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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

Instructor:
*Russell Frith*

This document presents fundamental details for the meaning of complex numbers. A definition is presented which shows the set of complex numbers is a superset of the set of real numbers. The arithmetic operations of complex addition, complex subtraction, complex multiplication, and complex division are presented.

**Complex numbers** are pretend numbers because they do not exist in reality. For example, the square root of -1. Complex numbers provide solutions to many math, science, and engineering problems that would otherwise have no solutions. For instance, consider finding the roots of the quadratic equation: *y* = *x*2 + 4*x* + 1. When graphed in the *x-y* plane, one readily sees that the graph never intersects the *x*-axis and thus has no real roots. With the use of complex numbers, however, this equation can be shown to have two complex roots.

A complex number is any number that can be written as *a + bi*, where *a* and *b* are real numbers and *i* is the square root of -1. In the complex number *a* + *bi*, *a* is called the real part and *b* is called the imaginary part. If *b* = 0, the complex number *a* + *bi* is simply a real number. If *b* is not zero and *a* = 0, the complex number 0 + *bi* (or just *bi*) is an imaginary number. An imaginary number is simply the square root of a negative number. Examples of imaginary numbers are:

Two complex numbers are said to be equal if both their real parts are equal and their imaginary parts are equal. Unlike real numbers, however, one complex number cannot be greater than or less than another.

To add two complex numbers, simply add the real parts of the complex numbers to get the real part of the sum and add the imaginary parts to get the imaginary part of the sum. Subtraction of two complex numbers is performed in the same manner, with the subtraction performed in place of addition. The distributive properties of addition and subtraction apply to complex numbers as well.

Let *z*1 = 4 + 7*i* and *z*2 = -2 + 6*i*. Then *z*1 + *z*2 = 2 + 13*i*.

Let *z*1 = 4 + 7*i* and *z*2 = -2 + 6*i*. Then *z*1 - *z*2 = 6 + *i*.

Let *z*1 = 4 + 7*i*, *z*2 = -2 + 6*i* and *z*3 = 5 - 8*i*. Then *z*1 - (*z*2 + *z*3) = *z*1 - (-7 - 3*i*) = *z*1 + 7 + 3*i* = 11 + 10*i*

Complex numbers may also be multiplied. Let *a* + *bi* be the first complex factor and let *c* + *di* be the second complex factor. When multiplying those two complex numbers, use the FOIL method for multiplying two binomials. The FOIL method stands for first, outer, inner, and last and is a mnemonic used to describe the order of operations in the multiplication of two binomials.

(*a* + *bi*)*(*c* + *di*) = *ac* + *adi* + *bci* + *bdi**2* = (*ac* - *bd*) + (*ad* + *bc*)*i*

Let *z*1 = 4 + 7*i* and *z*2 = -5 - 8*i*.

Then *z*1 * *z*2 = (-20 - 56) + (-32 - 35)*i* = -76 - 67*i*

Note the distributive property of multiplication with respect to addition or subtraction applies.

Let *z*1 = 4 + 7*i*, *z*2 = -5 - 8*i*, and *z*3 = 3 + 2*i*.

Then *z*1 * (*z*2 + *z*3) = *z*1 * *z*2 + *z*1 * *z*3 = 36 - 67*i* - 2 + 21*i* = 34 - 46*i*.

An example of multiplying two complex numbers in radical form is:

The process of dividing two complex numbers consists of multiplying the first complex number by the reciprocal of the second complex number. The symbology for dividing two complex numbers is: *z*1/*z*2 = (*a* + *bi*)/(*c* + *di*). The division may be viewed as a fraction, where the numerator is *a* + *bi* and the denominator is *c* + *di*. To achieve the division, multiply both the numerator and the denominator of the fraction by the complex conjugate of the denominator and then simplify the result. The complex conjugate is determined by flipping the given number's sign from either plus to minus or minus to plus. For example, the complex conjugate of *a* + *bi* is *a* - *bi*.

Let *z*1 = 2 - 3*i* and *z*2 = -7 - *i*.

*z*1/*z*2 = (2 - 3*i*)/(-7 - *i*)

Multiply the numerator and the denominator by -7 + *i* and the result becomes:

(2 - 3*i*)/(-7 - *i*) * (-7 + *i*)/(-7 + *i*).

Going through the multiplication, one gets:

(2 - 3*i*)*(-7 + *i*)/(-7 - *i*)*(-7 + *i*) = (-14 - 2*i* + 21*i* + 3)/(49 + 1) = (-11 + 19*i*)/50.

**Complex numbers** are pretend numbers because they do not exist in reality. Complex numbers fill gaps in math problems that cannot be readily solved by using real numbers. For example, complex numbers allow for the computation of the square roots of negative numbers. When allowances are made for this peculiar idea, then many math problems that previously had no solutions are now solvable. An example of such an application involves finding the roots of a quadratic equation where its discriminant is negative. Such a quadratic equation has no real solutions, but with the inclusion of complex numbers, then that equation can be shown to have two complex conjugate solutions.

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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