Learn on PengiOpenstax Intermediate Algebra 2EChapter 8: Roots and Radicals

Lesson 8.8: Use the Complex Number System

In this Openstax Intermediate Algebra 2E lesson on the complex number system, students learn to evaluate square roots of negative numbers using the imaginary unit i, where i² = −1, and to write results in standard form a + bi. The lesson covers adding, subtracting, multiplying, and dividing complex numbers, as well as simplifying powers of i. This chapter-closing section bridges real number concepts students already know with the broader complex number system used in higher mathematics.

Section 1

📘 Use the Complex Number System

New Concept

We're expanding our number system beyond real numbers. By defining the imaginary unit i=−1i = \sqrt{-1}, we create complex numbers in the form a+bia+bi. This allows us to perform arithmetic with the square roots of negative numbers.

What’s next

Now that you have the basics, you'll work through interactive examples of adding, subtracting, multiplying, and dividing complex numbers on our practice cards.

Section 2

Square Root of a Negative Number

Property

The imaginary unit ii is the number whose square is −1-1.

i2=−1ori=−1i^2 = -1 \quad \text{or} \quad i = \sqrt{-1}

Square Root of a Negative Number
If bb is a positive real number, then

−b=bi\sqrt{-b} = \sqrt{b} i

Complex Number
A complex number is of the form a+bia + bi, where aa and bb are real numbers.

Section 3

Add and Subtract Complex Numbers

Property

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, a+bia+bi.

Examples

  • Simplify (5−2i)+(3+8i)(5 - 2i) + (3 + 8i): (5+3)+(−2+8)i=8+6i(5 + 3) + (-2 + 8)i = 8 + 6i.
  • Simplify (7−6i)−(4−2i)(7 - 6i) - (4 - 2i): 7−6i−4+2i=(7−4)+(−6+2)i=3−4i7 - 6i - 4 + 2i = (7 - 4) + (-6 + 2)i = 3 - 4i.

Section 4

Multiply Complex Numbers

Property

Multiplying complex numbers is much like multiplying expressions with coefficients and variables. Use the Distributive Property or FOIL. The key is to simplify any instance of i2i^2 to −1-1. When multiplying square roots of negative numbers, first write them as complex numbers using −b=bi\sqrt{-b} = \sqrt{b}i.

Examples

  • Multiply 5i(3−2i)5i(3 - 2i): Distribute to get 15i−10i215i - 10i^2. Replace i2i^2 with −1-1: 15i−10(−1)=10+15i15i - 10(-1) = 10 + 15i.
  • Multiply (2+4i)(5−3i)(2 + 4i)(5 - 3i): Using FOIL, we get 10−6i+20i−12i210 - 6i + 20i - 12i^2. This simplifies to 10+14i−12(−1)=22+14i10 + 14i - 12(-1) = 22 + 14i.

Section 5

Product of Complex Conjugates

Property

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

Product of Complex Conjugates
If aa and bb are real numbers, then

(a−bi)(a+bi)=a2+b2(a - bi)(a + bi) = a^2 + b^2

Examples

  • Multiply (4−5i)(4+5i)(4 - 5i)(4 + 5i) using the pattern: a=4a=4 and b=5b=5. The result is a2+b2=42+52=16+25=41a^2 + b^2 = 4^2 + 5^2 = 16 + 25 = 41.

Section 6

Divide Complex Numbers

Property

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

Examples

  • Divide 2+i3−i\frac{2+i}{3-i}: Multiply by 3+i3+i\frac{3+i}{3+i} to get (2+i)(3+i)(3−i)(3+i)=6+5i−19+1=5+5i10=12+12i\frac{(2+i)(3+i)}{(3-i)(3+i)} = \frac{6+5i-1}{9+1} = \frac{5+5i}{10} = \frac{1}{2} + \frac{1}{2}i.
  • Divide 52+3i\frac{5}{2+3i}: Multiply by 2−3i2−3i\frac{2-3i}{2-3i} to get 5(2−3i)22+32=10−15i13=1013−1513i\frac{5(2-3i)}{2^2+3^2} = \frac{10-15i}{13} = \frac{10}{13} - \frac{15}{13}i.

Section 7

Simplify Powers of i

Property

The powers of ii repeat in a pattern:
i1=ii^1 = i
i2=−1i^2 = -1
i3=−ii^3 = -i
i4=1i^4 = 1
To simplify ini^n, rewrite it in the form in=(i4)q⋅iri^n = (i^4)^q \cdot i^r, where rr is the remainder when nn is divided by 4.

Examples

  • Simplify i35i^{35}: Divide 35 by 4. 35=4⋅8+335 = 4 \cdot 8 + 3. The remainder is 3, so i35=i3=−ii^{35} = i^3 = -i.
  • Simplify i102i^{102}: Divide 102 by 4. 102=4⋅25+2102 = 4 \cdot 25 + 2. The remainder is 2, so i102=i2=−1i^{102} = i^2 = -1.

