Operations with Complex Numbers

Learning Objective(s)

·         Subtract complex numbers.

·         Multiply complex numbers.

·         Find conjugates of complex numbers.

·         Divide complex numbers.

Introduction

Any time new kinds of numbers are introduced, one of the first questions that needs to be addressed is, “How do you add them?” In this topic, you’ll learn how to add complex numbers and also how to subtract, multiply, and divide them.

First, consider the following expression.

(6x + 8) + (4x + 2)

To simplify this expression, you combine the like terms, 6x and 4x. These are like terms because they have the same variable with the same exponents. Similarly, 8 and 2 are like terms because they are both constants, with no variables.

(6x + 8) + (4x + 2) = 10x + 10

In the same way, you can simplify expressions with radicals.

You can add  to  because the two terms have the same radical, , just as 6x and 4x have the same variable and exponent.

The number i looks like a variable, but remember that it is equal to . The great thing is you have no new rules to worry about—whether you treat it as a variable or a radical, the exact same rules apply to adding and subtracting complex numbers. You combine the imaginary parts (the terms with i), and you combine the real parts.

 Example Problem Add. (−3 + 3i) + (7 – 2i) −3 + 3i + 7 – 2i = −3 + 7 + 3i – 2i Rearrange the sums to put like terms together. Answer −3 + 7 = 4 and 3i – 2i = (3 – 2)i = i   (−3 + 3i) + (7 – 2i) = 4 + i Combine like terms.

 Example Problem Subtract. (−3 + 3i) – (7 – 2i) (−3 + 3i) – (7 – 2i) = −3 + 3i – 7 + 2i Be sure to distribute the subtraction sign to all terms in the subtrahend. −3 – 7 + 3i + 2i Rearrange the terms to put like terms together. Answer −3 – 7 = −10 and 3i + 2i = (3 + 2)i = 5i   (−3 + 3i) – (7 – 2i) = 10 + 5i Combine like terms.

 Subtract. (5 + 3i) – (3 – i)   A) 2 + 4i   B) 6   C) 2 + 2i   D) 8 + 2i   Show/Hide Answer

Multiplying Complex Numbers

Again, consider the following expression. Before reading further, consider how you would simplify it.

(5x)(3x)

You can simplify by multiplying the coefficients together, then the variables.

 (5x)( −3x) = (5)( −3)(x)(x) = −15x2

Multiplying two imaginary (but not complex!) numbers together works in a similar way, but there is an additional step. Start with the same method to multiply 5i and 3i.

 (5i)( −3i) = (5)( −3)(i)(i) = −15i2

This seems fine so far, but the i2 can be simplified further.

When you multiply a square root by itself, you get the number under the radical. This is what square root means.

Well, i is also a square root. It’s equal to .

So, the final step to simplifying (5i)(3i) = 15i2 is to replace i2 with 1.

 (5i)( −3i) = (5)( −3)(i)(i) = −15i2 = −15(−1) = 15

 Example Problem Multiply. (3i)(2i) (3i)(2i) = (3)(2)(i)(i) = 6i2 Multiply the coefficients of i together, and then multiply i times i. 6i2 = 6(−1)   6(−1) = −6 Replace i2 with –1.   Multiply. Answer (3i)(2i) = −6

Notice that the product of two imaginary numbers is a real number! We will see this again when we multiply two complex numbers.

 Multiply and simplify. (3i)( −i)   A) 3   B) −3   C) 3i   D) −3i2   Show/Hide Answer

Using the Distributive Property

The following expression is a little more complicated because two binomials are being multiplied. This means you have to use the Distributive Property of Multiplication. (Remember that multiplying using the FOIL method—First, Outside, Inside, Last—is an application of the distributive property of multiplication.) Once the binomials have been multiplied, simplify the expression by combining like terms.

 (6x + 8)(4x + 2) = 6x(4x + 2) + 8(4x + 2) = 6x(4x) + 6x(2) + 8(4x) + 8(2) = 24x2 + 12x + 32x + 16 = 24x2 + 44x + 16

Again, in the same way, you can multiply complex numbers. At the end, you will need to simplify i2.

 Example Problem Multiply and simplify. (6 + 8i)(4 + 2i) (6 + 8i)(4 + 2i) 6(4 + 2i) + 8i(4 + 2i) 6(4) + 6(2i) + 8i(4) + 8i(2i) 24 + 12i + 32i + 16i2 Two binomials are being multiplied, so you need to use the Distributive Property of Multiplication.   We could have used FOIL and gone directly to the line 6(4) + 6(2i) + 8i(4) + 8i(2i) . 24 + 44i + 16i2 Combine like terms. 24 + 44i + 16(-1) 24 + 44i – 16 8 + 44i Replace i2 with −1 and simplify. Answer (6 + 8i)(4 + 2i) = 8 + 44i

In this case, the product of two complex numbers is complex. But in the following example, the product is real, not complex. See if you can figure out why!

