Tuesday, December 18, 2012

Imagine a number — Imaginary number

by mindylin610, This post orignally appeared at Science Avengers.

Let’s start with a Math problem:

What is the answer of the square root of -1?

In the opinions of many calculators or normal human beings, the square root of -1 is simply undefined. However, there were some Mathematicians found the answer of the square root of -1 as i. That sounds like a random assumption, but it has essential applications in quantum mechanics, control theory, electrical engineering and many others.

The imaginary numbers can be shown in the two-dimensional number line line with the y-axis being the imaginary numbers and x-axis being the real numbers.

The following picture is an example of the imaginary number line:

When we convert the imaginary numbers back to the real numbers that we are familiar with, we change i to the square root of -1.
1+i = 1+√(-1)
2-2i = 2-√(-4)
-2-3/2i = -2-√(-9/4)
-2+i =-2+√(-1)

Now let’s do some simple calculations with i and get familiar with our new friend.

i = √(-1)                                 i^5 = i or √(-1)
i^2 = -1                                  i^6 = i^2 or -1
i^3 = -√(-1) or –I                     i^7 = i^3 or √(-1)
i^4 = 1                                   i^8 = i^4 or 1

* “^” means to the power of. E.g. i^2 is i to the power of two

The table shows a little secret of i that the value of i^n is equal to i^(n+4x) where x and n are any real numbers.

To make this more visually, let’s use a number line to depict the pattern. We start from 1, when multiply 1 by i, the result is i which turns 1 by 90 degrees. Then, we time i by i which is i^2 = -1, it then turns i by another 90 degrees. We do the same thing to -1 and the result –i turns -1 by 90 degrees again. Another 90 degrees will be turned when we multiply –i by i which is 1.

The idea of imaginary numbers was built from the ancient Greek and was developed gradually by several scientists and mathematicians. It is used widely in a variety of sciences in the real world. For instance, in electrical engineering, the imaginary numbers can describe the timing of voltage relative to current, or current relative to voltage, in an AC circuit.

Now, do you have an idea to imagine your own numbers? =)

About this contributor: Mindylin610 is a tenth grade high school student with special love of random things and a passionate heart toward math and science.  

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