2.3 - Complex Numbers

Imaginary number, i

The square of no real number can be negative. That, however, leaves certain equations (like x2+1=0) with no real solution. We mathematicians don't like things not to have an answer. So, we came up an imaginary number i such that i2 = -1. That makes i=sqrt(-1).

It is important to remember that i is an imaginary number. There will never be a real number whose square is negative. Since we graph equations in the real world, solutions that involve i will not appear on the graph.

Complex Numbers

Every complex number can be written as "a + bi" where a and b are real numbers. If b=0, then it is just an ordinary real number. If a=0 and the b<>0, then it is an imaginary number. When they are put together, you get a complex number.

It is important to remember, however, that every real number, and every imaginary number is also a complex number. This will come back into play in the chapter on polynomials when they state a theorem involving complex numbers. It is not true for just complex numbers involving i, it is true for all reals also.

Equality of Complex Numbers

If two complex numbers are equal, then the real component of the left side must equal the real component of the right side and the imaginary component of the left side must equal the imaginary component of the right side.

Powers of i

i should never be written to any power other than 1.

0 1 2 3
i0 = 1 i1 = i i2 = -1 i3 = -i
i4 = 1 i5 = i i6 = -1 i7 = -i
i8 = 1 i9 = i i10 = -1 i11 = -i

This pattern repeats. To simplify i to any power, just divide the exponent by 4 and look at the remainder. The remainder will either be 0, 1, 2, or 3, and that will correspond to 1, i, -1, and -i respectively. The column headings are the remaineder when dividing by 4.

A shortcut to dividing the exponent by four is to divide the last two digits of the exponent by four. The remainder will be the same.

Complex Conjugates

Consider the square of a complex number

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

Notice that when you square a complex number involving i, you get another complex number involving i.

However, now consider this product

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

(a+bi) and (a-bi) are called complex conjugates, and their product is a real number. Notice that the sign is only changed on the imaginary component. Do not go through and change the signs on the real component, only the imaginary component.

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

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

Graphing Complex Numbers

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