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. Thatmakes i=sqrt(-1).
It is important to remember that i is an imaginary number. There will never be a real numberwhose square is negative. Since we graph equations in the real world, solutions that involve iwill 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 theyare put together, you get a complex number.
It is important to remember, however, that every real number, and every imaginary number isalso a complex number. This will come back into play in the chapter on polynomials when theystate 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 realcomponent of the right side and the imaginary component of the left side must equal theimaginary 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 -irespectively.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 byfour. 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 numberinvolving 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 thatthe sign is only changed on the imaginary component. Do not go through and change the signson the real component, only the imaginary component.
To eliminate the imaginary component from a complex number, multiply by its complexconjugate.
This is how division with complex numbers is done. The numerator and denominator ismultiplied by the complex conjugate of the denominator.
Graphing Complex Numbers
To graph a complex number, the real component is the x-coordinate and the imaginarycomponent is the y-coordinate.
