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General Studies Math: Help & Review8 chapters | 85 lessons

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Lesson Transcript

Instructor:
*David Liano*

After completing this lesson, you will be able to describe and define complex conjugates. You will also be able to write complex conjugates and use them appropriately to solve problems.

Before we define complex conjugate and complete examples, you first need to understand conjugates and complex numbers. In math, a **conjugate** is formed by changing the sign between two terms in a binomial. For instance, the conjugate of the binomial *x* - *y* is *x* + *y*. These two binomials are conjugates of each other.

A **complex number** is a number that has a real part and an imaginary part. Real numbers contain all the rational numbers (e.g. the whole numbers 0 and 7, the integer -5, and the fraction 2/3) and all the irrational numbers (e.g. pi and square root of 3). The square root of a negative number does not exist within the system of real numbers. Therefore, mathematicians expanded the systems of numbers and created the imaginary unit *i*, which is defined as the square root of -1. This allows us to write the square root of any negative number and to solve problems that need the square root of a negative number. Notice that if we square the imaginary unit, we get -1. In other words, *i*^2 = -1.

The standard form of a complex number is *a* + *bi* where *a* and *b* are real numbers. The letter *a* represents the real part of the complex number, and the term *bi* represents the imaginary part of the complex number. Here is an example of a complex number: 8 + 3*i*. 8 is the real part of the number, and 3*i* is the imaginary part.

Now let's combine the above definitions. A **complex conjugate** is formed by changing the sign between two terms in a complex number. Let's look at an example: 4 - 7*i* and 4 + 7*i*. These complex numbers are a pair of complex conjugates. The real part (the number 4) in each complex number is the same, but the imaginary parts (7*i*) have opposite signs.

Let's multiply and simplify the following pair of complex conjugates:

(3 - 5*i*) (3 + 5*i*)

Use the FOIL (which stands for first, outer, inner, last) method to get 9 + 15*i* - 15i - 25*i*^2

Combine like terms to get 9 - 25*i*^2

Substitute -1 for *i*^2 to get 9 - 25(-1)

Simplify to get 9 + 25 = 34

Notice that the terms 15*i* and -15*i* cancel out and that *i*^2 is changed to -1. Therefore, we have eliminated the imaginary parts of the original pair of complex conjugates and are left with a real number; in this case a whole number. This will always happen when we multiply a pair of complex conjugates. The product of a pair of complex conjugates is always a real number.

Complex conjugates can be a useful tool when simplifying expressions with complex numbers. For instance, multiplying a complex number by its conjugate is very useful when simplifying certain fractions.

Let's consider the following fraction: (3 - 4*i*) / (1 + *i*)

This fraction is not simplified because there is an imaginary part in the denominator. An imaginary part in the numerator is all right but not in the denominator. We need to get rid of the *i* in the denominator. One reason for this rule is that fractions are usually easier to add and subtract when the denominator is not a complex number.

We now know that we could multiply the denominator by its conjugate to get rid of the *i*. However, that would change the value of the fraction. But we should also know that when you multiply the numerator and denominator of a fraction by the same number or expression we end up with a fraction that is equivalent to the original fraction. For example, let's start with the fraction of 1/2. Next, multiply the numerator and denominator by 3: (1 x 3) / (2 x 3) = 3/6. We end up with a fraction that is equivalent to 1/2.

Therefore, in our original problem, we must also multiply the numerator by the conjugate of the denominator.

After completing the multiplication and combining like terms, we'll end up with this expression:

We can now replace *i*^2 with -1 and simplify:

(3 - 7*i* + 4(-1)) / (1 - (-1)) = (3 - 7*i* - 4) / (1 + 1) = (-1 - 7*i*) / 2

There are no longer any imaginary parts in the denominator, and this is what we want. (Note: make sure to watch out for double negatives when simplifying.)

You may remember from your algebra classes that a quadratic equation sometimes does not intersect the *x*-axis. When this happens, the solutions to the quadratic equation are complex numbers. You usually calculate these types of solutions using the quadratic formula. Well, the complex solutions of a quadratic function that has real numbers as coefficients always occur in complex conjugate pairs.

For instance, if you are told that a quadratic function with real numbers as coefficients has a solution of -2 + *i*, then -2 - *i* must also be a solution. Let's complete an example: A quadratic equation with real numbers as coefficients has 5 - 2*i* as a solution. Write the quadratic equation in standard form.

If 5 - 2*i* is one solution of the quadratic equation, then 5 + 2*i* must be the other solution. We can now write the equation in intercept, or factored, form: *y* = (*x* - (5 - 2*i*)) (*x* - (5 + 2*i*))

We need to get rid of the inner parentheses in each factor and get *y* = (*x* - 5 + 2*i*) (*x* - 5 - 2*i*)

If we complete the multiplication, we get the following *y* = *x*^2 - 5*x* - 2*xi* - 5*x* + 25 + 10*i* + 2*xi* - 10*i* - 4*i*^2

Next, change *i*^2 to -1 so that -4*i*^2 becomes -4(-1) and equals +4: *y* = *x*^2 - 5*x* - 2*xi* - 5*x*+ 25 + 10*i* + 2*xi* - 10*i* + 4

Now we can combine like terms (some terms cancel each other out) and get the following quadratic equation: *y* = *x*^2 -10*x* + 29.

Complex conjugates are a simple concept but are valuable when simplifying some types of fractions. A pair of **complex conjugates** is formed by changing the sign between two terms in a complex number. To create a conjugate of a complex number, just rewrite it and change the sign of the imaginary part

Some other things to remember include:

- A
**complex number**is a number that has an imaginary part. - The product of a pair of complex conjugates is always a real number.
- An imaginary part in the numerator of a fraction is alright but not in the denominator.
- The complex solutions of a quadratic function that has real numbers as coefficients always occur in complex conjugate pairs.

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General Studies Math: Help & Review8 chapters | 85 lessons

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