Comprehensive Notes on Complex Numbers

Objectives and Overview of Complex Numbers

• Primary Learning Objectives:

  • Utilize the imaginary unit $i$ to write complex numbers.

  • Perform addition, subtraction, and multiplication of complex numbers.

  • Utilize complex conjugates to write the quotient of two complex numbers in standard form.

  • Determine complex solutions of quadratic equations.

The Imaginary Unit $i$

• Motivation and Definition:

  • Certain quadratic equations do not possess real solutions. For instance, the quadratic equation $x^2 + 1 = 0$ has no solution within the real number system because there is no real number $x$ whose square equals $-1$.

  • To overcome this structural limitation, an expanded number system was defined using the imaginary unit $i$.

  • The imaginary unit $i$ is defined as:     i = \text{\sqrt{-1}}     where $i^2 = -1$.

Definition and Standard Form of Complex Numbers

• Standard Form Structure:

  • Let $a$ and $b$ be real numbers. The number $a + bi$ is defined as a complex number written in standard form.

  • The real number $a$ is designated as the real part (parte real).

  • The real number $b$ is designated as the imaginary part (parte imaginaria).

• Subsets and Terminology:

  • When $b = 0$, the expression $a + bi$ simplifies to $a + 0i = a$, which represents a real number. Consequently, the set of real numbers is a proper subset of the set of complex numbers.

  • When $b \neq 0$, the expression $a + bi$ is called an imaginary number.

  • A number written in the form $bi$, where $b \neq 0$ and $a = 0$, is specifically defined as a pure imaginary number.

• Expressing Radicals with Negative Radicands in Standard Form:

  • To express a complex number containing a radical with a negative radicand in standard form $a + bi$, simplify the radical by factoring out \text{\sqrt{-1}} = i

  • Example: Express -5 + \text{\sqrt{-9}} in standard form.     -5 + \text{\sqrt{-9}} = -5 + \text{\sqrt{3^2(-1)}} = -5 + 3\text{\sqrt{-1}} = -5 + 3i

Equality of Complex Numbers

• Criterion for Equality:

  • Two complex numbers $a + bi$ and $c + di$, written in standard form, are equal if and only if their real parts are equal and their imaginary parts are equal:     $a + bi = c + di \iff a = c \text{ and } b = d$

Operations with Complex Numbers

• Addition and Subtraction:

  • To add or subtract two complex numbers, sum or subtract their real parts and imaginary parts independently.

  • Sum Definition:     $(a + bi) + (c + di) = (a + c) + (b + d)i$

  • Difference Definition:     $(a + bi) - (c + di) = (a - c) + (b - d)i$

• Identity and Inverse Elements:

  • The additive identity in the complex number system is zero, written as $0 + 0i = 0$

  • The additive inverse of a complex number $a + bi$ is $-(a + bi) = -a - bi$

  • Adding a complex number to its additive inverse yields the additive identity:     $(a + bi) + (-a - bi) = (a - a) + (b - b)i = 0 + 0i = 0$

• Step-by-Step Addition and Subtraction Examples:

  • Example 1a: Calculate $(4 + 7i) + (1 - 6i)$

  • Remove parentheses: $4 + 7i + 1 - 6i$

  • Group like terms: $(4 + 1) + (7 - 6)i$

  • Standard form result: $5 + i$

  • Example 1b: Calculate $(1 + 2i) + (3 - 2i)$

  • Remove parentheses: $1 + 2i + 3 - 2i$

  • Group like terms: $(1 + 3) + (2 - 2)i$

  • Simplify: $4 + 0i$

  • Standard form result: $4$

  • Example 1c: Calculate $(3 - 2i) + (3 + 2i) + (5 + 2i)$

  • Remove parentheses and group terms: $(3 + 3 + 5) + (-2 + 2 + 2)i$

  • Standard form result: $11 + 2i$

  • Example 1d: Calculate $(3 + 2i) - (4 + 7i)$

  • Distribute subtraction: $3 + 2i - 4 - 7i$

  • Group like terms: $(3 - 4) + (2 - 7)i$

  • Standard form result: $-1 - 5i$

• Algebraic Properties Preserved in Complex Multiplication:

  • Associative property of addition and multiplication.

