Question Video: Finding the Modulus of Complex Numbers in Algebraic Form Mathematics • 12th Grade

What is the modulus of the complex number 3 + 7𝑖?

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Video Transcript

What is the modulus of the complex number three plus seven 𝑖?

The definition of the modulus of a complex number of the form 𝑎 plus 𝑏𝑖 is the square root of 𝑎 squared plus 𝑏 squared. 𝑎 is the real part of the complex number whilst 𝑏 is the imaginary part. Let’s compare this form to our complex number three plus seven 𝑖. This complex number has a real part of three and an imaginary part of seven. Don’t confuse this with seven 𝑖. The imaginary part is essentially the coefficient of 𝑖. We can substitute these values into the formula for the modulus. And we get the square root of three squared plus seven squared. Three squared is nine and seven squared is 49. So the modulus is the square root of 58.

We would usually try to simplify this surd. However, there are no factors of 58 which are also square numbers. So we’re done; it’s in its simplest form. And we can say that the modulus of the complex number three plus seven 𝑖 is root 58.

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