The addition of pi to the number system made rationals not encompass all numbers. Therefore the system including both rational and irrational numbers is called the real numbers. Once the real numbers were discovered it seemed that they encompassed all numbers. But a closer look at square roots can change that.

Note: The equations below may be confusing. You do not need to fully understand the equations to understand complex numbers. Just try to understand the theory of it.

The square of a positive or a negative number is positive (the square of 0 is just 0). Therefore it seems that no number squares to a negative number, namely -1. Obviously, mathematicians had to come up with a new number for it. Therefore the quantity i was introduced. i was set to equal the square root of -1 forming the equation i^2=-1. i and other multiples of it make up the system called imaginary numbers. Also, the system encompassing all numbers is called the complex numbers. Complex numbers are in the form a+bi where a and b are ordinary real numbers. Make a=0 and you get bi, an imaginary number. Make b=0 and you get the constant a, a real number. In general, any square root of a negative number can be calculated by the simple equation (where square root of x=x^1/2) (-x)^1/2=i(x^1/2) where x is a positive number.

Complex numbers have other applications as well. For example, complex numbers provide solutions to any equation of the form

ax^0+bx^1+cx^2...+zx^n=0

In case the equation above confuses you, here are the basics: A complex number x can make the equations equal 0, and a, b, c, etc. are any numbers. Some of these equations couldn't previously be solved, which fills gaps that real numbers do not fill. The identity equation of this form is

x^2+1=0

This equation is solved by i (i^2=-1, -1+1=0, 0=0) which is the one of the identity complex numbers.

Before reaching the end of this post, one more topic must be covered: the complex plane. The complex plane is grid consisting of a real x-axis (a number line that increases left to right) and an imaginary y-axis (a number line that increases from down to up, except in multiples of i). Say you wanted to go to the point 3+2i. First, go three units right on the real x-axis, and then go two units up on the imaginary y-axis.

Today, phenomena on complex numbers are found in nature and used for physics as well as mathematics.

## Sunday, January 25, 2009

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