝑍 be equal to π‘Ž plus 𝑏𝑖, where π‘Ž and 𝑏 are real numbers. So, the complex conjugate of 𝑍 is π‘Ž minus 𝑏𝑖. We can use these expressions to find an expression for 𝑍 multiplied by the complex conjugate of 𝑍. It’s equal to π‘Ž plus 𝑏𝑖 multiplied by π‘Ž minus 𝑏𝑖. And we can evaluate this product by distributing over the parentheses. We will use the FOIL method. We multiply the first two terms together. That’s π‘Ž times π‘Ž, which is equal to π‘Ž squared. We then multiply the outer two terms together, giving us negative π‘Žπ‘π‘–. Next, we need to add on the product of the inner two terms; that’s π‘Žπ‘π‘–. Finally, we need to add on the product of our outer two terms; that’s negative 𝑏 squared times 𝑖 squared.

We can then simplify this expression. First, negative π‘Žπ‘π‘– plus π‘Žπ‘π‘– is equal to zero. Next, we know that 𝑖 is the square root of negative one, so 𝑖 squared is equal to negative one. This then gives us π‘Ž squared minus 𝑏 squared multiplied by negative one, which we can then rewrite as π‘Ž squared plus 𝑏 squared. Remember, we want to show this is equal to the modulus of 𝑍 squared. So, let’s recall what we mean by the modulus of 𝑍.

The modulus of a complex number is its distance from the origin in an Argand diagram. In particular, for a complex number written in algebraic form, the modulus of π‘Ž plus 𝑏𝑖 is the square root of π‘Ž squared plus 𝑏 squared. We can simplify this equation by squaring both sides. We get the modulus of 𝑍 squared is equal to π‘Ž squared plus 𝑏 squared. In other words, the modulus of any complex number squared is the sum of the squares of its real and imaginary parts. In particular, we can see this is exactly the same as the expression we got for 𝑍 multiplied by the conjugate of 𝑍. So, we’ve shown 𝑍 multiplied by its conjugate is equal to the modulus of 𝑍 squared. And we know the modulus of 𝑍 is three, so three squared is equal to nine.

Therefore, we were able to show if the modulus of a complex number 𝑍 is equal to three, then the value of 𝑍 multiplied by its complex conjugate is equal to nine.

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