University Study Notes: Number Systems and Complex Numbers Sets, Intervals, and Complex Numbers

Real Number System Hierarchy and Classification

• The set of real numbers, denoted by the symbol $\mathbb{R}$, is the comprehensive set containing all rational and irrational numbers.

• The Hierarchy of Number Sets:

  • Natural Numbers ($\mathbb{N}$): The most basic set of counting numbers.

  • Whole Numbers ($\mathbb{W}$): The set containing all natural numbers plus zero ($0$).

  • Integers ($\mathbb{Z}$): Includes all whole numbers and their negative counterparts.

  • Rational Numbers ($\mathbb{Q}$): Numbers that can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. They have finite (terminating) or repeating decimal representations.

  • Irrational Numbers ($\mathbb{\bar{Q}}$): Numbers that cannot be expressed as a simple fraction and have non-terminating, non-repeating decimal representations.

• Set Relationships:

  • The relationships are defined by the subset chain: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

  • The union of rational and irrational numbers constitutes the real numbers: $\mathbb{Q} \cup \mathbb{\bar{Q}} = \mathbb{R}$.

• Refining Real Numbers Examples (Set $S$):

  • Given the set $S = \{-9, -7, 2, \pi, \frac{3}{1}, 0.4, 5.125125\}$, classifications are as follows:

    • Natural numbers ($\mathbb{N}$): $\frac{3}{1}$ (which equals $3$) and $2$.

    • Whole numbers ($\mathbb{W}$): $2$ and $\frac{3}{1}$.

    • Integers ($\mathbb{Z}$): $-9, -7, 2, \frac{3}{1}$.

    • Rational numbers ($\mathbb{Q}$): $-9, -7, 2, \frac{3}{1}, 0.4, 5.125125$.

    • Irrational numbers ($\mathbb{\bar{Q}}$): $\pi$.

    • Real numbers ($\mathbb{R}$): All elements in the set $S$.

Intervals and Number Line Representations

• Notation Conventions:

  • An empty circle ($\circ$) on a number line represents an open end point, meaning the value itself is not included in the set. This corresponds to the brackets $($ or $)$.

  • A dense circle (solid point) represents a closed end point, meaning the value is included. This corresponds to the brackets $[$ or $]$.

  • A dotted line is used specifically when representing sets of integers ($\mathbb{Z}$), whole numbers ($\mathbb{W}$), or natural numbers ($\mathbb{N}$) across a range, rather than a continuous real number interval.

• Types of Intervals:

  • Open Interval ($a, b$): Given by $a < x < b$. Both endpoints are excluded.

  • Closed Interval $[a, b]$: Given by $a \leq x \leq b$. Both endpoints are included.

  • Half-Open Intervals:

    • $(a, b]$: Given by $a < x \leq b$. Endpoint $a$ is excluded; $b$ is included.

    • $[a, b)$: Given by $a \leq x < b$. Endpoint $a$ is included; $b$ is excluded.

  • Infinite Intervals:

    • $(-\infty, b)$: Represented by $-\infty < x < b$.

    • $(-\infty, b]$: Represented by $-\infty < x \le b$.

    • $(a, \infty)$: Represented by $a < x < \infty$.

    • $[a, \infty)$: Represented by $a \le x < \infty$.

• Examples of Conversions:

  • The interval $[2, 5]$ represents all real numbers from $2$ to $5$ inclusive and is a closed interval.

  • The set $\{x : x \leq 0, x \in \mathbb{R}\}$ is equivalent to the interval $(-\infty, 0]$.

Intersection and Union of Intervals

• Intersection ($\cap$): Represents the region where two intervals overlap (indicated by the word "and").

  • Example: Finding the intersection of two intervals involves identifying common values shared by both sets on the real number line.

• Union ($\cup$): Represents the total region covered by either or both intervals (indicated by the words "all" or "or").

  • Example: The union of $(2, 4]$ and $(4, 5)$ results in the continuous open/half-open segment from the two sets.

• Complex Set Simplification:

  • If $A = \{x : 1 < x < 7, x \in \mathbb{Z}\}$, then $A = \{2, 3, 4, 5, 6\}$.

