Numbers and Elementary Calculations

This section covers the fundamental mathematical operations and calculations that form the foundation of mathematical study.

Numbers and Number Systems

Numbers are the basic building blocks of mathematics.

Basic Types of Numbers

Numbers can be categorized into different types based on their properties, forming a hierarchical structure:

Natural Numbers: $\mathbb{N}$ - Positive integers starting from 1: $\{1, 2, 3, 4, \ldots\}$
Whole Numbers: $\mathbb{N}_0$ - Natural numbers including zero: $\{0, 1, 2, 3, \ldots\}$
Integers: $\mathbb{Z}$ - Whole numbers including negatives: $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$
Rational Numbers: $\mathbb{Q}$ - Numbers expressible as fractions: $\left\{\frac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0\right\}$
Irrational Numbers: Numbers that cannot be expressed as fractions: $\{\sqrt{2}, \pi, e, \ldots\}$
Real Numbers: $\mathbb{R}$ - All rational and irrational numbers combined
Imaginary Numbers: Multiples of the imaginary unit $i$: $\{bi \mid b \in \mathbb{R}, i^2 = -1\}$
Complex Numbers: $\mathbb{C}$ - Numbers with real and imaginary parts: $\{a + bi \mid a, b \in \mathbb{R}\}$

Extended Types of Numbers

In addition to the basic types, there are several extended types of numbers used in advanced mathematics:

Algebraic Number: Roots of polynomial equations with integer coefficients (e.g., $\sqrt{2}$, which is a root of $x^2 - 2 = 0$, a quadratic equation)
Transcendental Numbers: Irrational numbers that are not algebraic (e.g., $e$, $\pi$)
Ordinal Numbers: Numbers indicating position in a sequence (e.g., first, second, third)
Cardinal Numbers: Numbers indicating quantity (e.g., one, two, three)
Transfinite Numbers: Numbers extending beyond finite quantities, used in set theory (e.g., $\aleph_0$, cardinality of natural numbers)
Hyperreal Numbers: Extension of reals including infinitesimals and infinite numbers, used in non-standard analysis
Surreal Numbers: Class including reals, infinitesimals, and infinite numbers, used in game theory
P-adic Numbers: Number system based on divisibility by prime $p$, used in number theory
Quaternions: Four-dimensional extension of complex numbers: $a + bi + cj + dk$
Octonions: Eight-dimensional extension of quaternions, used in physics and computer graphics
Hypercomplex Numbers: Generalizations of complex numbers including quaternions and octonions
Dual Numbers: Extension of reals with infinitesimal unit $\varepsilon$ where $\varepsilon^2 = 0$

Important Number Categories

Even Numbers: Integers divisible by 2 (e.g., -4, -2, 0, 2, 4, ...)
Odd Numbers: Integers not divisible by 2 (e.g., -3, -1, 1, 3, 5, ...)
Prime Numbers: Natural numbers greater than 1 with no divisors other than 1 and themselves (e.g., 2, 3, 5, 7, 11, ...)
Composite Numbers: Natural numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9, 10, ...)
Perfect Numbers: Positive integers equal to the sum of their proper divisors (e.g., 6, 28, 496)

Arithmetic Operations

Basic arithmetic operations form the foundation of mathematical computation and follow specific rules and properties.

Fundamental Operations

The four basic arithmetic operations:

Addition: $(+)$ - Combining quantities
Subtraction: $(-)$ - Finding the difference between quantities
Multiplication: $(\times \text{ or } \cdot)$ - Repeated addition
Division: $(\div \text{ or } /)$ - Repeated subtraction or inverse of multiplication

Order of Operations (PEMDAS/BODMAS)

Operations must be performed in the correct sequence:

Parentheses/Brackets: Operations inside grouping symbols first
Exponents/Orders: Powers and roots second
Multiplication and Division: From left to right, third
Addition and Subtraction: From left to right, last

The BODMAS acronym is used in some regions, where "B" stands for Brackets and "O" for Orders (exponents).