 Example Problem Multiply and simplify. (6 + 8i)(6 – 8i) (6 + 8i)(6 – 8i) 6(6) + 6(–8i) + 8i(6) + 8i(–8i) 36 – 48i + 48i – 64i2 Use FOIL to expand the product. 36 – 64i2 Combine like terms. 36 – 64(−1) 36 + 64 100 Replace i2 with −1 and simplify. Answer (6 + 8i)(6 – 8i) = 100

Just as  and  are conjugates, 6 + 8i and 6 – 8i are conjugates. (Again, i is a square root, so this isn’t really a new idea.) When the numbers are complex, they are called complex conjugates. Because conjugates have terms that are the same except for the operation between them (one is addition and one is subtraction), the i terms in the product will add to 0. In the example above, 48i was added to 48i, and that sum is 0, so there was no i term in the final product. That means the product of complex conjugates will always be a real (not complex) number.

 Multiply. (9 + i)(9 – i)   A) 82 + 18i   B) 80 – 18i   C) 80   D) 82   Show/Hide Answer

Division of Complex Numbers

So far, each operation with complex numbers has worked just like the same operation with radical expressions. This should no longer be a surprise—the number i is a radical, after all, so complex numbers are radical expressions!

Let’s look at division in two parts, like we did multiplication. First, let’s look at a situation in which the divisor is a monomial.

 Example Problem Simplify. −24i ÷ 6 Treat the division as a fraction. Simplify the fraction using a factor that the numerator and denominator have in common. Answer −24i ÷ 6 = −4i Since the result has no denominator, no more simplification is needed.

 Example Problem Simplify. 32i ÷ 6i Treat the division as a fraction. Simplify the fraction using a factor that the numerator and denominator have in common. Note in this case, i is part of the common factor. Answer 32i ÷ 6i  = The fraction is in simplest form.

 Example Problem Simplify. 56 ÷ −7i Treat the division as a fraction. Simplify the fraction using a factor that the numerator and denominator have in common. In this case, the denominator still has i in it. Since i is a radical, you should simplify further by rationalizing the denominator. Since the denominator is just one term, you don’t need to think about complex conjugates. Just multiply by 1 in the form  and simplify. (Remember, the product of two imaginary numbers is real, so the denominator is real.) Answer 56 ÷ −7i  = 8i

 Simplify. 12 ÷ 10i   A)   B)   C)   D)   Show/Hide Answer

When the divisor (that is, the denominator in the fraction) is a complex number with non-zero real and imaginary parts, you must rationalize the denominator using the complex conjugate. Remember that the product of a complex number with its complex conjugate is always a real number, so the denominator will be a real number. That means the result will be equivalent, but rationalized.

 Example Problem Simplify. (56 – 8i) ÷ (14 + 10i) Treat the division as a fraction. Simplify the fraction using a factor that the numerator and denominator have in common, if any.   Be careful to use the distributive property—the numbers must be a factor of all terms. In this case, the denominator still has i in it. To rationalize the denominator, multiply by the complex conjugate of the denominator. In this case, the complex conjugate is (7 – 5i).   (For complex conjugates, the real parts are equal and the imaginary parts are additive inverses.) Expand the numerator and the denominator. Remember, the denominator should be a real number (no i term) if you chose the correct complex conjugate and performed the multiplication correctly. Replace i2 with −1 and simplify. Be sure to replace i2 in both the numerator and the denominator! The quotient can be written in the form a + bi using fractions for both a and b. Always check the final product to see if you can simplify further. In this case, both fractions can be simplified. Answer (56 – 8i) ÷ (14 + 10i) =

 Simplify. (10 + 6i) ÷ (5 – 3i)   A)   B) 2 – 2i   C)   D)   Show/Hide Answer

 Operations with Complex Numbers   To add or subtract, combine like terms.   To multiply monomials, multiply the coefficients and then multiply the imaginary numbers i. If i2 appears, replace it with −1.   To multiply complex numbers that are binomials, use the Distributive Property of Multiplication, or the FOIL method. Multiply the resulting terms as monomials.   To divide, treat the quotient as a fraction. ·         Simplify the numerical parts, and then rationalize the denominator, if needed. ·         Replace i2 in both the numerator and the denominator with −1, as needed. ·        Write the answer in a + bi form, which may require further simplification of a and b when they are fractions.

Summary

Complex numbers can be added, subtracted, multiplied, and divided using the same ideas you used for radicals and variables. With multiplication and division, you may need to replace i2 with −1 and simplify further.