MTH 111: Complex Numbers and Number Systems

Number Systems and the Hierarchy of Mathematical Sets

The mathematical universe is structured through different number systems, each building upon the other to form a comprehensive framework for analysis. The most foundational set is the Set of Natural Numbers, denoted as $N$, which consists of the positive counting numbers: $N = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \dots\}$. Extending this set to include zero and negative whole numbers results in the Set of Integers, denoted by $Z$. The specific representation for the set of integers is $Z = \{\dots, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, \dots\}$. From these foundations, we develop the Set of Rational Numbers ($Q$). A rational number is defined as any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are both integers ($p, q \in Z$) and $q \neq 0$. Examples of rational values include $0, \pm 1, \pm 2, \pm 3, \pm \frac{1}{4}, \pm \frac{1}{3}, \pm \frac{1}{2}, \pm \frac{2}{3}, \dots$. Rational numbers are also identified by their decimal representations: they are either terminating decimals or non-terminating but repeating decimals.

Numbers that cannot be expressed as a ratio of two integers are classified as the Set of Irrational Numbers ($I$ or $Q'$). These numbers, such as $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$, are characterized as non-terminating and non-repeating decimals. The combination of all rational and irrational sets forms the Set of Real Numbers ($R$). Within this hierarchy, the complex number system ($C$) represents the broadest category, containing both real numbers and imaginary numbers. The visual representation of this hierarchy shows the Set of Natural Numbers nested within Whole Numbers, which are nested within Integers, which are nested within Rational Numbers, all of which are contained within the Real Number system alongside Irrational Numbers. The Complex Number system encompasses both the Real Numbers and the Imaginary Numbers.

The Conceptual Necessity of Complex Numbers

In algebra, quadratic equations typically possess two solutions or roots. The nature of these roots depends heavily on the discriminant. For instance, consider the following equations: 1) $x^2 - 3x + 2 = 0$ 2) $x^2 - 2x + 1 = 0$ 2) $5x^2 - 6x + 5 = 0$

While many quadratic equations can be solved using real numbers, certain equations present a challenge that the real number system alone cannot address. For example, consider the equation $x^2 + 1 = 0$. This equation implies that $x^2 = -1$. Within the system of real numbers, there is no solution because no real number, when squared, results in a negative value. This shortcoming necessitated an extension of the number system through the introduction of a new number known as the imaginary number, denoted as $i$.

The imaginary number $i$ is explicitly defined as $i = \sqrt{-1}$. This definition implies that $i^2 = -1$. Consequently, the square root of any negative real number can be expressed in terms of $i$. For instance, $\sqrt{-1} = i$, $\sqrt{-2} = \sqrt{2}i$, and $\sqrt{-4} = 2i$.

Cyclic Patterns of the Powers of $i$

The powers of the imaginary unit $i$ follow a specific repeating cycle of length four. This cyclic behavior is essential for simplifying expressions involving higher powers of $i$. The first four powers are calculated as follows:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = i^2 \times i = (-1) \times i = -i$

• $i^4 = (i^2)^2 = (-1)^2 = 1$

As the powers increase beyond 4, the pattern repeats:

• $i^5 = i^4 \times i = 1 \times i = i$

• $i^6 = i^4 \times i^2 = 1 \times (-1) = -1$

• $i^7 = i^4 \times i^3 = 1 \times (-i) = -i$

• $i^8 = (i^4)^2 = 1^2 = 1$

Generalizing this observations, powers of $i$ categorized by remainders when divided by 4 yield consistent results:

• $i^{1, 5, 9, 13} = i$

• $i^{2, 6, 10, 14} = -1$

• $i^{3, 7, 11, 15} = -i$

• $i^{4, 8, 12, 16} = 1$

Definition and Cartesian Form of Complex Numbers

A complex number is defined as any number that can be expressed in the form $z = x + iy$, where $x$ and $y$ are real numbers and $i = \sqrt{-1}$. The value $x$ is referred to as the real part of the complex number, while $y$ is referred to as the imaginary part. This expression style is known as the Cartesian form of a complex number. Examples include $4 + 5i$ and $3.7 + 2.5i$.

