Pre-Calculus

Task #3A

Imaginary numbers are simply an expansion of the system of real numbers, invented by mathematicians to provide solutions to equations requiring an even root of a negative number. The imaginary number is defined by taking the square root of a negative one. Thus, and . By using , we can simplify and perform operations with expressions such as . For example,

CHECK:

The realm of complex numbers includes all numbers that possess a real and imaginary part and can be written in the form where a is the real part and b is the imaginary part. This means that real numbers and imaginary numbers can both be considered complex numbers. Real numbers are simply complex numbers in which and imaginary numbers are those in which . For example, the real number 8 can be written as just as the imaginary number could be written as . In the complex number below, the real and imaginary parts are labeled.

canon

When we add or subtract complex numbers, we simply combine the real parts and the imaginary parts of each complex number as shown in the equations below.

and

The algebra is no different than if were replaced with a variable such as x. If we were to add we would simply combine like terms by adding the 4 and the 5 as well as the -2x and the 3x as shown below.

The method looks identical with complex numbers. For example,

To divide complex numbers algebraically, we must use the complex conjugate of the denominator. The complex conjugate of is and the complex conjugate of is . Thus, the complex conjugates are multiplied the result is the sum of the square of the real part and the square of the imaginary part as shown below.

and

When dividing complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator. To simplify we must multiply by as shown below.

When graphing complex numbers we use...