Question

What are imaginary numbers?

Answer 1

The Real numbers can be represented as a line.

The Complex numbers can be represented by a plane whose `x` axis is the Real numbers.

Imaginary numbers are the numbers on the `y` axis of that plane.

Explanation:

Suppose we start with Whole numbers `0, 1, 2, 3,...`

We can add them them quite happily and we always get another Whole number.

We can solve problems like `x + 2 = 5`, but when we try to solve problems like `x + 5 = 2` we find our Whole numbers are insufficient.

So we can introduce the idea of a negative number and expand our idea of what a number is to include all of the Integers:

`ZZ = { 0, 1, -1, 2, -2, 3, -3,...}`

We can add and multiply any two Integers and we always get an Integer.

We can solve problems like `2x + 6 = 0`, but when we try to solve problems like `6x + 2 = 0` we find our Integers are insufficient.

So we can introduce the idea of a Rational number and expand our idea of what a number is to include all numbers of the form `m / n` where `m, n in ZZ` and `n != 0`.

...

To cut a long story short, in order to be able to solve problems like `x^2 + 1 = 0` we introduce the imaginary unit `i`, with the property `i^2 = -1`.

A square root of a negative Real number is a pure imaginary number.

In fact, we define the principal square root of a negative Real number as:

`sqrt(x) = i sqrt(-x)`

Any Complex number `z in CC` can be represented as `z = a + ib` where `a` and `b` are Real numbers. `a` is called the Real part of `z` and `ib` the Imaginary part. This can be pictured as the point `(a, b)` on a plane.

Please note that Imaginary numbers are no more imaginary than Real numbers. Sir Isaac Newton seemed to dislike working with negative numbers, which he called "imaginary" numbers.