1.1 Real Numbers: Algebra Essentials

Real numbers form the foundation of algebra, encompassing rational and irrational numbers. Understanding their subsets and how they relate to each other is crucial for everything else in this course, from solving equations to graphing functions.

This section also covers algebraic expressions and the rules for manipulating them. Getting comfortable with order of operations, properties of real numbers, and simplification techniques now will pay off throughout the semester.

Real Number System

Subsets of Real Numbers

The real number system is organized like a set of nested boxes, where each smaller subset fits inside a larger one.

• Natural numbers (counting numbers): $1, 2, 3, ...$ These are the numbers you'd use to count objects.
• Whole numbers: $0, 1, 2, 3, ...$ Same as natural numbers, but with zero included.
• Integers: $..., -3, -2, -1, 0, 1, 2, 3, ...$ Whole numbers plus their negatives.
• Rational numbers: Any number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. This includes fractions, but also terminating decimals like $0.25$ and repeating decimals like $0.\overline{3} = 0.333...$
• Irrational numbers: Numbers that cannot be written as a fraction of two integers. Their decimal expansions go on forever without repeating. Common examples are $\pi$, $\sqrt{2}$, and $e$.

Every number on this list is a real number. The real numbers are the union of all rational and irrational numbers together.

A common point of confusion: every integer is also a rational number (for example, $5 = \frac{5}{1}$), and every whole number is also an integer. The subsets nest inside each other.

Representation of Real Numbers

• Number line: A horizontal line where each point corresponds to exactly one real number. Points to the right are greater; points to the left are smaller.
• Interval notation: A compact way to describe a range of numbers. Brackets $[ \ ]$ mean the endpoint is included; parentheses $( \ )$ mean it's excluded. For example, $[2, 5)$ means all real numbers from 2 to 5, including 2 but not 5.
• Set-builder notation: Describes a set by stating a condition its members must satisfy, such as $\{x \mid x > 3\}$, which reads "the set of all $x$ such that $x$ is greater than 3."

Algebraic Operations and Expressions

Order of Operations

The PEMDAS mnemonic guides the order in which you perform operations:

1. Parentheses (and other grouping symbols like brackets or fraction bars): Simplify inside these first.
2. Exponents: Evaluate powers and roots.
3. Multiplication and Division: Work left to right. These have equal priority, so you handle whichever comes first as you read left to right.
4. Addition and Subtraction: Work left to right, same equal-priority rule.

A common mistake is treating multiplication as always coming before division (or addition before subtraction). They're done left to right within their pair.

For example, $8 \div 2 \times 4 = 4 \times 4 = 16$, not $8 \div 8 = 1$.

Properties of Real Numbers

These properties are the rules that let you rearrange and simplify expressions. They may seem obvious, but being able to name them matters when you're justifying steps.

• Commutative property: You can swap the order of operands.
  • Addition: $a + b = b + a$
  • Multiplication: $a \times b = b \times a$
• Associative property: You can regroup operands without changing the result.
  • Addition: $(a + b) + c = a + (b + c)$
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$
• Distributive property: Multiplication distributes over addition.
  • $a(b + c) = ab + ac$
• Identity property: Operating with the identity element leaves a value unchanged.
  • Addition: $a + 0 = a$ (0 is the additive identity)
  • Multiplication: $a \times 1 = a$ (1 is the multiplicative identity)
• Inverse property: Operating with an inverse produces the identity element.
  • Addition: $a + (-a) = 0$
  • Multiplication: $a \times \frac{1}{a} = 1$ for $a \neq 0$

Note that subtraction and division are not commutative or associative. $5 - 3 \neq 3 - 5$, and $12 \div 4 \neq 4 \div 12$.

Evaluation of Algebraic Expressions

To evaluate an expression for given values of its variables:

1. Substitute the given values in place of each variable.
2. Apply order of operations (PEMDAS) to simplify.
3. Calculate to get a single numerical answer.

For example, evaluate $3x^2 - 2x + 1$ when $x = 4$:

1. Substitute: $3(4)^2 - 2(4) + 1$

2. Exponent first: $3(16) - 2(4) + 1$

3. Multiply: $48 - 8 + 1$

4. Add/subtract left to right: $41$

Simplification of Complex Expressions

Several techniques come up repeatedly when simplifying:

• Combine like terms: Terms with the same variable raised to the same power can be added or subtracted. For example, $5x^2 + 3x^2 = 8x^2$, but $5x^2 + 3x$ cannot be combined.
• Factor out common factors: If every term shares a factor, pull it out. For example, $6x + 9 = 3(2x + 3)$.
• Simplify fractions: Divide the numerator and denominator by their greatest common factor (GCF).

Properties of exponents are used constantly in simplification:

• Multiply powers with the same base: $a^m \times a^n = a^{m+n}$
• Divide powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$
• Power of a power: $(a^m)^n = a^{mn}$
• Power of a product: $(ab)^m = a^m b^m$
• Power of a quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ for $b \neq 0$

Rationalizing denominators: When a denominator contains a radical, multiply the numerator and denominator by the conjugate (or by the radical itself for a single-term denominator) to eliminate the radical from the denominator. For example, to rationalize $\frac{1}{\sqrt{3}}$, multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ to get $\frac{\sqrt{3}}{3}$.

Absolute value represents the distance of a number from zero on the number line. It's always non-negative: $|5| = 5$ and $|-5| = 5$.

Arithmetic Operations and Polynomials

A polynomial is an expression made up of terms that involve variables raised to non-negative integer exponents, combined using addition, subtraction, and multiplication. For example, $4x^3 - 2x + 7$ is a polynomial, but $3x^{-1}$ is not (because the exponent is negative).

You can add, subtract, and multiply polynomials using the same properties covered above. Adding and subtracting polynomials comes down to combining like terms. Multiplying polynomials requires distributing each term in one polynomial across every term in the other.