R.1 Real Numbers and Fundamental Properties

R.1 Real Numbers

Working with Sets

In mathematics, a set is a collection of distinct objects or elements defined by a rule. Sets are fundamental to understanding number classifications and algebraic structures.

• Example: The set of digits is D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

• Sets with a finite number of elements can be listed explicitly.

• For sets with infinite or complex membership, set-builder notation is used: A = {x | x ∈ ℝ, x ≥ 3} means "A is the set of real numbers x such that x is greater than or equal to 3."

Classifying Numbers

Numbers are classified into various types based on their properties. Understanding these classifications is essential for algebraic reasoning.

• Natural Numbers (Counting Numbers): ℕ = {1, 2, 3, 4, ...}

• Whole Numbers: ℕ₀ = {0, 1, 2, 3, ...}

• Integers: ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Rational Numbers: Numbers that can be expressed as a quotient $\frac{a}{b}$, where a and b are integers and b ≠ 0. ℚ = {x | x = $\frac{a}{b}$, a, b ∈ ℤ, b ≠ 0}

• Irrational Numbers: Numbers that cannot be written as $\frac{a}{b}$, where a and b are integers and b ≠ 0. Their decimal forms are nonterminating and nonrepeating (e.g., $\pi$, $e$, $\sqrt{2}$).

• Real Numbers: The set of all rational and irrational numbers.

Example: -8 is an integer and a real number, but not a natural or whole number.

Ellipsis (...): Indicates that a pattern continues indefinitely.

Rational vs. Irrational Numbers

• Rational Numbers: $\frac{1}{4} = 0.25$, $-\frac{2}{3} = -0.666...$, $\sqrt{49} = 7 = \frac{7}{1}$

• Irrational Numbers: $\pi = 3.14159...$, $e = 2.71828...$, $\sqrt{2} = 1.414213...$

The Order of Operations

PEMDAS Rule

When evaluating mathematical expressions, operations must be performed in a specific order, summarized by the acronym PEMDAS:

• P: Parentheses or grouping symbols. Perform operations within the innermost parentheses first.

• E: Exponents. Evaluate powers and roots.

• MD: Multiplication or Division. Perform these operations from left to right.

• AS: Addition or Subtraction. Perform these operations from left to right.

Example: Evaluate the expression $-6 + 3(15 - 4)^2 \div 18 - 3 \cdot 7$ by following PEMDAS.

Properties of Real Numbers

Reciprocal

The reciprocal of a nonzero real number b is $\frac{1}{b}$. For example, the reciprocal of 7 is $\frac{1}{7}$, and the reciprocal of $-\frac{2}{3}$ is $-\frac{3}{2}$.

Distributive Property

The distributive property of multiplication over addition states:

• $a \cdot (b + c) = a \cdot b + a \cdot c$

• Example: $4(7x - 3) = 28x - 12$

Rules of Signs

Negative signs in fractions can be written in several equivalent ways:

• $-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$

• Example: $-\frac{1}{4} = \frac{-1}{4} = \frac{1}{-4} = -0.25$

• When there are two terms in the numerator or denominator, apply the negative sign carefully: $-\frac{x}{x+5} = \frac{-x}{x+5} = \frac{x}{-(x+5)} = \frac{x}{-x-5}$

Multiplying and Adding Fractions

Multiplying Fractions

To multiply fractions, multiply the numerators and denominators directly. Simplify before multiplying if possible.

• Example: $\frac{4}{6} \cdot \frac{15}{16}$

Adding Fractions

To add fractions with the same denominator, add the numerators:

• $\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$

If denominators differ, find the least common multiple (LCM) of the denominators before adding:

• Example: $\frac{8}{15} + \frac{11}{12}$

• List multiples of 15 and 12 to find the LCM (e.g., 60 is the LCM of 15 and 12).

Type of Number	Definition	Example
Natural Numbers	Counting numbers	1, 2, 3, ...
Whole Numbers	Natural numbers plus zero	0, 1, 2, ...
Integers	Whole numbers and their negatives	-3, 0, 2, ...
Rational Numbers	Numbers expressible as $\frac{a}{b}$	$\frac{1}{4}$, $-\frac{2}{3}$
Irrational Numbers	Non-repeating, non-terminating decimals	$\pi$, $\sqrt{2}$