  • If $B = \{-6, -4, -2, 0, 2, 4, 6\}$, then the union $A \cup B = \{-6, -4, -2, 0, 2, 3, 4, 5, 6\}$.

  • The intersection $B \cap C$, where $C = \{x : 1 < x < 5, x \in \mathbb{R}\}$, extracts integers from set $B$ that fall between $1$ and $5$, resulting in $\cap = \{2, 4\}$.

Cartesian Form of Complex Numbers

• Fundamental Definition: A complex number $z$ is written in Cartesian form as $z = a + bi$, where:

  • $a$ is the real part ($\text{Re}(z)$).

  • $b$ is the imaginary part ($\text{Im}(z)$).

  • $i$ is the imaginary unit, defined such that $i^2 = -1$ or $i = \sqrt{-1}$.

• Equality of Complex Numbers:

  • Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are equal ($z_1 = z_2$) if and only if their real parts are equal ($a = c$) and their imaginary parts are equal ($b = d$).

  • Example: If $3 + 2i = m + ni$, then $m = 3$ and $n = 2$.

  • Example: Given $x + 4 + i = 10 - (2y - 7)i$, equating parts gives:

    • Real: $x + 4 = 10 \Rightarrow x = 6$.

    • Imaginary: $1 = -(2y - 7) \Rightarrow 1 = -2y + 7 \Rightarrow 2y = 6 \Rightarrow y = 3$.

Algebraic Operations and Conjugates

• The Conjugate: The conjugate of a complex number $z = a + bi$, denoted as $\bar{z}$, is obtained by reversing the sign of the imaginary part ($z = a - bi$).

  • Example: If $z = 5 + 2i$, then $\bar{z} = 5 - 2i$.

  • Example: If $z = -7 - 7i$, then $\bar{z} = -7 + 7i$.

  • Example: If $z = 5i$ (purely imaginary), then $\bar{z} = -5i$.

• Division and Rationalization:

  • A complex number cannot remain in the denominator. To simplify, multiply both the numerator and the denominator by the conjugate of the denominator.

  • Example Calculation: $\frac{4 + 7i}{2 + 5i}$

    • Multiply by denominator conjugate: $\frac{(4 + 7i)(2 - 5i)}{(2 + 5i)(2 - 5i)}$

    • Expand numerator: $8 - 20i + 14i - 35i^2 = 8 - 6i + 35 = 43 - 6i$

    • Expand denominator: $4 - 25i^2 = 4 + 25 = 29$

    • Result: $\frac{43}{29} - \frac{6}{29}i$

Polar Form of Complex Numbers

• General Expression: Polar form is expressed as $z = r(\cos(\theta) + i\sin(\theta))$, where:

  • $r$ is the modulus (magnitude).

  • $\theta$ is the argument (angle in radians).

• The iMAAP Method for Polar Conversion:

  1. Identity (I): Identify the real part $a$ and imaginary part $b$.

  2. Modulus (M): Calculate $r = |z| = \sqrt{a^2 + b^2}$. Note that $r > 0$.

  3. Alpha (A): Calculate the basic angle $\alpha = \tan^{-1}\left(\frac{|b|}{|a|}\right)$.

  4. Argand Diagram (A): Plot the complex number to determine which quadrant it lies in to solve for $\theta$.

     • Quadrant I: $\theta = \alpha$

     • Quadrant II: $\theta = \pi - \alpha$

     • Quadrant III: $\theta = -(\pi - \alpha)$

     • Quadrant IV: $\theta = -\alpha$

  5. Polar Form (P): Substitute $r$ and $\theta$ into the formula $z = r(\cos(\theta) + i\sin(\theta))$.

• Example: Converting $z = -1 - i\sqrt{3}$ to Polar Form:

  • I: $a = -1$, $b = -\sqrt{3}$.

  • M: $r = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2$.

  • A: $\alpha = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$.

  • A: The point lies in Quadrant III ($a < 0$, $b < 0$).

    • $\theta = -(\pi - \frac{\pi}{3}) = -\frac{2\pi}{3}$.

  • P: $z = 2(\cos(-\frac{2\pi}{3}) + i\sin(-\frac{2\pi}{3}))$.

• Argument Range: The argument $\theta$ must satisfy the condition $-\pi < \theta \leq \pi$.