  • Commutative property of addition and multiplication.

  • Distributive property of multiplication over addition.

• Multiplication Derivation:

  • Applying the distributive property to two complex numbers $(a + bi)$ and $(c + di)$, and replacing $i^2$ with $-1$:     $(a + bi)(c + di) = a(c + di) + bi(c + di)$     $= ac + (ad)i + (bc)i + (bd)i^2$     $= ac + (ad)i + (bc)i + (bd)(-1)$     $= (ac - bd) + (ad + bc)i$

• Step-by-Step Multiplication Examples:

  • Example 2a: Calculate $4(-2 + 3i)$

  • Distribute $4$: $4(-2) + 4(3i)$

  • Result: $-8 + 12i$

  • Example 2b: Calculate $(2 - i)(4 + 3i)$

  • Apply distributive property: $2(4 + 3i) - i(4 + 3i)$

  • Expand: $8 + 6i - 4i - 3i^2$

  • Substitute $i^2 = -1$: $8 + 6i - 4i - 3(-1)$

  • Group like terms: $(8 + 3) + (6 - 4)i$

  • Standard form result: $11 + 2i$

  • Example 2c: Calculate $(3 + 2i)(3 - 2i)$

  • Expand: $3(3 - 2i) + 2i(3 - 2i) = 9 - 6i + 6i - 4i^2$

  • Substitute $i^2 = -1$: $9 + 4$

  • Standard form result: $13$

  • Example 2d: Calculate $(3 + 2i)^2$

  • Apply square of a binomial: $(3 + 2i)(3 + 2i) = 3(3 + 2i) + 2i(3 + 2i)$

  • Expand: $9 + 6i + 6i + 4i^2 = 9 + 12i + 4(-1)$

  • Simplify: $9 - 4 + 12i$

  • Standard form result: $5 + 12i$

Complex Conjugates and Division

• Definition and Property of Complex Conjugates:

  • Pair of complex numbers written in the form $a + bi$ and $a - bi$ are called complex conjugates.

  • Multiplication of complex conjugates yields a non-negative real number:     $(a + bi)(a - bi) = a^2 - abi + abi - b^2 i^2 = a^2 - b^2(-1) = a^2 + b^2$

• Quotient of Complex Numbers in Standard Form:

  • To write the quotient of $a + bi$ and $c + di$ in standard form (where $c$ and $d$ are not both zero), multiply the numerator and denominator by the complex conjugate of the denominator, $(c - di)$.

• Step-by-Step Quotient Example:

  • Example 4: Write the quotient $\frac{2 + 3i}{4 - 2i}$ in standard form.

  • Step 1: Multiply numerator and denominator by conjugate of denominator $(4 + 2i)$:       $\frac{2 + 3i}{4 - 2i} \left(\frac{4 + 2i}{4 + 2i}\right)$

  • Step 2: Expand numerator and denominator:       $\frac{8 + 4i + 12i + 6i^2}{16 - 4i^2}$

  • Step 3: Substitute $i^2 = -1$ and combine like terms:       $\frac{8 - 6 + 16i}{16 + 4} = \frac{2 + 16i}{20}$

  • Step 4: Write in standard form $a + bi$:       $\frac{2}{20} + \frac{16}{20}i = \frac{1}{10} + \frac{4}{5}i$

Complex Solutions of Quadratic Equations

• Principal Square Root of a Negative Number:

  • Let $a$ be a positive real number. The principal square root of $-a$ is defined as:     \text{\sqrt{-a}} = \text{\sqrt{a}}i

  • The imaginary unit $i$ must be placed outside the square root symbol.

