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19 Complex Number System

Topics Covered:

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:
  1. Evaluate Square Root of a Negative Number
  2. Add or Subtract Complex Numbers
  3. Multiply Complex Numbers
  4. Divide Complex Numbers
  5. Negative Roots
  6. Key Concepts

Evaluate the Square Root of a Negative Number

Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify \sqrt{-1},  we are looking for a real number x so that x2 = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.

Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like \sqrt{5}. All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.

But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the imaginary unit i as the number whose square is –1.

Imaginary Unit

The imaginary unit i is the number whose square is –1.

i2 = −1 or i = \sqrt{-1}
We will use the imaginary unit to simplify the square roots of negative numbers.
Square Root of a Negative Number

If b is a positive real number, then

\sqrt{-b} = \sqrt{b}i

We will use this definition in the next example. Be careful that it is clear that the i is not under the radical. Sometimes you will see this written as \sqrt{-b} = i\sqrt{b} to emphasize the i is not under the radical. But the \sqrt{-b} = \sqrt{b}i is considered standard form.

Try it!

Write each expression in terms of i and simplify if possible:

a. \sqrt{-25}  b. \sqrt{-7}  c. \sqrt{-12}

Solution:

a.

Steps Algebraic
Expression \sqrt{-25}
Use the definition of the square root of negative numbers. \sqrt{25}i
Simplify. 5i

b.

Steps Algebraic
Expression \sqrt{-7}
Use the definition of the square root of negative numbers. \sqrt{7}i
Simplify. Be careful that it is clear that i is not under the radical sign.

c.

Steps Algebraic
Expression \sqrt{-12}
Use the definition of the square root of negative numbers. \sqrt{12}i
Simplify 2\sqrt{3}i

Now that we are familiar with the imaginary number i, we can expand the real numbers to include imaginary numbers. The complex number system includes the real numbers and the imaginary numbers. A complex number is of the form a + bi, where a, b are real numbers. We call a the real part and b the imaginary part.

Complex Number

A complex number is of the form a + bi, where a and b are real numbers.

The image shows the expression a plus b i. The number a is labeled “real part” and the number b i is labeled “imaginary part”.

A complex number is in standard form when written as  a + bi ,where a and b are real numbers.

If b = 0,   then a + bi becomes a + 0 · i = a, and is a real number.

If b ≠ 0,   then a + bi is an imaginary number.

If a = 0,    then a + bi becomes 0 + bi = bi, and is called a pure imaginary number.

We summarize this here.

a + bi
b = 0 a + 0 · i = a Real number (a)
b ≠ 0 a + bi Imaginary number
a = 0 0 + bi = bi Pure imaginary number

The standard form of a complex number is a + bi, so this explains why the preferred form is \displaystyle\sqrt{-b}=\sqrt{b}i when b > 0.

The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.

The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and “Real number”. The third row contains b is not equal to 0, a plus b i, and “Imaginary number”. The fourth row contains a = 0, 0 plus b i, and “Pure imaginary number”.
Expressing an Imaginary Number in Standard Form

Try it!

Express \sqrt{-9} in standard form.

Solution:
\sqrt{-9} =\sqrt{9} \sqrt{-1}
= 3i

In standard form, this is 0 + 3i.

Add or Subtract Complex Numbers

We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers—just as we did with the real numbers.

Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.

Try it!

Add: \sqrt{-12} +\sqrt{-27}.

Solution:

Steps Algebraic
Expression \sqrt{-12} +\sqrt{-27}
Use the definition of the square root of negative numbers. \sqrt{12}i +\sqrt{27}i
Simplify the square roots. 2\sqrt{3}i + 3\sqrt{3}i
Add. 5\sqrt{3}i

Remember to add both the real parts and the imaginary parts in this next example.

Adding complex numbers:

                                                               (a + bi) + (c + di) = (a + c) + (b + d)i

Subtracting complex numbers:

                                                               (a + bi) − (c + di) = (a − c) + (b − d)i

Given two complex numbers, find the sum or difference.

  1. Identify the real and imaginary parts of each number.
  2. Add or subtract the real parts.
  3. Add or subtract the imaginary parts.

Try it!

Add or subtract as indicated.

  1. (3 − 4i) + (2 + 5i)
  2. (−5 + 7i) − (−11 + 2i)

Solution:

We add the real parts and add the imaginary parts.

