What are imaginary numbers?

1 Answer
Oct 3, 2015

Answer:

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.