Find the complex conjugate of the complex number one plus 𝑖 and the product of this number with its complex conjugate.
So, there’re two parts to this question. Firstly, we’re asked to find the complex conjugate of this complex number one plus 𝑖. Well, we can recall that the complex conjugate of a complex number is the complex number we get when we simply change the sign of its imaginary part. So, in general, the complex conjugate of the complex number 𝑧 equals 𝑎 plus 𝑏𝑖 is the complex number 𝑧 star, which is equal to 𝑎 minus 𝑏𝑖. We’ve changed the sign of the complex part. It’s no longer positive 𝑏. It’s now negative 𝑏.
So if we let 𝑧 be our complex number, one plus 𝑖, then to find its complex conjugate 𝑧 star, we simply change the sign of the imaginary part. So previously, we had plus 𝑖, which is plus one 𝑖. And we change it to negative 𝑖 or negative one 𝑖. The complex conjugate of one plus 𝑖 is therefore one minus 𝑖. The second part of this question asks us to find the product of this number. So that’s our original complex number with its complex conjugate. So we’re looking for the product of 𝑧 and 𝑧 star. As we’ve just found the complex conjugate to be one minus 𝑖, we’re therefore looking for the product of one plus 𝑖 and one minus 𝑖.
We can go ahead and distribute the parentheses. One multiplied by one gives one. And then, one multiplied by negative 𝑖 gives negative 𝑖. 𝑖 multiplied by one gives positive 𝑖. And then, 𝑖 multiplied by negative 𝑖 gives negative 𝑖 squared. So we have one minus 𝑖 plus 𝑖 minus 𝑖 squared. Now, of course, in the centre of our expression, negative 𝑖 plus 𝑖 simplifies to zero. So these two terms cancel out. And we’re left with one minus 𝑖 squared. We need to recall here that 𝑖 squared is equal to negative one. We therefore have one minus negative one or one plus one, which is equal to two. And so we found that the product of our complex number with its complex conjugate is two.
In fact, there is actually a general result that we could’ve used here, which is that, for the complex number 𝑧 equals 𝑎 plus 𝑏𝑖, the product of 𝑧 with its complex conjugate 𝑎 minus 𝑏𝑖 will always be equal to 𝑎 squared plus 𝑏 squared. We can see that this is certainly the case for our complex number one plus 𝑖. Both the real and imaginary parts are equal to one. And one squared plus one squared is equal to one plus one, which is equal to two.
To see why this is the case, we just need to distribute the parentheses in the product 𝑎 plus 𝑏𝑖 multiplied by 𝑎 minus 𝑏𝑖. And we see that, in the general case, just as it did in our specific example, the imaginary parts of this expansion cancel, leaving 𝑎 squared minus 𝑏 squared 𝑖 squared. That’s 𝑎 squared minus 𝑏 squared multiplied by negative one, which is 𝑎 squared plus 𝑏 squared. So we’ve completed the problem.
The complex conjugate of one plus 𝑖 is one minus 𝑖. And the product of one plus 𝑖 with its complex conjugate is two.