Perform computations involving complex numbers.
Some functions have zeros that are not real numbers. In order to find the zeros of such functions, we must consider the complex-number system.
We know that the square root of a negative number is not a real number. For example, is not a real number because there is no real number x such that . This means that certain equations, like or , do not have real-number solutions, and certain functions, like , do not have real-number zeros. Consider the graph of .
We see that the graph does not cross the x-axis and thus has no x-intercepts. This illustrates that the function has no real-number zeros. Thus there are no real-number solutions of the corresponding equation .
We can define a nonreal number that is a solution of the equation .
To express roots of negative numbers in terms of i, we can use the fact that
when p is a positive real number.
Express each number in terms of i.
Now Try Exercises 1, 7, and 9.
The complex numbers are formed by adding real numbers and multiples of i.
Note that either a or b or both can be 0. When , , so every real number is a complex number. A complex number like or 17i, in which , is called an imaginary number. A complex number like 17i or −4i, in which and , is sometimes called a pure imaginary number. The relationships among various types of complex numbers are shown in the following figure.
The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials. We collect the real parts and the imaginary parts of complex numbers just as we collect like terms in binomials.
Add or subtract and simplify each of the following.
Now Try Exercises 11 and 21.
When set in mode, most graphing calculators can perform operations on complex numbers. The operations in Example 2 are shown in the following window. Some calculators will express a complex number in the form (a, b) rather than .
When and are real numbers, , but this is not true when and are not real numbers. Thus,
But
Keeping this and the fact that in mind, we multiply with imaginary numbers in much the same way that we do with real numbers.
Multiply and simplify each of the following.
Now Try Exercises 31, 39, and 55.
We can multiply complex numbers on a graphing calculator set in mode. The products found in Example 3 are shown below.
Recall that −1 raised to an even power is 1, and −1 raised to an odd power is −1. Simplifying powers of i can then be done by using the fact that and expressing the given power of i in terms of . Consider the following:
Note that the powers of i cycle through the values i, −1, −i, and 1.
Simplify each of the following.
Now Try Exercises 79 and 83.
These powers of i can also be simplified in terms of rather than . Consider in Example 4(a), for instance. When we divide 37 by 4, we get 9 with a remainder of 1. Then , so
The other examples shown above can be done in a similar manner.
Conjugates of complex numbers are defined as follows.
The conjugate of a complex number is . The numbers and are complex conjugates.
Each of the following pairs of numbers are complex conjugates:
The product of a complex number and its conjugate is a real number.
Multiply each of the following.
Now Try Exercise 49.
Conjugates are used when we divide complex numbers.
Divide by .
We write fraction notation and then multiply by 1, using the conjugate of the denominator to form the symbol for 1.
Now Try Exercise 69.
With a graphing calculator set in mode, we can divide complex numbers and express the real and imaginary parts in fraction form, just as we did in Example 6.
Express the number in terms of i.
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Simplify. Write answers in the form , where a and b are real numbers.
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Simplify.
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89. Write a slope–intercept equation for the line containing the point (3, −5) and perpendicular to the line . [1.4]
Given that and , find each of the following. [2.2]
90. The domain of
91. The domain of
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94. For the function , construct and simplify the difference quotient
Determine whether the statement is true or false.
95. The sum of two numbers that are conjugates of each other is always a real number.
96. The conjugate of a sum is the sum of the conjugates of the individual complex numbers.
97. The conjugate of a product is the product of the conjugates of the individual complex numbers.
Let and .
98. Find a general expression for .
99. Find a general expression for .
100. Solve for z.
101. Multiply and simplify: