Imaginary Numbers Explained in 5 Minutes

Imaginary Numbers Explained in 5 Minutes

Imaginary numbers appeared when mathematicians studied polynomial equations. These equations are simple, they're just equations that contain powers, for example..

x^2-9=0 or x^3+x^2+x=9

It turned out that although these equations were easy to write down they were not so easy to solve.

Using integers or whole numbers {..,-3,-2,-1,0,1,2,3,..} it was possible to solve some of these equations, but many were still unsolvable.

Using rational numbers or fractions, a/b where a and b are integers, it was possible to solve many more but still not all.

A major breakthrough occurred when mathematicians introduced a third type of number called irrational numbers which are not integers and not rational numbers. This allowed many more polynomial equations to be solved - but still not all!

In fact, the situation was a bit embarrassing, because one of the simplest polynomial equations could not be solved even with three types on numbers available. It was this..

x^2+1=0

This incredibly simple equation cannot be solved by an integer, it cannot be solved by a rational number, and it cannot be solved by an irrational number!

So what did mathematicians do? No problem, they just invented a new number, and they called it i..

i=square root of -1

It's easy to see that this number solves the original equation, that is..

i^2+1=0

Of course i can be used to define an infinite amount of new numbers, just take any number and multiply it by i. Mathematicians called the new class of numbers complex or imaginary numbers. Not a great name.

Finally, it turned out that with 4 types of numbers available (integer, rational number, irrational numbers and complex numbers) it was possible to solve all polynomial equations.

Of course the name imaginary number sometimes causes non-mathematicians to ask questions like "do imaginary numbers really exist?" This of course is a useless question. In mathematics if you can define something it exists!

There was a big bonus. In addition to helping solve all polynomial equations, the number i turned out to be amazingly useful in other areas of mathematics and it's also used in many areas of physics.

So the number i, defined as "that number which when multiplied by itself gives -1", turned out to be an amazing invention!

But the name? Not so much.

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Content written and posted by Ken Abbott abbottsystems@gmail.com
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