  1. Steps Algebraic
    (3 − 4i) + (2 + 5i) = 3 −4i + 2 + 5i
    Group real and imaginary parts
    		 = 3 + 2 + (−4i) + 5i
     Simplify = (3 + 2) + (−4 + 5)i
     Solution = 5 + i
  2. Steps Algebraic
    (−5 + 7i) − (−11 + 2i) = −5 + 7i + 11 − 2i
    Group real and imaginary parts
    		 = −5 + 11 + 7i − 2i
     Simplify = (−5 + 11) + (7 − 2)i
     Solution = 6 + 5i

Multiply Complex Numbers

Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.

Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example, 3(6 + 2i):

Multiplication of a real number and a complex number. The 3 outside of the parentheses has arrows extending from it to both the 6 and the 2i inside of the parentheses. This expression is set equal to the quantity three times six plus the quantity three times two times i; this is the distributive property. The next line equals eighteen plus six times i; the simplification.

Try it!

Multiply: 2i (7 − 5i).

Solution:

Steps Algebraic
Expression 2i (7 − 5i)
Distribute. 14i − 10i2
Simplify i2. 14i − 10(−1)
Multiply. 14i + 10
Write in standard form. 10 + 14i

Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term, i2, it equals −1.

               (a + bi)(c + di) = ac + adi + bci + bdi2
                                   = ac + adi + bci − bd           i2 = −1
                                   = (ac − bd) + (ad + bc)i           Group real terms and imaginary terms
How to multiply a complex number by a complex number.
  • Use the distributive property or the FOIL method.
  • Remember that i2 = −1
  • Group together the real terms and the imaginary terms

Try it!

Multiply:  (3 + 2i)(4 − 3i).

Solution:

Steps Algebraic
Expression (3 + 2i)(4 − 3i)
Use FOIL. 12 − 9i + 8i − 6i2
Simplify i2 and combine like terms. 12 − i − 6(−1)
Multiply. 12 − i +6
Combine the real parts. 18 − i

In the next example, we could use FOIL or the Product of Binomial Squares Pattern.

Try it!

Multiply: (3 + 2i)2

Solution:

Steps Algebraic
Example
Use the Binomial Squares Pattern formula the quantity a plus b in parentheses squared equals a squared plus 2 a b plus b squared. Image shows parenthesis 3 with an a above it, plus 2i with a b above it, parenthesis and it is squared.
Use the Product of Binomial Squares Pattern,

(a + b)2 = a2 + 2ab + b2.
Applied to this example we get the expression 3 squared plus 2 times 3 times 2 i plus the quantity 2 i in parentheses squared.
Simplify.
Simplifying we get 9 plus 12 i plus 4 i squared.
Simplify i2.
Simplifying further we get 9 plus 12 i plus 4 times negative 1.
Simplify.
The final simplified version is 5 plus 12

Divide Complex Numbers

Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form a + bi. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of a + bi is a − bi.

We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form (a − b), (a + b).

A complex conjugate pair is very similar. For a complex number of the form a + bi, its conjugate is  a − bi. Notice they have the same first term and the same last term, but one is a sum and one is a difference.

Complex Conjugate Pair

 A complex conjugate pair is of the form  a + bi, a − bi.

From our study of polynomials, we know the product of conjugates is always of the form (a − b)(a + b) = a2 − b2. The result is called a difference of squares. We can multiply a complex conjugate pair using this pattern.

Earlier for multiplying complex numbers we used FOIL. Now we will use the Product of Conjugates Pattern.

he quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.When we multiply complex conjugates, the product of the last terms will always have an i2 which simplifies to −1.

(a − bi)(a + bi)
a2 − (bi)2
a2 − b2i2
a2 − b2(−1)
a2 + b2
This leads us to the Product of Complex Conjugates Pattern: (a − bi)(a + bi)= a2 + b2
Complex Conjugates

If a and b are real numbers, then

(a − bi)(a + bi) = a2 + b2

Note that complex conjugates have an opposite relationship: The complex conjugate of a + bi is a − bi, and the complex conjugate of a − bi is a + bi.

The complex conjugate of a complex number a + bi is a − bi. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

  • When a complex number is multiplied by its complex conjugate, the result is a real number.
  • When a complex number is added to its complex conjugate, the result is a real number.

Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.

How to divide complex numbers.

  1. Write both the numerator and denominator in standard form.
  2. Multiply the numerator and denominator by the complex conjugate of the denominator.
  3. Simplify and write the result in standard form.

Try it!

