Video Transcript
What is the real part of the complex number three plus four 𝑖?
In this question, we’re given a complex number, and we’re asked to find the real part of this complex number. And there’s a few different ways we could do this. The first thing we need to notice is the complex number we’re given is in algebraic form. And remember, we say if a complex number is given in the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers, then this is called the algebraic form of a complex number.
And this is useful for our question because it will help us discuss exactly what we mean by the real part of a complex number. The real part of a complex number given in algebraic form is the value 𝑎. And one way of seeing why this might be true is 𝑏𝑖 is 𝑏 multiplied by 𝑖. We’re multiplying 𝑏 by the imaginary value of 𝑖. So we could think of 𝑎 as being the real part of our complex number and 𝑏 times 𝑖 as being the nonreal part of our complex number.
And in fact we have names and terminologies for both of these. First, we represent the real part of a complex number in the following notation. This says the real part of the complex number 𝑎 plus 𝑏𝑖. And if 𝑎 plus 𝑏𝑖 is in algebraic form, then this is just going to be equal to our value of 𝑎. Similarly, we also have something called the imaginary part of a complex number. This is represented in the following notation. This says the imaginary part of 𝑎 plus 𝑏𝑖. And if this is given in algebraic form, then the imaginary part of this number is the coefficient of 𝑖, which is 𝑏.
Let’s now apply this to the complex number given to us in the question. We want to find the real part of the complex number three plus four 𝑖. We can see this is given in algebraic form. So the real part of this complex number is going to be equal to three. And this gives us the answer to our question is three. However, we could also find the imaginary part of this complex number. The imaginary part of the complex number three plus four 𝑖 is going to be equal to the coefficient of 𝑖 because this is given in algebraic form. And we can see this is equal to four. Therefore, we were able to find the real part of the complex number three plus four 𝑖. We saw that this was equal to three.
