Squares and Square Roots
I Like Abstract Stuff; Why Should I Care?
We've seen a number of equations over these pages that have no real number solutions. Some of these equations have no solutions because all of mathematics would come crashing down if they did. Scary. Hey, at least that would free up an extra hour of your school day.
An example is the equation x = x + 1.
This can only be true if 0 = 1, which is a contradiction when we're dealing with the usual everyday integers. Although, if you ever have the chance, ask an algebraist about the field of characteristic 1, in which 0 and 1 are the same thing. That will be one memorable conversation. Or, you know, zero memorable conversations.
A more interesting case is when an equation has no real number solutions, not because it's merely stating a contradiction, but because there is no real number that can possibly fit the job description. For example, the equation x2 = -1 has no real number solutions because there's no real number we can square to give us a negative value.
Remember how we needed to introduce the concept of negative numbers to answer questions like 3 – 5 = ? and then had to do the same thing with rational numbers to answer questions like 2 ÷ 9 = ?
Now we need imaginary numbers to answer questions like
Mathematicians are like people (we could end the sentence right here) who are in possession of a genie in a magic lamp. Every time they need something to make an equation work, they simply rub the lamp and ask for whatever new type of number they want, and poof! their wish is granted. You'd think they would have asked for a billion dollars or to be ruler of the world, but their desires are modest and mainly confined to their work.
The most popular imaginary number, i, can be defined as the square root of -1. If we allow imaginary numbers as solutions to equations, we can solve many more equations than we can with real numbers. This keeps the mathematicians, and their genies, happy. As an example, i is a solution to the equation x2 = -1, an equation which has no real number solutions.
Although i and company are referred to as "imaginary," they're perfectly good numbers. You can't sit down and count out i of something, but i can still be extremely useful in solving complex equations in which we would otherwise be lost. If "lost" is how you're feeling right now, then you know exactly how we'd feel without imaginary numbers.