Real Numbers, Exponents, and Cartesian Coordinates

The Real Number System

Real numbers, denoted by the set $\mathbb{R}$, are numbers that can be expressed as decimal expansions. An element $a$ belonging to the real numbers is written as $a \in \mathbb{R}$.

• Natural Numbers ($\mathbb{N}$): The set of positive integers: $\mathbb{N} = \{1, 2, 3, \ldots\}$.

• Integers ($\mathbb{Z}$): The set of whole numbers and their negatives: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, 3, \ldots\}$.

• Rational Numbers ($\mathbb{Q}$): Numbers expressible in the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$. Rational numbers have decimal expansions that are either finite (e.g., $\frac{1}{4} = 0.25$) or infinite repeating (e.g., $\frac{1}{3} = 0.3333\ldots = 0.\overline{3}$).

• Irrational Numbers: Real numbers that cannot be written as a ratio of two integers (e.g., $\pi$, $\sqrt{2}$, historically identified with Hippasus). Their decimal expansions are infinite and nonrepeating.

Decimal expansions are unique with one exception: every finite decimal is identical to an infinite decimal with repeating $9$s (e.g., $\frac{1}{2} = 0.5 = 0.4999\ldots$ and $1 = 0.9999\ldots$).

Exponentiation and Radical Rules

For base $b$ and real exponents $x, y$:

• Exponent Zero: $b^0 = 1$ (for $b \neq 0$, e.g., $\pi^0 = 1$)

• Product Rule: $b^x b^y = b^{x+y}$

• Quotient Rule: $\frac{b^x}{b^y} = b^{x-y}$

• Negative Exponent Rule: $b^{-x} = \frac{1}{b^x}$

• Power to a Power: $(b^x)^y = b^{xy}$

• Radicals and Fractional Exponents: $b^{\frac{1}{n}} = \sqrt[n]{b}$

Key simplifications:

• $\frac{(-4)^{17}}{2^{40}} = \frac{(-1)^{17} (2^2)^{17}}{2^{40}} = -\frac{2^{34}}{2^{40}} = -\frac{1}{2^6} = -\frac{1}{64}$

• $25^{\frac{1}{2}} = (5^2)^{\frac{1}{2}} = 5$

• $(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y}) = (\sqrt{x})^2 - (\sqrt{y})^2 = x - y$ (for $x, y \ge 0$)

• $\sqrt{32} + \sqrt{18} = 4\sqrt{2} + 3\sqrt{2} = 7\sqrt{2}$

Absolute Value and Distance

The absolute value of a real number $a$, denoted $|a|$, represents its directed distance from the origin on the real number line:

$|a| = \begin{cases} a & \text{if } a \ge 0 \\ -a & \text{if } a < 0 \end{cases}$

• Distance Between Two Numbers: The distance between $a$ and $b$ on the real line is given by the length of the connecting segment: $|b - a| = |a - b|$.

• Non-additivity: In general, $|a + b| \neq |a| + |b|$. For example, if $a = -1$ and $b = 2$, then $|a + b| = |-1 + 2| = 1$, whereas $|a| + |b| = |-1| + |2| = 3$.

• Triangle Inequality: For all real numbers $a$ and $b$:

$|a + b| \le |a| + |b|$

Interval Notation

For real numbers $a$ and $b$ with $a < b$:

• Closed Interval: $[a, b] = \{x \in \mathbb{R} \mid a \le x \le b\}$

• Open Interval: $(a, b) = \{x \in \mathbb{R} \mid a < x < b\}$

• Half-Open Intervals: $[a, b) = \{x \in \mathbb{R} \mid a \le x < b\}$ and $(a, b] = \{x \in \mathbb{R} \mid a < x \le b\}$

• Ray / Infinite Intervals:

  • $[a, \infty) = \{x \in \mathbb{R} \mid x \ge a\}$

  • $(a, \infty) = \{x \in \mathbb{R} \mid x > a\}$

  • $(-\infty, b] = \{x \in \mathbb{R} \mid x \le b\}$

  • $(-\infty, b) = \{x \in \mathbb{R} \mid x < b\}$

Infinity ($\infty$) is a concept indicating unboundedness and is not a real number. Exponent signs affect set membership; for example, $-2^{40} \notin [-2, \infty)$ because $-2^{40} < -2$, whereas $(-2)^{40} \in [-2, \infty)$ because $(-2)^{40} > 0$.

Cartesian Coordinate System

The Cartesian coordinate plane is named after René Descartes (associated with the philosophical phrase "Cogito, ergo sum").

• Axes and Origin: Constructed from two perpendicular real number lines: the horizontal $x$-axis and vertical $y$-axis. Their point of intersection is the origin $(0, 0)$, dividing the plane into four quadrants.

• Coordinates: Every point corresponds to an ordered pair $(x, y)$, where the $x$-coordinate is the directed distance from the $y$-axis and the $y$-coordinate is the directed distance from the $x$-axis.