3.4 Complex Numbers
3.4.1 Learning Objectives
By the end of this section, you should be able to:
- identify the real and imaginary parts of a complex number;
- simplify powers of \(i\);
- add, subtract, multiply, and divide complex numbers;
- use complex conjugates; and
- solve quadratic equations with non-real solutions.
3.4.2 The Imaginary Unit and Complex Numbers
The imaginary unit \(i\) is defined by
\[ i^2=-1. \]
A complex number has standard form
\[ a+bi, \]
where \(a\) and \(b\) are real. The real part is \(a\), and the imaginary part is \(b\). A real number has \(b=0\), while a pure imaginary number has \(a=0\).
Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.
Example 3.16 Identifying Complex Components
State the real and imaginary parts of \(z=-7+4i\).
Solution
The real term is \(-7\), and the coefficient of \(i\) is \(4\).
\[ \boxed{\operatorname{Re}(z)=-7,\quad \operatorname{Im}(z)=4} \]
Exercise 3.16 State the real and imaginary parts of \(z=9-3i\).
Show answer
Answer: \(\boxed{\operatorname{Re}(z)=9,\quad \operatorname{Im}(z)=-3}\)
3.4.3 Powers of \(i\)
The powers of \(i\) repeat in a cycle of length \(4\):
\[ i^0=1,\qquad i^1=i,\qquad i^2=-1,\qquad i^3=-i,\qquad i^4=1. \]
To simplify \(i^n\), divide \(n\) by \(4\) and use the remainder to identify the matching power.
Example 3.17 Simplifying a Large Power
Simplify \(i^{58}\).
Solution
Since \(58=4(14)+2\),
\[ i^{58}=i^2=-1. \]
\[ \boxed{-1} \]
Exercise 3.17 Simplify \(i^{103}\).
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Answer: Since \(103=4(25)+3\), \(\boxed{i^{103}=-i}\).
3.4.4 Adding, Subtracting, and Multiplying
Add or subtract complex numbers by combining real parts and imaginary parts separately. Multiply using the distributive property, then replace \(i^2\) with \(-1\).
Example 3.18 Performing Complex Arithmetic
Simplify \((3-2i)(4+5i)\).
Solution
\[ \begin{aligned} (3-2i)(4+5i) &=12+15i-8i-10i^2\\ &=12+7i+10\\ &=22+7i. \end{aligned} \]
\[ \boxed{22+7i} \]
Exercise 3.18 Simplify \((6+3i)-(2-5i)\).
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Answer: \(\boxed{4+8i}\)
3.4.5 Conjugates and Division
The complex conjugate of \(a+bi\) is \(a-bi\). Their product is real:
\[ (a+bi)(a-bi)=a^2+b^2. \]
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator.
Example 3.19 Dividing Complex Numbers
Write
\[ \frac{5+2i}{1-3i} \]
in standard form.
Solution
\[ \begin{aligned} \frac{5+2i}{1-3i} &=\frac{(5+2i)(1+3i)}{(1-3i)(1+3i)}\\ &=\frac{5+15i+2i+6i^2}{1+9}\\ &=\frac{-1+17i}{10}\\ &=-\frac{1}{10}+\frac{17}{10}i. \end{aligned} \]
\[ \boxed{-\frac{1}{10}+\frac{17}{10}i} \]
Exercise 3.19 Write \(\dfrac{3-i}{2+i}\) in standard form.
Show answer
Answer: Multiply by \(2-i\) to obtain \(\boxed{1-i}\).
3.4.6 Square Roots of Negative Numbers
For \(r>0\),
\[ \sqrt{-r}=i\sqrt r. \]
Simplify the positive factor before attaching \(i\). When multiplying square roots of negative numbers, first convert each one to imaginary form. This avoids misusing real-number radical rules.
Example 3.20 Simplifying a Negative Radicand
Simplify \(\sqrt{-72}\).