  1. Divide: \displaystyle\frac{4+3i}{3-4i}

Solution:


tep 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.
2. Divide, writing the answer in standard form: \displaystyle\frac{-3}{5+2i}

Solution:

Steps Algebraic
Example \displaystyle\frac{-3}{5+2i}
Multiply the numerator and denominator by the complex conjugate of the denominator. \displaystyle\frac{-3(5-2i)}{(5+2i)(5-2i)}
Multiply in the numerator and use the Product of Complex Conjugates Pattern in the denominator. \displaystyle\frac{-15+6i}{5^{2}+2^{2}}
Simplify. \displaystyle\frac{-15+6i}{29}
Write in standard form. –\displaystyle\frac{15}{29} + \displaystyle\frac{6}{29}i
3. Divide:  (2 + 5i) by (4 − i).

Solution:

Steps Algebraic
We begin by writing the problem as a fraction. \displaystyle\frac{(2+5i)}{(4−-i)}
Then we multiply the numerator and denominator by the complex conjugate of the denominator. \displaystyle\frac{(2+5i)}{(4-−i)} ⋅ \displaystyle\frac{(4+i)}{(4+i)}
To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL). \displaystyle\frac{(2+5i)}{(4-−i)} ⋅ \displaystyle\frac{(4+i)}{(4+i)} = \displaystyle\frac{8+2i+20i+5i^2}{16+4i−-4i-i^2}
Because i2 = −1 \displaystyle\frac{8+2i+20i+5(-1)}{16+4i-4i-(-1)}
Simplify = \displaystyle\frac{3+22i}{17}
Separate real and imaginary parts. \displaystyle\frac{3}{17} + \displaystyle\frac{22}{17}i 
Note that this expresses the quotient in standard form.
4. Divide:  \displaystyle\frac{5+3i}{4i}

Solution:

Steps Algebraic
Example
\displaystyle\frac{5+3i}{4i}
Write the denominator in standard form.
\displaystyle\frac{5+3i}{0+4i}
Multiply the numerator and denominator bythe complex conjugate of the denominator.
\displaystyle\frac{(5+3i)(0-4i)}{(0+4i)(0-4i)}
Simplify.
\displaystyle\frac{(5+3i)(-4i)}{(4i)(-4i)}
Multiply.
\displaystyle\frac{-20i-12i^2}{-16i^2}
Simplify the i2.
\displaystyle\frac{-20i+12}{16}
Rewrite in standard form. \displaystyle\frac{12}{16}-\frac{20}{16}i
Simplify the fractions. \displaystyle\frac{3}{4} – \displaystyle\frac{5}{4}i

Negative Roots

Since the square root of a negative number is not a real number, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using \sqrt{-b} = \sqrt{b}i. This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.

Principal Square Root of a Negative Number

 \sqrt{-b} = i\sqrt{b}

Try it!

Multiply:  \sqrt{-36} • \sqrt{-4}

Solution:

To multiply square roots of negative numbers, we first write them as complex numbers.

Steps Algebraic
Example  \sqrt{-36} • \sqrt{-4}
Write as complex numbers using \sqrt{-b} = \sqrt{b}i.  \sqrt{36}i • \sqrt{4}i
Simplify. 6i · 2i
Multiply. 12i2
Simplify i2 and multiply. −12

In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.

Try it!

Multiply: (3−\sqrt{-12})(5+\sqrt{-27}).

Solution:

To multiply square roots of negative numbers, we first write them as complex numbers.

Steps Algebraic
Example (3−\sqrt{-12})(5+\sqrt{-27})
Write as complex numbers using \sqrt{-b} = \sqrt{b}i (3 − 2\sqrt{3}i)(5 + 3\sqrt{3}i)
Use FOIL. 15 + 9\sqrt{3}i − 10\sqrt{3}i − 6 · 3i2
Combine like terms and simplify i2. 15 − \sqrt{3}i − 6 · (−3)
Multiply and combine like terms. 33 − \sqrt{3}i

Key Concepts

  • Square Root of a Negative Number
    • If b is a positive real number, then \sqrt{-b} = \sqrt{b}i
a + bi
b = 0 a + 0 · i = a Real number (a)
b ≠ 0 a + bi Imaginary number
a = 0 0 + bi = bi Pure imaginary number
    • A complex number is in standard form when written as a + bi, where a, b are real numbers.
      he diagram has a rectangle with the labels “Complex Numbers” and a plus b i. A second rectangle has the labels “Real Numbers”, a plus b i, b = 0. A third rectangle has the labels “Imaginary Numbers”, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.
  • Product of Complex Conjugates
    • If a, b are real numbers, then (a − b)(a + b) = a2 + b2
  • How to Divide Complex Numbers
    1. Write both the numerator and denominator in standard form.
    2. Multiply the numerator and denominator by the complex conjugate of the denominator.
    3. Simplify and write the result in standard form.
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