If given a specific complex number such as $z = 6 - 4i$, the real part ($x$) is $6$ and the imaginary part ($y$) is $-4$. Two complex numbers, $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$, are considered equal if and only if their real parts are identical ($x_1 = x_2$) and their imaginary parts are identical ($y_1 = y_2$). The set of complex numbers satisfies several algebraic properties, including the closure, associative, commutative, and distributive properties.

Operations on Complex Numbers

The set of complex numbers is closed under standard arithmetic operations, meaning that the sum, difference, product, or quotient of two or more complex numbers will always result in another complex number.

To add two complex numbers, one must add their real and imaginary parts respectively. Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. The sum is given by: $z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2)$ For example, $(6 + 3i) + (8 + 2i) = (6 + 8) + (3 + 2)i = 14 + 5i$.

To subtract two complex numbers, one must subtract their real and imaginary parts respectively. The difference is given by: $z_1 - z_2 = (x_1 + iy_1) - (x_2 + iy_2) = (x_1 - x_2) + i(y_1 - y_2)$ For example, $(6 + 3i) - (8 + 2i) = (6 - 8) + (3 - 2)i = -2 + i$.

Multiplication is defined by the rule: $z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2) = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)$ Expanding the product of $(6 + 3i)$ and $(8 + 2i)$ gives: $48 + 12i + 24i + 6i^2 = 48 + 36i + 6(-1) = 48 - 6 + 36i = 42 + 36i$.

The Complex Conjugate and Division

Before performing division, it is necessary to define the complex conjugate. The conjugate of a complex number $z = x + iy$, denoted by $\bar{z}$, is defined as $\bar{z} = x - iy$. It is important to note that the conjugate is not the same as the negative of the complex number, as the negative would be $-z = -(x + iy) = -x - iy$. The conjugate of a conjugate returns the original number ($\bar{\bar{z}} = z$).

Properties of the complex conjugate include:

1. $\overline{z_1 \times z_2} = \bar{z}_1 \times \bar{z}_2$

2. $\overline{\frac{z_1}{z_2}} = \frac{\bar{z}_1}{\bar{z}_2}$

3. $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$

To divide two complex numbers, the numerator and denominator are both multiplied by the conjugate of the denominator. If dividing $z_1 = x_1 + iy_1$ by $z_2 = x_2 + iy_2$, the process is: $\frac{z_1}{z_2} = \frac{x_1 + iy_1}{x_2 + iy_2} \times \frac{x_2 - iy_2}{x_2 - iy_2} = \frac{x_1x_2 + y_1y_2}{x_2^2 + y_2^2} + i\frac{x_2y_1 - x_1y_2}{x_2^2 + y_2^2}$ An example division of $\frac{6 + 3i}{8 + 2i}$ involves multiplying the top and bottom by $8 - 2i$: $\frac{(6 + 3i)(8 - 2i)}{8^2 + 2^2} = \frac{48 - 12i + 24i - 6i^2}{64 + 4} = \frac{48 + 6 + 12i}{68} = \frac{54 + 12i}{68} = \frac{54}{68} + i\frac{12}{68} = \frac{27}{34} + i\frac{3}{17}$ (Note: While the calculation process is consistent, the values derived in specific examples in source materials may vary due to arithmetic simplification steps).

Questions & Discussion

1. Which of the following is an irrational number? a) $\dots$ b) $\sqrt{2}$ c) $0.75$ d) $5$ Answer: (b) $\sqrt{2}$

2. Which set of numbers includes both positive and negative integers as well as zero? a) Natural Numbers b) Whole Numbers c) Integers d) Rational Numbers Answer: (c) Integers

3. What is the result of dividing two integers, 7 and 3? a) Always an integer b) Always a rational number c) Always an irrational number d) None of the above Answer: (b) Always a rational number

4. Which of the following statements is true? a) All rational numbers are integers. b) All integers are whole numbers. c) All whole numbers are natural numbers. d) All natural numbers are whole numbers. Answer: (d) All natural numbers are whole numbers.

