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🧮 Complex Numbers Overview
Brief Overview
This note covering complex number operations was created from the Complex Numbers
- Basic Operations YouTube video. It covers graphing, absolute values, simplifying square roots, powers of i, and operations like addition, multiplication, and division.
Key Points
Understand how to plot complex numbers in the standard form and compute their absolute values. Master simplification of roots with negative numbers and the cyclical pattern of powers of i. Learn the rules for adding, subtracting, multiplying, and dividing complex numbers, including the use of conjugates. Apply complex number concepts to solve quadratic equations and reconstruct polynomials from given roots.
📊 Graphing Complex Numbers
A complex number in standard form is written as $a + bi$, where: $a$ = real part (plotted on the horizontal axis) $bi$ = imaginary part (plotted on the vertical axis) Definition: The real axis is the horizontal $x$-axis, and the imaginary axis is the vertical $y$-axis.
Example: Graphing $3 + 4i$
Step Action Result 1 Move 3 units right on real axis $x = 3$ 2 Move 4 units up on imaginary axis $y = 4$ 3 Plot the point $(3, 4)$ in complex plane
📏 Absolute Value of Complex Numbers Definition: The absolute value (or modulus) of $a + bi$ represents the distance from the origin to the point in the complex plane. $\text{Absolute value} = |a + bi| = \sqrt{a^2 + b^2}$ This is equivalent to the hypotenuse of the right triangle formed by $a$ and $b$.
Examples
Common Pythagorean Triples
🔢 Simplifying Square Roots of Negative Numbers Key Principle: $\sqrt{-1} = i$, so any negative under an even root produces an imaginary number. Complex Number Calculation Absolute Value $3 + 4i$ $\sqrt{3^2 + 4^2} = \sqrt{
$-5 + 12i$ $\sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169}$
$8 - 15i$ $\sqrt{8^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289}$
Triangle Sides Check 3-4-5 $3^2 + 4^2 = 5^2$ (^) $9 + 16 = 25$ ✓ 5-12-13 $5^2 + 12^2 = 13^2$ $25 + 144 = 169$ ✓ 8-15-17 $8^2 + 15^2 = 17^2$ (^) $64 + 225 = 289$ ✓ 7-24-25 $7^2 + 24^2 = 25^2$ $49 + 576 = 625$ ✓
Pattern: The powers of $i$ cycle every 4. After $i^4$, the pattern repeats: $i^5 = i$, $i^6 = -1$, etc.
Simplifying Higher Powers
Method: Break the exponent into a multiple of 4 plus remainder.
Systematic Method for Large Exponents
To find $i^n$ when $n$ is large:
- Divide $n$ by 4: $n \div 4 = \text{quotient}.\text{remainder}$ Power Value Derivation $i^1$ $i$ $\sqrt{-1}$ $i^2$ $-1$ $i \times i = -1$ $i^3$ $-i$ $i^2 \times i = -1 \times i = -i$ $i^4$ $\mathbb{1}$ $i^2 \times i^2 = (-1) \times (-1) = 1$ Expression Breakdown Simplification Result $i^6$ $i^4 \times i^2$ $1 \times (-1)$ $-1$ $i^8$ $(i^4)^2$ $1^2$ $\mathbb{1}$ $i^{12}$ $(i^4)^3$ $1^3$ $\mathbb{1}$ $i^{15}$ $i^{12} \times i^3 = (i^4)^3 \times i^3$ $1 \times (-i)$ $-i$ $i^{29}$ $i^{28} \times i = (i^4)^7 \times i$ $1 \times i$ $i$ $i^{62}$ $i^{60} \times i^2 = (i^4)^{15} \times i^2$ $1 \times (-1)$ $-1$ $i^{201}$ $i^{200} \times i = (i^4)^{50} \times i$ $1 \times i$ $i$
- First number: quotient $\times$ 4 (the multiple of 4)
- Second number: decimal portion $\times$ 4 (the remainder) Example: $i^{201}$ $201 \div 4 = 50.25$ First number: $50 \times 4 = 200$ Second number: $0.25 \times 4 = 1$ So: $i^{201} = i^{200} \times i^1 = (i^4)^{50} \times i = 1 \times i = i$ Key Insight: There are only four possible answers when simplifying powers of $i$: $i$, $-1$, $-i$, or $\mathbb{1}$. ➕➖ Adding and Subtracting Complex Numbers Rule: Combine like terms — add real parts together and imaginary parts together.