  • Derivation example for \text{\sqrt{-3}}:     \text{\sqrt{-3}} = \text{\sqrt{3(-1)}} = \text{\sqrt{3}}\text{\sqrt{-1}} = \text{\sqrt{3}}i

• General Quadratic Solution via Quadratic Formula:

  • For a general quadratic equation $ax^2 + bx + c = 0$, solutions are found using:     x = \frac{-b \pm \text{\sqrt{b^2 - 4ac}}}{2a}

• Step-by-Step Quadratic Example:

  • Example 6: Solve $3x^2 - 2x + 5 = 0$

  • Step 1: Substitute $a = 3$, $b = -2$, and $c = 5$ into quadratic formula:       x = \frac{-(-2) \pm \text{\sqrt{(-2)^2 - 4(3)(5)}}}{2(3)}

  • Step 2: Simplify terms under radical:       x = \frac{2 \pm \text{\sqrt{4 - 60}}}{6} = \frac{2 \pm \text{\sqrt{-56}}}{6}

  • Step 3: Express \text{\sqrt{-56}} using principal square root definition:       \text{\sqrt{-56}} = \text{\sqrt{56}}i = \text{\sqrt{4 \times 14}}i = 2\text{\sqrt{14}}i

  • Step 4: Substitute simplified radical back into fraction:       x = \frac{2 \pm 2\text{\sqrt{14}}i}{6}

  • Step 5: Separate terms into standard form $a + bi$:       x = \frac{1}{3} \pm \frac{\text{\sqrt{14}}}{3}i

Supplemental Practice Exercises and Solutions

• Exercise 1: Express 3 + \text{\sqrt{-16}} in standard form $a + bi$

  • Solution:     \text{\sqrt{-16}} = \text{\sqrt{16}}i = 4i     Standard form: $3 + 4i$

• Exercise 2: Simplify $-5i + i^2$

  • Solution:     Substitute $i^2 = -1$     Result in standard form: $-1 - 5i$

• Exercise 3a: Perform subtraction $(7 - 2i) - (3 - 8i)$ and output in standard form.

  • Solution:     $(7 - 3) + (-2 - (-8))i = 4 + 6i$

• Exercise 3b: Perform multiplication $(4 + i)(3 - 10i)$ and output in standard form.

  • Solution:     $4(3) + 4(-10i) + i(3) + i(-10i) = 12 - 40i + 3i - 10i^2$     $= 12 - 37i - 10(-1) = 22 - 37i$

• Exercise 4: Expand $(5 - 9i)^2$ and output in standard form.

  • Solution:     $5^2 - 2(5)(9i) + (9i)^2 = 25 - 90i + 81i^2$     $= 25 - 90i + 81(-1) = -56 - 90i$

• Exercise 5: Write quotient $\frac{4}{1 - 2i}$ in standard form.

  • Solution:     $\frac{4}{1 - 2i} \left(\frac{1 + 2i}{1 + 2i}\right) = \frac{4(1 + 2i)}{1^2 + 2^2} = \frac{4 + 8i}{1 + 4} = \frac{4 + 8i}{5}$     Standard form: $\frac{4}{5} + \frac{8}{5}i$

• Exercise 6: Solve $4x^2 + 4x + 7 = 0$ using the quadratic formula.

  • Solution:

  • Identify coefficients: $a = 4$, $b = 4$, $c = 7$

  • Apply formula:       x = \frac{-4 \pm \text{\sqrt{4^2 - 4(4)(7)}}}{2(4)}

  • Simplify discriminant:       \text{\sqrt{16 - 112}} = \text{\sqrt{-96}} = \text{\sqrt{96}}i = \text{\sqrt{16 \times 6}}i = 4\text{\sqrt{6}}i

  • Substitute back into expression:       x = \frac{-4 \pm 4\text{\sqrt{6}}i}{8}

  • Standard form result:       x = -\frac{1}{2} \pm \frac{\text{\sqrt{6}}}{2}i