5. Classwork 1: Determine the following square roots:

• $\sqrt{-16} = 4i$

• $\sqrt{-49} = 7i$

• $\sqrt{-a} = i\sqrt{a}$

• $\sqrt{-64} = 8i$

6. Classwork 2: Solve the following quadratic equations: 1) $x^2 - 6x + 13 = 0$ 2) $x^2 + 1 = 0$ 3) $5x^2 - 6x + 5 = 0$ 4) $x^2 - 6x + 34 = 0$

7. Classwork 3: Evaluate the following expressions: 1) $i^9 + 2i^{11} + i^{13}$ 2) $i^{10} + i^{15} + 3i^8$ 3) $i^{12} + i^{25} + 3i^{18}$

8. Classwork 4: Addition Operations: 1) $(4 + 5i) + (3 + 2i) = 7 + 7i$ 2) $(3 + 5i) + (5 + 4i) = 8 + 9i$ 3) $(5 + 7i) + (3 + 4i) = 8 + 11i$

9. Classwork 6: Subtraction Operations: 1) $(4 + 5i) - (3 + 2i) = 1 + 3i$ 2) $(3 + 5i) - (5 + 4i) = -2 + i$ 3) $(5 + 7i) - (3 + 4i) = 2 + 3i$

10. Classwork 7: Multiplication Operations: 1) $(4 + 5i)(3 + 2i) = 12 + 8i + 15i + 10i^2 = 2 + 23i$ 2) $(3 + 5i)(5 + 4i) = 15 + 12i + 25i + 20i^2 = -5 + 37i$ 3) $(5 + 7i)(3 + 4i) = 15 + 20i + 21i + 28i^2 = -13 + 41i$

11. Find the complex conjugate of $z$ if $z = -i$:Options: (A) $1$ (B) $-1$ (C) $i$ (D) $-i$ Answer: (C) $i$

12. Find the complex conjugate of $z$ if $z = 2 - 2 \times 3i$:Answer: $2 + 6i$

Past Questions on Complex Numbers

1. The imaginary part of $\frac{1+i}{1-i}$ is: a) (A) $1$ b) (B) $\frac{1}{2}$ c) (C) $2$ d) (D) $-2$

2. Find the complex conjugate of $z$ if $z = (2 + 3i)(4 + 8i)$: a) (A) $32 - 4i$ b) (B) $32 + 4i$ c) (C) $4 - 32i$ d) (D) $4 + 32i$

3. Express $\frac{1}{1 + 3i}$ in the form of $x + iy$: a) (A) $\frac{1}{10} - \frac{3}{10}i$ b) (B) $\frac{1}{10} + \frac{3}{10}i$ c) (C) $\frac{1}{7} - \frac{3}{10}i$ d) (D) $1 - 3i$

4. Find $a$ if $\frac{4 + i}{3 + i} = a + ib$: a) (A) $\frac{7}{10}$ b) (B) $\frac{11}{10}$ c) (C) $\dots$ d) (D) $\dots$ *(Note: Calculation indicates $a = \frac{(4+i)(3-i)}{10} = \frac{12-4i+3i+1}{10} = \frac{13}{10}$; note the options provided in context).

5. Find the real part of $z = \frac{1}{1 - i}$: a) (A) $1$ b) (B) $\frac{1}{2}$ c) (C) $\frac{1}{2}i$ d) (D) $1$

6. Find $x, y \in R$ such that $\frac{2y + xi}{2} + \frac{4 + x + i}{3} = 1$:Options: (A) $x=1; y=2$ (B) $x=0; y=1$ (C) $x=1; y=2$ (D) $x=0, y=3$

7. Find $a, b \in R$ if $a + ib = (2 - i)^2$: a) (A) $3, 4$ b) (B) $3, -4$ c) (C) $-3, -4$ d) (D) $3, 4$ (Calculated: $(2-i)^2 = 4 - 4i + i^2 = 3 - 4i$; implies $a=3$, $b=-4$).

8. Which of the following is a square root of $z = -2i$? a) (A) $\pm(2 + i)$ b) (B) $\pm(1 - i)$ c) (C) $\pm(1 + i)$ d) (D) $\pm(-1 + i)$ (Calculated: $(1-i)^2 = 1 - 2i + i^2 = -2i$; hence $\pm(1-i)$ is correct).