Addition Examples
Subtraction Examples
Critical: Distribute the negative sign to all terms in the second parentheses!
Simplifying with Square Roots
Problem Real Parts Imaginary Parts Result $(5 + 2i) + (3 + 7i)$ $5 + 3 = 8$ $2i + 7i = 9i$ $8 + 9i$ $(7 + 3i) + (6 + 5i)$ $7 + 6 = 13$ $3i + 5i = 8i$ $13 + 8i$ Problem After Distribution Real Parts Imaginary Parts Result $(4 + 8i) - (3 - 5i)$ $4 + 8i - 3 + 5i$ $4 - 3 = 1$ $8i + 5i = 13i$ $1 + 13i$ $7(4 + 3i) - 5(
- 6i)$ $28 + 21i - 10 + 30i$ $28 - 10 = 18$ $21i + 30i = 51i$ $18 + 51i$
More FOIL Examples:
Complex Conjugates
Definition: The conjugate of $a + bi$ is $a - bi$. The real parts are identical; the imaginary parts have opposite signs. Key Property: When you multiply a complex number by its conjugate, the result is always a real number (the imaginary terms cancel). Full FOIL Verification for $(3 + 4i)(3 - 4i)$: Step First Outer Inner Last Multiply $5 \times 4 = 20$ $5 \times 7i = 35i$ $3i \times 4 = 12i$ $3i \times 7i = 21i^2$ Combine $20 + 35i + 12i
- 21i^2$ Simplify $i^2$ $20 + 47i + 21(-1) = 20 + 47i - 21$ Final Answer $41 + 47i$ Problem Work Result $(6 - 5i)(3 + 8i)$ $18 + 48i - 15i - 40i^2 = 18
- 33i + 40$ $58 + 33i$ $(4 + 5i)^2$ $16 + 20i + 20i + 25i^2 = 16 + 40i - 25$ $-9 + 40i$ Original Conjugate Product (Quick Method) Result $3 + 4i$ $3 - 4i$ $3^2 + 4^2 = 9 + 16$
$5 - 2i$ $5 + 2i$ $5^2 + 2^2 = 25 + 4$
The middle terms $-12i + 12i = 0$ cancel, leaving $9 + 16 = 25$. ➗ Dividing Complex Numbers
Method: Multiply by the Conjugate
Rule: To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. Example: $\frac{4 + 3i}{5 - 2i}$
More Division Examples
Term Calculation First $3 \times 3 = 9$ Outer $3 \times (-4i) = -12i$ Inner $4i \times 3 = 12i$ Last $4i \times (-4i) = -16i^2 = +16$ Step Action Result 1 Multiply by conjugate $\frac{5 + 2i}{5 + 2i}$ $\frac{(4 + 3i)(5 + 2i)}{(5 - 2i)(5 + 2i)}$ 2 FOIL numerator $20 + 8i + 15i + 6i^2 = 20 + 23i - 6 = 14 + 23i$ 3 Multiply denominator (conjugate shortcut)
$5^2 + 2^2 = 25 + 4 = 29$
4 Combine $\frac{14 + 23i}{29}$ 5 Standard form $\frac{14}{29} + \frac{23} {29}i$ Problem Work Result $\frac{8}{6 + i}$ $\frac{8(6-i)}{(6+i)(6-i)} = \frac{48-8i}{36+1} = $\frac{48}{37} - \frac{8} {37}i$
Key Rule: Multiplication/division inside: distribute the exponent Addition/subtraction inside: FOIL — you cannot distribute! 🔢 Solving Quadratic Equations with Complex Solutions
Sum and Product of Roots
Key Formulas: For a quadratic equation $ax^2 + bx + c = 0$: Sum of roots = $-\frac{b}{a}$ Product of roots = $\frac{c}{a}$
Example: $x^2 + 36 = 0$
Sum of roots: $-\frac{0}{1} = 0$ Product of roots: $\frac{36}{1} = 36$ Verification by factoring: $x^2 + 36 = (x + 6i)(x - 6i) = 0$ Roots: $x = 6i$ and $x = -6i$ Sum: $6i + (-6i) = 0$ ✓ Product: $(6i)(-6i) = -36i^2 = -36(-1) = 36$ ✓ \times 49i^2 = 441(-1)$ $(3 + 7i)^2$ Binomial squared FOIL or $(a+b)^2 = a^2 + 2ab + b^2$: $9 + 42i + 49i^2 = 9