Solution
\[ \sqrt{-72}=i\sqrt{72}=i\sqrt{36\cdot2}=6i\sqrt2. \]
\[ \boxed{6i\sqrt2} \]
Exercise 3.20 Simplify \(\sqrt{-45}\).
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Answer: \(\boxed{3i\sqrt5}\)
3.4.7 Quadratic Equations with Complex Solutions
For \(ax^2+bx+c=0\), the quadratic formula remains valid:
\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. \]
If the discriminant \(b^2-4ac\) is negative, the equation has two non-real complex solutions. For a quadratic with real coefficients, these solutions form a conjugate pair.
Example 3.21 Solving a Quadratic with Non-Real Roots
Solve \(2x^2-4x+7=0\).
Solution
\[ \begin{aligned} x&=\frac{4\pm\sqrt{(-4)^2-4(2)(7)}}{4}\\ &=\frac{4\pm\sqrt{-40}}{4}\\ &=\frac{4\pm2i\sqrt{10}}{4}\\ &=1\pm\frac{\sqrt{10}}{2}i. \end{aligned} \]
\[ \boxed{x=1\pm\frac{\sqrt{10}}{2}i} \]
Exercise 3.21 Solve \(x^2+6x+13=0\).
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Answer: \(\boxed{x=-3\pm2i}\)
3.4.8 Conceptual Takeaways
- Complex numbers extend the real numbers by defining \(i^2=-1\).
- Real and imaginary parts behave like separate like terms under addition and subtraction.
- Powers of \(i\) repeat every four exponents.
- Conjugates create a real product and make complex division possible.
- Non-real solutions of real-coefficient quadratics occur in conjugate pairs.
3.4.9 Skills You Should Be Able to Do
- Write complex numbers in standard form.
- Simplify powers of \(i\).
- Perform all four arithmetic operations with complex numbers.
- Simplify square roots of negative numbers.
- Solve quadratic equations with complex solutions.
3.4.10 Practice Problems with Solutions
State the real and imaginary parts of \(z=-5-8i\).
Show Solution
In \(a+bi\), \(a=-5\) and \(b=-8\). Thus,
\[ \boxed{\operatorname{Re}(z)=-5,\quad \operatorname{Im}(z)=-8}. \]
Simplify \(i^{74}\).
Show Solution
Since \(74=4(18)+2\),
\[ \boxed{i^{74}=i^2=-1}. \]
Simplify \((4+7i)+(-6+2i)\).
Show Solution
Combine like parts:
\[ (4-6)+(7+2)i=\boxed{-2+9i}. \]
Simplify \((2-3i)(5+i)\).
Show Solution
\[ \begin{aligned} (2-3i)(5+i)&=10+2i-15i-3i^2\\ &=13-13i. \end{aligned} \]
\[ \boxed{13-13i} \]
Write \(\dfrac{4+3i}{2-i}\) in standard form.
Show Solution
Multiply by the conjugate \(2+i\):
\[ \frac{(4+3i)(2+i)}{(2-i)(2+i)} =\frac{5+10i}{5} =\boxed{1+2i}. \]
Simplify \(\sqrt{-108}\).
Show Solution
\[ \sqrt{-108}=i\sqrt{108}=i\sqrt{36\cdot3}. \]
Therefore, \(\boxed{6i\sqrt3}\).
Solve \(x^2-8x+25=0\).
Show Solution
\[ x=\frac{8\pm\sqrt{64-100}}{2} =\frac{8\pm6i}{2}. \]
Thus, \(\boxed{x=4\pm3i}\).
Find real numbers \(a\) and \(b\) if \((a+bi)(2-i)=11+2i\).
Show Solution
Expanding gives
\[ (a+bi)(2-i)=(2a+b)+(2b-a)i. \]
Equate real and imaginary parts:
\[ 2a+b=11,\qquad -a+2b=2. \]
Solving gives \(a=4\) and \(b=3\).
\[ \boxed{a=4,\quad b=3} \]