Pure Imagination

When you square a real number, whether it be positive or negative, you will always get a positive number. 82 = 64. (-8)2 = 64. It is quite simple to square root a positive number. But what if we want to square root a negative number?

The answer is: you can. It seems kind of silly to be able to create new numbers that don’t particularly make sense, but mathematicians do it all the time. One of the new numbers that mathematicians created is i, where i is defined to be the square root of negative 1:

CodeCogsEqn (23)

You will never need to use powers of i. This is because when you simplify an expression involving powers of i, everything can be reduced to just i:

CodeCogsEqn (24)

CodeCogsEqn (26)

CodeCogsEqn (27)

Above i4, the pattern just repeats itself again.

i is called the imaginary unit (hence the letter i), and is an example of an imaginary number. 2i is also an imaginary number, and so is 5.8i.

Now we have the ability to square root negative numbers! For instance the square root of -4 is 2i, and the square root of -100 is 10i.

What if we combined real and imaginary numbers? We get a new type of a number called a complex number. Complex numbers have the form a + bi, where a and b have to be real numbers. For example, 3 + 7i is a complex number. Real numbers and imaginary numbers are both types of complex numbers, as you can obtain them by setting one of a or b to be equal to zero.

A pictorial way of representing complex numbers is using an Argand diagram. This works similarly to the common (x,y) coordinates (Cartesian coordinates). In an Argand diagram, the real component goes on the x-axis and the imaginary component goes on the y-axis.

Argand Diagram.png

The length of the line is called the modulus, and represents the ‘size’ of the complex number, and can be calculated using the Pythagorean theorem. The angle that the line makes to the x-axis is called the argument.

For imaginary things, complex numbers have a surprisingly wide range of uses in maths and physics and can do much more than allow us to square root negative numbers. In mathematics, complex numbers introduce many functions and methods that enable us to solve problems that were previously unsolvable. In physics, complex numbers underpin the maths behind fields of study such as electromagnetism, signal processing and quantum mechanics (see The Quantum) .

With Easy as 1, 2, e and A Slice of π, we have now investigated all the numbers involved in Euler’s identity!

CodeCogsEqn (25)

That is, assuming you know about 0 and 1.

Yanhao

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