Key Properties of Arithmetic Operations

Essential properties that govern arithmetic operations:

Commutative Property:
- Addition: $a + b = b + a$
- Multiplication: $a \times b = b \times a$
- Note: Subtraction and division are NOT commutative
Associative Property:
- Addition: $(a + b) + c = a + (b + c)$
- Multiplication: $(a \times b) \times c = a \times (b \times c)$
- Note: Subtraction and division are NOT associative
Distributive Property:
- Allows distributing multiplication over addition: $a(b + c) = ab + ac$
Identity Elements:
- Additive identity: $a + 0 = a$
- Multiplicative identity: $a \times 1 = a$
Inverse Elements:
- Additive inverse: $a + (-a) = 0$
- Multiplicative inverse: $a \times \frac{1}{a} = 1$ (for $a \neq 0$)
Special Properties:
- Zero property of multiplication: $a \times 0 = 0$
- Division by zero is undefined
- Absolute value: $|a|$ represents distance from zero

Working with Signed Numbers

Rules for operations with positive and negative numbers:

Addition:
- Same signs: Add absolute values, keep the sign
  - Example: $(-3) + (-5) = -(3 + 5) = -8$
  - Example: $7 + 4 = 11$
- Different signs: Subtract smaller absolute value from larger, use sign of larger
  - Example: $(-7) + 3 = -(7 - 3) = -4$
  - Example: $8 + (-5) = 8 - 5 = 3$
Subtraction:
- subtracting equals adding the opposite: $a - b = a + (-b)$
  - Example: $5 - 8 = 5 + (-8) = -3$
  - Example: $(-4) - (-7) = (-4) + 7 = 3$
Multiplication and Division:
- Same signs: Result is positive
  - Example: $(-6) \times (-4) = 24$
  - Example: $15 \div 3 = 5$
- Different signs: Result is negative
  - Example: $(-8) \times 3 = -24$
  - Example: $20 \div (-4) = -5$

Fractions

Fractions represent parts of a whole and are fundamental to understanding rational numbers.

Basic Fraction Concepts

Essential terminology and notation:

Fraction Form: Written as $\frac{a}{b}$ where $a$ is the numerator and $b$ is the denominator ($b \neq 0$)
Proper Fraction: Numerator less than denominator (e.g., $\frac{3}{4}$)
Improper Fraction: Numerator greater than or equal to denominator (e.g., $\frac{5}{3}$)
Mixed Number: Whole number plus proper fraction (e.g., $1\frac{2}{3} = \frac{5}{3}$)
Unit Fraction: Numerator equals 1 (e.g., $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{5}$)

Equivalent Fractions

Fractions representing the same value:

Fundamental Principle: Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent fraction
Examples: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \ldots$
Cross-Multiplication Test: $\frac{a}{b} = \frac{c}{d}$ if and only if $ad = bc$

Operations with Fractions

Systematic methods for operating with fractions:

Addition and Subtraction:
- Same denominator: Add/subtract numerators, keep denominator
  - Example: $\frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7}$
  - Example: $\frac{5}{8} - \frac{1}{8} = \frac{5-1}{8} = \frac{4}{8} = \frac{1}{2}$
- Different denominators: Find common denominator first
  - Example: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$
  - Example: $\frac{3}{5} - \frac{1}{3} = \frac{9}{15} - \frac{5}{15} = \frac{4}{15}$
- General form: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$
Multiplication:
- Multiply numerators and denominators: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
  - Example: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$
  - Example: $\frac{3}{4} \times \frac{2}{9} = \frac{6}{36} = \frac{1}{6}$ (after simplification)
- Simplify by canceling common factors before multiplying
  - Example: $\frac{4}{9} \times \frac{3}{8} = \frac{4 \times 3}{9 \times 8} = \frac{1}{6}$ (canceling 4 and 8, then 3 and 9)
Division:
- Multiply by reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
  - Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$
  - Example: $\frac{7}{8} \div \frac{3}{4} = \frac{7}{8} \times \frac{4}{3} = \frac{28}{24} = \frac{7}{6}$
- Remember: "invert and multiply"

Fraction Simplification

Reducing fractions to lowest terms using systematic methods:

Find the GCD:
- Find the greatest common divisor of numerator and denominator
- Example: For $\frac{12}{18}$, $\gcd(12,18) = 6$, so $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
Prime factorization method:
- Factor both parts and cancel common prime factors
- Example: $\frac{24}{36} = \frac{2^3 \times 3}{2^2 \times 3^2} = \frac{2 \times 3}{3^2} = \frac{6}{9} = \frac{2}{3}$
Visual method:
- Look for obvious common factors (2, 3, 5, etc.)
- Example: $\frac{15}{25} = \frac{3 \times 5}{5 \times 5} = \frac{3}{5}$
Euclidean algorithm:
- Systematic method for finding GCD using the Euclidean algorithm
- For $\gcd(48,18)$: $48 = 18 \times 2 + 12$, then $18 = 12 \times 1 + 6$, then $12 = 6 \times 2 + 0$, so $\gcd = 6$

Comparing Fractions

Methods for determining which fraction is larger:

Cross-multiplication:
- To compare $\frac{a}{b}$ and $\frac{c}{d}$, compare $ad$ with $bc$
- Example: Compare $\frac{2}{3}$ and $\frac{3}{5}$: $2 \times 5 = 10$, $3 \times 3 = 9$, so $\frac{2}{3} > \frac{3}{5}$
Common denominator:
- Convert to same denominator and compare numerators
- Example: $\frac{2}{3}$ vs $\frac{3}{5}$ becomes $\frac{10}{15}$ vs $\frac{9}{15}$, so $\frac{2}{3} > \frac{3}{5}$
Decimal conversion:
- Convert to decimals for easy comparison
- Example: $\frac{2}{3} \approx 0.667$, $\frac{3}{5} = 0.6$, so $\frac{2}{3} > \frac{3}{5}$

Powers and Exponents

Powers and exponents provide a compact way to represent repeated multiplication and form the basis for more advanced mathematical concepts.

Basic Exponent Notation

Understanding the components and meaning:

Base: The number being multiplied repeatedly
Exponent: How many times the base multiplies itself
Power: The result of the exponentiation
Exponential Form: $a^n$ where $a$ is base, $n$ is exponent

Fundamental Exponent Laws

These rules apply to all real number bases (with appropriate restrictions):

Product Rule: $a^m \times a^n = a^{m+n}$
- Example: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$
- Example: $x^5 \times x^2 = x^7$
Quotient Rule: $a^m \div a^n = a^{m-n}$ (where $a \neq 0$)
- Example: $3^6 \div 3^2 = 3^{6-2} = 3^4 = 81$
- Example: $y^8 \div y^3 = y^5$
Power Rule: $(a^m)^n = a^{mn}$
- Example: $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$
- Example: $(x^4)^3 = x^{12}$
Product to Power: $(ab)^n = a^n b^n$
- Example: $(3 \times 4)^2 = 3^2 \times 4^2 = 9 \times 16 = 144$
- Example: $(xy)^3 = x^3 y^3$
Quotient to Power: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (where $b \neq 0$)
- Example: $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$
- Example: $\left(\frac{x}{y}\right)^4 = \frac{x^4}{y^4}$

Special Exponent Cases

Important special values and their meanings:

Zero Exponent: $a^0 = 1$ (for $a \neq 0$)
Unit Exponent: $a^1 = a$
Negative Exponents: $a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$)
Fractional Exponents: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Sign Rules for Exponents

How signs behave with different exponent types:

Even Exponents: Always produce non-negative results: $a^{2n} \geq 0$
Odd Exponents: Preserve the sign of the base
Negative Bases: Use parentheses to clarify: $(-2)^3 = -8$ vs $-2^3 = -8$

Systematic Approach to Simplifying Exponential Expressions

Step 1: Identify like bases

Example 1: $2^3 \times 2^5 = 2^{3+5} = 2^8 = 256$
Example 2: $x^4 \times x^2 \times x = x^{4+2+1} = x^7$
Example 3: $\frac{3^7}{3^4} = 3^{7-4} = 3^3 = 27$

Step 2: Apply exponent laws

Product rule: $(2x)^3 \times (2x)^2 = (2x)^{3+2} = (2x)^5 = 32x^5$
Power rule: $(x^3)^4 = x^{3 \times 4} = x^{12}$
Quotient rule: $\frac{y^8}{y^3} = y^{8-3} = y^5$

Step 3: Handle negative exponents

Convert to positive: $x^{-3} = \frac{1}{x^3}$
Simplify fractions: $\frac{2^{-2}}{3^{-1}} = \frac{1/4}{1/3} = \frac{1}{4} \times \frac{3}{1} = \frac{3}{4}$
Mixed terms: $2x^{-2}y^3 = \frac{2y^3}{x^2}$

Step 4: Simplify complex expressions

Example: $(2x^3)^2 \times (3x^{-1})^3$
- Expand each: $(2x^3)^2 = 4x^6$ and $(3x^{-1})^3 = 27x^{-3}$
- Combine: $4x^6 \times 27x^{-3} = 108x^{6-3} = 108x^3$

Step 5: Evaluate numerical powers

Simple powers: $2^4 = 16$, $3^3 = 27$, $5^2 = 25$
Zero exponents: $7^0 = 1$, $(2x)^0 = 1$ (for $x \neq 0$)
Unit exponents: $x^1 = x$

Step 6: Convert between forms when helpful

Radical to exponential: $\sqrt{x^3} = x^{3/2}$
Exponential to radical: $x^{2/3} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2$
Mixed conversion: $2\sqrt{x} \times x^{3/2} = 2x^{1/2} \times x^{3/2} = 2x^{1/2+3/2} = 2x^2$

Roots and Radicals

Roots are the inverse operations of powers and are essential for solving equations and simplifying expressions.

