Math CentralQuandaries & Queries


Question from David, a student:

What is the only number that when it's squared becomes negative?

Hi David.

I think you are mistaken!

Squared real numbers are always positive, except for zero, which isn't negative either.

Only if you are dealing with imaginary numbers do you get a square that is negative. And then there are as many possibilities as there are for positive squares! We write imaginary numbers with the letter i as a suffix. For example, 5i is an imaginary number representing 5 times the square root of -1. Thus, its square is -25. Similarly, ( 1.4i )2 is -1.96.

There are also "complex" numbers that are a kind of composite of imaginary numbers and real numbers. 5i + 3 has an imaginary part (5i) and a real part (3). But the composite can be squared as well: (5i + 3)(5i + 3) = -25 + 15i + 15i + 9 = 30i - 16. As you can see, we squared a complex number and got a different complex number.

Stephen La Rocque.

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