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Ethan said that the square of any pure imaginary number is a negative real number. Do you agree with Ethan? Justify your answer.

I agree with Ethan

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 4

The Complex Numbers

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

Campbell University

McMaster University

Harvey Mudd College

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Okay, So what this question is asking is is if you take any pure imaginary number and you square it well, that turned out to be a negative riel number. And so basically, we could do a couple of examples. We know that a pure imaginary number is some really number, eh? That's not zero times this eye here. And so let's do a couple of examples. If I said okay, well, what about five? I if I squared that well, what would that be? That would be we've got to distribute this, too. So that would be five squared, And that would be I squared. And we know that I squared is a negative one and five square goes to 25. And so this would be a negative 25. So in this particular case, it does seem true that this would become a negative real number. But let's try another number. Let's try another pure imaginary number. What if I had a negative five? I So here, if we square this again, we're gonna square. We're going to distribute that to, and I'm going to get a negative five squared and I'm hit again. Haven't I squared And that's going to be This goes to a negative one this year and negative times and negative is a positive. So this is goes to 25 we're back again to a negative rial number. So I guess we're going to agree with Ethan on this one.

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