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.