Basic Root Concepts

Fundamental terminology and notation:

Root Definition: If $r^n = x$, then $r$ is an $n$-th root of $x$
Radical Notation: $\sqrt[n]{x}$ represents the $n$-th root of $x$
Radicand: The expression under the radical sign
Index: The small number indicating which root (2 for square root, 3 for cube root, etc.)
Principal Root: The non-negative root (for even indices of positive numbers)

Types of Roots

Different roots have different properties:

Square Roots: $(\sqrt{\phantom{x}})$ - Most common; every positive number has two square roots $(\pm)$
Cube Roots: $(\sqrt[3]{\phantom{x}})$ - Every real number has exactly one real cube root
Even Roots: Require non-negative radicands for real results
Odd Roots: Defined for all real numbers, preserve sign of radicand

Radical Properties

Rules governing radical operations:

Product Property: $\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$
Quotient Property: $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ (where $b \neq 0$)
Power Property: $\sqrt[n]{a^m} = a^{m/n}$
Root of Root: $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$

Systematic Approach to Simplifying Radicals

Step 1: Factor out perfect powers

Example 1: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
Example 2: $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$
Example 3: $\sqrt[3]{24} = \sqrt[3]{8 \times 3} = 2\sqrt[3]{3}$

Step 2: Combine like radicals

Example 1: $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$
Example 2: $4\sqrt{3} - 7\sqrt{3} = -3\sqrt{3}$
Example 3: $2\sqrt{5} + 3\sqrt{2}$ (cannot be simplified further)

Step 3: Simplify complex expressions

Example: $2\sqrt{12} + 3\sqrt{27}$
- Simplify each: $2\sqrt{12} = 2(2\sqrt{3}) = 4\sqrt{3}$ and $3\sqrt{27} = 3(3\sqrt{3}) = 9\sqrt{3}$
- Combine: $4\sqrt{3} + 9\sqrt{3} = 13\sqrt{3}$

Step 4: Rationalize denominators

Simple: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
Complex: $\frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$

Step 5: Use radical identities

Multiplication: $\sqrt{6} \times \sqrt{10} = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$
Division: $\sqrt{20} \div \sqrt{5} = \sqrt{\frac{20}{5}} = \sqrt{4} = 2$

Special Cases of Roots

Important root behaviors:

Even roots of positive numbers: Two real solutions (±)
Even roots of negative numbers: Complex solutions
Odd roots: One real solution for any real number
Principal roots: Non-negative by convention

Algebraic Expression Simplification

Algebraic expressions can be simplified using various systematic techniques.

Key Simplification Techniques

Essential methods for reducing expressions:

Combining like terms: Group terms with same variables and exponents
Distributive property: $a(b + c) = ab + ac$
Factoring: Find common factors to simplify
Expanding: Multiply out parentheses when beneficial

Common Algebraic Identities

Useful patterns for simplification:

Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
Perfect square trinomials: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
Sum/difference of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

Example Simplifications

Practical applications with step-by-step solutions:

Combining Like Terms

Simple case: $3x + 5x = (3 + 5)x = 8x$
Mixed terms: $4x^2 + 2x + 7x^2 - 3x = (4 + 7)x^2 + (2 - 3)x = 11x^2 - x$
Multiple variables: $2xy + 3x + 5xy - x = (2 + 5)xy + (3 - 1)x = 7xy + 2x$

Distribution and Expansion

Simple distribution: $2(x + 3) = 2x + 6$
Negative distribution: $-3(2x - 5) = -6x + 15$
FOIL method: $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$
Complex expansion: $(2x - 1)(3x + 4) = 6x^2 + 8x - 3x - 4 = 6x^2 + 5x - 4$

Factoring Applications

Common factor: $6x^2 + 9x = 3x(2x + 3)$
Difference of squares: $x^2 - 4 = (x - 2)(x + 2)$
Perfect square trinomial: $x^2 + 6x + 9 = (x + 3)^2$
Factoring by grouping: $x^3 + 2x^2 + 3x + 6 = x^2(x + 2) + 3(x + 2) = (x^2 + 3)(x + 2)$

Complex Simplification Example

Simplify: $(3x + 2)(x - 1) - 2(x^2 - 3x + 1)$

Step 1: Expand $(3x + 2)(x - 1) = 3x^2 - 3x + 2x - 2 = 3x^2 - x - 2$
Step 2: Expand $-2(x^2 - 3x + 1) = -2x^2 + 6x - 2$
Step 3: Combine: $(3x^2 - x - 2) + (-2x^2 + 6x - 2) = x^2 + 5x - 4$