- 42i - 49$ $-40 + 42i$ Component Value Calculation $a$ 1 coefficient of $x^2$ $b$ 0 no $x$ term $c$ 36 constant term
Example: $x^2 - 4x + 29 = 0$
Solving with quadratic formula: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(29)}}{2(1)} = \frac{4 \pm \sqrt{16 - 116}}{2} = \frac{4 \pm \sqrt{-100}}{2}$ $\sqrt{-100} = 10i, \text{ so: } x = \frac{4 \pm 10i}{2} = \frac{4}{2} \pm \frac{10i}{2} = 2 \pm 5i$ Solutions: $x = 2 + 5i$ and $x = 2 - 5i$ Verification: Sum: $(2 + 5i) + (2 - 5i) = 4$ ✓ Product: $(2 + 5i)(2 - 5i) = 4 - 25i^2 = 4 + 25 = 29$ ✓ 🔄 Working Backwards: Finding Equations from Solutions
Pure Imaginary Solutions
Given solution: $x = 3i$ Key Insight: Imaginary solutions always come in conjugate pairs. If $3i$ is a solution, then $-3i$ is also a solution. Using Formulas Value Sum of roots $-\frac{-4}{1} = 4$ Product of roots $\frac{29}{1} = 29$ Step Action Result 1 Write factored form $(x - 3i)(x + 3i) = 0$ 2 Multiply (conjugate shortcut) $x^2 - (3i)^2 = x^2 - 9i^2$ 3 Simplify $i^2 = -1$ $x^2 - 9(-1) = x^2 + 9$
Canceling terms: $3xi - 3xi = 0$ and $-12i + 12i = 0$ Remaining: $x^2 - 8x + 16 - 9i^2 = x^2 - 8x + 16 + 9 = x^2 - 8x + 25$ Method 2: Shortcut (Recommended) Recognize the pattern: $(x - 4 - 3i)(x - 4 + 3i) = [(x-4) - 3i][(x-4) + 3i]$ This is $(a - b)(a + b) = a^2 - b^2$ where $a = (x-4)$ and $b = 3i$
Another Example: Solutions $5 + 2i$ and $5 - 2i$
Final equation: $x^2 - 10x + 29 = 0$ 🔍 Solving Equations with Radicals and Complex Numbers
Example: $x = \sqrt{7} + 3i$
Key Insight: The conjugate solution is $\sqrt{7} - 3i$ (change the sign of the imaginary part only). Factored form: $[x - (\sqrt{7} + 3i)][x - (\sqrt{7} - 3i)] = 0$ Last $(-3i) \cdot (3i) = -9i^2$ Step Calculation $(x-4)^2$ $x^2 - 8x + 16$ $-(3i)^2$ $-9i^2 = +9$ Final $x^2 - 8x + 25$ Shortcut Method Work Pattern $(x - 5)^2 - (2i)^2$ Expand $x^2 - 10x + 25 - 4i^2$ Simplify $x^2 - 10x + 25 + 4 = x^2 - 10x + 29$
Rewrite: $(x - \sqrt{7} - 3i)(x - \sqrt{7} + 3i) = 0$ Using the shortcut: $(x - \sqrt{7})^2 - (3i)^2$ 📐 Equating Real and Imaginary Parts
Solving for Two Variables
Key Principle: If $a + bi = c + di$, then real parts are equal ($a = c$) and imaginary parts are equal ($b = d$).
Example: $6x + 2i = 18 + 12yi$
Solving for $x$: $6x = 18 \Rightarrow x = \frac{18}{6} = 3$ Solving for $y$: $2i = 12yi \Rightarrow 2 = 12y \Rightarrow y = \frac{2}{12} = \frac{1} {6}$ Solution: $x = 3$ and $y = \frac{1}{6}$
Example: $3x + 4i = 15 + 16yi$
Step Calculation $(x - \sqrt{7})^2$ $x^2 - 2\sqrt{7}x + 7$ $-(3i)^2$ $-9i^2 = +9$ Final $x^2 - 2\sqrt{7}x + 16$ Component Left Side Right Side Equation Real part $6x$ $\mathbb{18}$ $6x = 18$ Imaginary part $2i$ $12yi$ $2i = 12yi$ Component Equation Solution Real parts $3x = 15$ $x = 5$ Imaginary parts $4i = 16yi \Rightarrow 4 = 16y$ $y = \frac{4}{16} = \frac{1} {4}$