Book overview

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Chapter 8: Roots and Radicals

  1. Lesson 1

    Lesson 8.1: Simplify Expressions with Roots

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  2. Lesson 2

    Lesson 8.2: Simplify Radical Expressions

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  3. Lesson 3

    Lesson 8.3: Simplify Rational Exponents

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  4. Lesson 4

    Lesson 8.4: Add, Subtract, and Multiply Radical Expressions

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  5. Lesson 5

    Lesson 8.5: Divide Radical Expressions

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  6. Lesson 6

    Lesson 8.6: Solve Radical Equations

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  7. Lesson 7

    Lesson 8.7: Use Radicals in Functions

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  8. Lesson 8Current

    Lesson 8.8: Use the Complex Number System

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Use the Complex Number System

New Concept

We're expanding our number system beyond real numbers. By defining the imaginary unit i=−1i = \sqrt{-1}, we create complex numbers in the form a+bia+bi. This allows us to perform arithmetic with the square roots of negative numbers.

What’s next

Now that you have the basics, you'll work through interactive examples of adding, subtracting, multiplying, and dividing complex numbers on our practice cards.

Section 2

Square Root of a Negative Number

Property

The imaginary unit ii is the number whose square is −1-1.

i2=−1ori=−1i^2 = -1 \quad \text{or} \quad i = \sqrt{-1}

Square Root of a Negative Number
If bb is a positive real number, then

−b=bi\sqrt{-b} = \sqrt{b} i

Complex Number
A complex number is of the form a+bia + bi, where aa and bb are real numbers.

Section 3

Add and Subtract Complex Numbers

Property

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, a+bia+bi.

Examples

  • Simplify (5−2i)+(3+8i)(5 - 2i) + (3 + 8i): (5+3)+(−2+8)i=8+6i(5 + 3) + (-2 + 8)i = 8 + 6i.
  • Simplify (7−6i)−(4−2i)(7 - 6i) - (4 - 2i): 7−6i−4+2i=(7−4)+(−6+2)i=3−4i7 - 6i - 4 + 2i = (7 - 4) + (-6 + 2)i = 3 - 4i.

Section 4

Multiply Complex Numbers

Property

Multiplying complex numbers is much like multiplying expressions with coefficients and variables. Use the Distributive Property or FOIL. The key is to simplify any instance of i2i^2 to −1-1. When multiplying square roots of negative numbers, first write them as complex numbers using −b=bi\sqrt{-b} = \sqrt{b}i.

Examples

  • Multiply 5i(3−2i)5i(3 - 2i): Distribute to get 15i−10i215i - 10i^2. Replace i2i^2 with −1-1: 15i−10(−1)=10+15i15i - 10(-1) = 10 + 15i.
  • Multiply (2+4i)(5−3i)(2 + 4i)(5 - 3i): Using FOIL, we get 10−6i+20i−12i210 - 6i + 20i - 12i^2. This simplifies to 10+14i−12(−1)=22+14i10 + 14i - 12(-1) = 22 + 14i.

Section 5

Product of Complex Conjugates

Property

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

Product of Complex Conjugates
If aa and bb are real numbers, then

(a−bi)(a+bi)=a2+b2(a - bi)(a + bi) = a^2 + b^2

Examples

  • Multiply (4−5i)(4+5i)(4 - 5i)(4 + 5i) using the pattern: a=4a=4 and b=5b=5. The result is a2+b2=42+52=16+25=41a^2 + b^2 = 4^2 + 5^2 = 16 + 25 = 41.

Section 6

Divide Complex Numbers

Property

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

Examples

  • Divide 2+i3−i\frac{2+i}{3-i}: Multiply by 3+i3+i\frac{3+i}{3+i} to get (2+i)(3+i)(3−i)(3+i)=6+5i−19+1=5+5i10=12+12i\frac{(2+i)(3+i)}{(3-i)(3+i)} = \frac{6+5i-1}{9+1} = \frac{5+5i}{10} = \frac{1}{2} + \frac{1}{2}i.
  • Divide 52+3i\frac{5}{2+3i}: Multiply by 2−3i2−3i\frac{2-3i}{2-3i} to get 5(2−3i)22+32=10−15i13=1013−1513i\frac{5(2-3i)}{2^2+3^2} = \frac{10-15i}{13} = \frac{10}{13} - \frac{15}{13}i.

Section 7

Simplify Powers of i

Property

The powers of ii repeat in a pattern:
i1=ii^1 = i
i2=−1i^2 = -1
i3=−ii^3 = -i
i4=1i^4 = 1
To simplify ini^n, rewrite it in the form in=(i4)q⋅iri^n = (i^4)^q \cdot i^r, where rr is the remainder when nn is divided by 4.

Examples

  • Simplify i35i^{35}: Divide 35 by 4. 35=4⋅8+335 = 4 \cdot 8 + 3. The remainder is 3, so i35=i3=−ii^{35} = i^3 = -i.
  • Simplify i102i^{102}: Divide 102 by 4. 102=4⋅25+2102 = 4 \cdot 25 + 2. The remainder is 2, so i102=i2=−1i^{102} = i^2 = -1.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Roots and Radicals

  1. Lesson 1

    Lesson 8.1: Simplify Expressions with Roots

    LearnPractice
  2. Lesson 2

    Lesson 8.2: Simplify Radical Expressions

    LearnPractice
  3. Lesson 3

    Lesson 8.3: Simplify Rational Exponents

    LearnPractice
  4. Lesson 4

    Lesson 8.4: Add, Subtract, and Multiply Radical Expressions

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  5. Lesson 5

    Lesson 8.5: Divide Radical Expressions

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  6. Lesson 6

    Lesson 8.6: Solve Radical Equations

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  7. Lesson 7

    Lesson 8.7: Use Radicals in Functions

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  8. Lesson 8Current

    Lesson 8.8: Use the Complex Number System

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