Video: Finding the Product and Quotient of Complex Numbers

If 𝑧₁ = (βˆ’1/√2) βˆ’ 𝑖 and 𝑧₂ = (βˆ’1/2) + √2𝑖, is it true that 𝑧₁² = 𝑧₂?

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Video Transcript

If 𝑧 one is equal to negative one over root two minus 𝑖 and 𝑧 two is equal to negative one-half plus root two 𝑖, is it true that 𝑧 one squared is equal to 𝑧 two?

Here, we’ve been given two complex numbers represented in algebraic or rectangular form. And we’re being asked to decide whether the statement 𝑧 one squared is equal to 𝑧 two is true. To work this out, we’re going to need to first evaluate 𝑧 one squared. That’s negative one over root two minus 𝑖 all squared.

We recall that squaring a number is the same as multiplying it by itself. So 𝑧 one squared is equal to negative one over root two minus 𝑖 multiplied by negative one over root two minus 𝑖. And once we don’t want to consider 𝑖 as a variable, we can distribute these brackets as normal.

Let’s look at the FOIL method. F stands for first. We multiply the first term in the first bracket by the first term in the second bracket. One multiplied by one is one. And root two multiplied by root two is two. And since we’re multiplying a negative by a negative, we end up with positive one-half. We then multiply the outer term in each bracket. That’s negative one over root two multiplied by negative 𝑖. That gives us positive 𝑖 over root two.

We repeat this process with the inner terms, which once again gives us 𝑖 over root two. And L stands for last. We multiply the last term in each bracket. That’s 𝑖 squared. Now remember, we said 𝑖 is not a variable. It’s in fact the square root of negative one. This means that 𝑖 squared is actually equal to negative one.

So our expression for 𝑧 one squared becomes one-half plus two lots of 𝑖 over root two minus one. Now, one-half minus one is negative one-half. So currently, we can see that 𝑧 one squared is equal to negative one-half plus two 𝑖 over root two. And we need to rationalize the denominator of this second fraction.

To do this, we multiply both the numerator and the denominator by the square root of two. Two 𝑖 multiplied by root two is two root two 𝑖. And the square root of two multiplied by the square root of two is two. So this second fraction becomes two root two 𝑖 over two. But of course, we can simplify this further by dividing through by two. So we see that 𝑧 one squared is equal to negative one-half plus root two 𝑖. And we said that 𝑧 two was negative one-half plus root two 𝑖.

So we can see that 𝑧 one squared is equal to 𝑧 two. And this statement is indeed true.

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