An imaginary number is represented as ai where a is any positive or negative number, and i is a special constant whose square equals -1 (This means i*i = -1). While this constant may not exist in the normal number system, we may, for convenience, imagine it may, and we may then take the square root out of any negative number (sqrt(-4) == sqrt(4)*sqrt(-1) == 2*i). This has been very useful in physics.

The basic imaginary number i is the number whose square equals -1. Since it's impossible to multiply two identical real numbers together to produce a negative number, some mathematician just pulled the idea of imaginary numbers out of thin air, called it i, and proceeded to describe their attributes based on what is known about real numbers.

If you picture the imaginary number line as being perpendicular to the real number line, then powers of i go in a circle -- something unique in mathematics, or at least in the part of it I've studied. How is this possible? Take a look at the chart below, remembering that i * i = -1 by definition:

  • i^1 = i
  • i^2 = i * i = -1
  • i^3 = i * i * i = -1 * i = -i
  • i^4 = i * i * i * i = -1 * -1 = 1
  • i^5 = i * i * i * i * i = 1 * i = i
  • i^6 = i * i * i * i * i * i = 1 * -1 = -1
You get the idea. It keeps looping, and raising i to a power of a multiple of 4 always yields a result of 1.

To represent an imaginary number other than i itself, treat i as any ordinary variable by sticking a numerical coefficient in front of it to represent how big an imaginary quantity you've got. For example, 3i is the square root of -9. Don't confuse this with something like 3 + i, though -- that's a complex number, not on the real or imaginary number lines but on the same plane as both.

In basic mathematics we are taught the concept of the number line.This line extends horizontally from minus infinity on the left to plus infinity on the right with 0 being the center.

Any point on this line represents what is called a real number.The sum, difference, product or quotient of any two real numbers is also a real number.

However, there are some special numbers which are not represented on this line.These are called imaginary numbers. The imaginary numbers can be thought of as existing on another line perpendicular to the real number line.

The first integer on this imaginary number line is denoted by the symbol i, which represents the square root of -1(In electrical engineering math, the symbol j is used instead as i usually is the symbol for current).

The sum of an imaginary number and real number gives us a class of numbers called complex numbers. These are numbers which lie anywhere on the plane defined by these two number lines. They are usually written as (a + bi) or alternately, they can be represented by a length and phase angle which are the distance and angle of the point from the intersection of the two number lines.

There are various rules which govern how to perform basic operations on complex numbers.

Complex numbers and imaginary numbers are used extensively in problems involving alternating current circuits.

Believe it or not, imaginary numbers aren't these mystical things that don't exist except in some obscure mathematician's head. They are real things that you can find real life examples of. So my goal in this write-up is to explain what they are without resorting to weirdness like "the imaginary number is the square root of negative one".

So let's start easy. Suppose I ask you to show me the number three. Now, you can't show me "three", but you can certainly show me three of something, like three apples, three dollars, or three steps forward. So numbers exist in the real world as a count or a measure of something.

Let's try something a little harder: show me negative three. Well... you can't really show me three negative apples, but you can give me negative three dollars by taking three dollars from me, and you can show me negative three steps forward by walking three steps backwards. A negative number is a positive number done backwards; to do something a negative number of times is to do the opposite a positive number of times.

Why do we have negative numbers? Because sometimes it's more convenient to say "minus three" than to say "do the opposite three times." For example, suppose that you're filling out a form and there's a blank which says, "How much money have you given to charities this year?" And let's say that you are actually a cradle-robber who steals candy from impoverished babies. Rather than wasting your time scratching off the words "given to" and writing in "taken from", you can instead lazily write in the negative of the dollar amount of the candy you stole and mean the same thing.

So if negative numbers are just the opposite of what you would do with a positive number, what's an imaginary number? An "imaginary number" is when you do something countable, but it was neither what you were expecting to count, nor the opposite of what you were expecting to count. So say you wanted to count how many steps a girl was going to walk forward, but then she surprises you by walking five steps left instead. Since she did not travel either forwards or backwards, neither a positive nor a negative number will work here. At this point you have two options. First, you could scratch out the word "forwards" on your notepad and replace it with "left". Or... if you are too lazy, you could instead write down that she walked an imaginary five steps forward. So just as negative numbers are a convenient way of indicating we mean the opposite of whatever positive means, imaginary numbers are a convenient way of indicating we mean the "left" of what positive meant.

Now, here's a cute trick: If walking left five steps is walking forward an imaginary five steps, what would it mean to walk an imaginary number of steps left? Well, since in this case imaginary already means "left", that means you are walking five steps left of left, or five steps backwards! So if you do the "imaginary" thing an "imaginary" number of times, you get the opposite of what you are doing! That this works is nothing more magical than saying when you do the opposite of the opposite of what you meant to do, then you are really just doing what you meant to do in the first place. Just like many actions have natural opposites, so do some actions have a natural "imaginary".

When we want to say using fancy mathematical equations that the opposite of the opposite is the original, we say that (-1) * (-1) = 1, or equivalently (-1)2=1. Likewise, when we want to say that the imaginary of an imaginary is the opposite, we can say that i * i = i2 = -1. There is no mathematical trickery here; the only trickery lay in finding an "imaginary" way of doing something, which you cannot always do, just like you can't always find an "opposite". It is as puzzling to figure out what it means to give you an imaginary amount of money as it is to figure out what how one can physically hold a negative number of apples.

So in conclusion, the imaginary number, i, is more than just some crazy symbol with the property that i2=-1; it is rather real concept that has practical meaning behind it, just like positive and negative numbers.

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