#### Theory of Complex Numbers and Equality of Complex Numbers

Complex Numbers:

Imaginary number, i:

The square of non real number may be negative. That, though, leaves some equations (such as x2 + 1 = 0) with no real solution. The mathematicians do not like things which have no answer. Therefore they came up an imaginary number i like that i2 = -1. That makes i = sqrt(-1).

It is significant to keep in mind that i is an imaginary number. There will be no real number whose square is negative. As we graph equations in the real world, solutions which include i will not show on the graph.

Complex Numbers:

Each and every complex number can be written as ‘a + bi’ where a and b are the real numbers. When b = 0, then it is simply an ordinary real number. When a = 0 and b<>0, then it is an imaginary number. Whenever they are put altogether, you obtain a complex number.

It is very important to keep in mind, though, that each and every real number, and each and every imaginary number is as well a complex number. This will come back to play the chapter on polynomials whenever they define a theorem comprising complex numbers. This is not true for just complex numbers including i, it is true for all real as well.

Equality of Complex Numbers:

When two complex numbers are equivalent, then the real component of left side should equal the real component of the right side and imaginary component of the left side should equal the imaginary component of the right side.

Powers of i:

i must never be written to any power other than 1. This pattern repeats. To simplify i to any power, simply divide the exponent by 4 and observe the remainder. The remainder will either be 0, 1, 2 and 3 and that will correspond to 1, i, -1, and -i correspondingly. The column headings are remainder whenever dividing by 4.

A shortcut to divide the exponent by 4 is to divide the last two digits of the exponent by 4. The remainder will be similar.

Complex Conjugates:

Let consider the square of complex number:

(a + bi)2 = a2 + 2ab i + b2 i2 = (a2 - b2) + 2ab i

Note that whenever you square a complex number including i, you get the other complex number including i.

Now consider this product

(a + bi) (a - bi) = a2 - b2 i2 = a2 + b2

(a + bi) and (a - bi) are termed as complex conjugates, and their product is the real number. Note that the sign is just changed on the imaginary component. Don’t go via and modify the signs on the real component, just the imaginary component.

To remove the imaginary component from the complex number, multiply by its complex conjugate.

This is how the division with complex numbers is completed. The numerator and denominator are multiplied by the complex conjugate of denominator.

Graphing Complex Numbers:

To graph a complex number, real component is the x-coordinate and imaginary component is the y-coordinate.

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