Topics Mathematics

Question What do you believe to be the square root of -1? Is it a flaw in our complex math system? Is it really a non-existent and incomprehendable thing? Or are our brains simply too basic to understand this notion?

Accepted:
October 24, 2005

Comments

Mathematicians have defined many different number systems--integers, rational numbers, real numbers, complex numbers, quaternions, etc. The answer to your question will be different depending on which number system you use.

In the real numbers, there is no number whose square is -1. In the complex numbers, there are two numbers whose squares are -1; in the usual notation for complex numbers, they are called i and -i. Just as the real numbers are often represented as the points on a line, the complex numbers are represented as the points on a plane. In this picture, the real numbers lie along a horizontal axis, and the numbers i and -i are 1 unit above and below 0.

But this answer may not satisfy you. Perhaps you want to know what the square root of -1 is really, independent of any choice of number system. I am inclined to say that that question is meaningless. To make sense of your question, you have to have a number system containing the number -1, in which there is a squaring operation. Only then does it make sense to ask for a number whose square is equal to -1.

Question What do you believe to be the square root of -1? Is it a flaw in our complex math system? Is it really a non-existent and incomprehendable thing? Or are our brains simply too basic to understand this notion? Accepted:October 24, 2005

Question What do you believe to be the square root of -1? Is it a flaw in our complex math system? Is it really a non-existent and incomprehendable thing? Or are our brains simply too basic to understand this notion?

Accepted:
October 24, 2005