Algebraic Expressions and Sets of Numbers

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations (such as addition, subtraction, multiplication, and division). Algebraic expressions do not contain equal signs.

• Numbers: Constants or coefficients (e.g., 5 in 5x).

• Variables: Letters that represent unknown values (e.g., x, y, z).

• Operations: Addition (+), subtraction (−), multiplication (×), division (÷).

Examples:

• 2x

• 3x + 5

• \( \frac{x+5}{6} \)

• z^3

In the expression 5x + 2:

• x is the variable

• 5 is the coefficient

• 2 is the constant

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given value(s) for the variable(s) and perform the operations in the correct order.

• Step 1: Substitute the given values for each variable.

• Step 2: Follow the order of operations (PEMDAS/BODMAS).

Example: Evaluate \( 3x - y \) when \( x = 15,\ y = 4 \)

• Substitute: \( 3(15) - 4 \)

• Multiply: \( 45 - 4 \)

• Answer: \( 41 \)

Sets of Numbers

A set is a collection of numbers or objects. In algebra, we classify numbers into several important sets:

• Natural Numbers: Counting numbers: \( 1, 2, 3, 4, 5, \ldots \)

• Whole Numbers: Natural numbers plus zero: \( 0, 1, 2, 3, 4, 5, \ldots \)

• Integers: Positive and negative whole numbers, including zero: \( \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \)

• Rational Numbers: Numbers that can be written as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). Examples: \( \frac{1}{2},\ \frac{3}{4},\ -5,\ 0.25 \)

• Irrational Numbers: Numbers that cannot be written as a fraction of two integers. Their decimals never end and never repeat. Examples: \( \sqrt{2},\ \sqrt{3},\ \pi \)

• Real Numbers: All rational and irrational numbers.

Number Set Hierarchy

The sets of numbers are nested as follows:

\( \boxed{\text{Natural} \subset \text{Whole} \subset \text{Integers} \subset \text{Rational} \subset \text{Real}} \)

Think of each set as a box inside a larger box. Irrational numbers are real numbers but not rational.

Absolute Value

The absolute value of a number is its distance from zero on the number line. Absolute value is always non-negative.

• Symbol: \( |x| \)

• Examples:

  • \( |5| = 5 \)

  • \( |-5| = 5 \)

  • \( |0| = 0 \)

\( \boxed{\text{Absolute value = distance from zero}} \)

Opposites

The opposite of a number is the same distance from zero but on the other side of the number line.

• Examples:

  • Opposite of 5 is -5

  • Opposite of -8 is 8

  • Opposite of 3 is -3

  • Opposite of 0 is 0

• The opposite of \( x \) is \( -x \).

Translating Words Into Algebraic Expressions

Many algebra problems require translating verbal phrases into algebraic expressions. Let the unknown number be \( x \).

Example: "Five more than twice a number"

• Twice a number: \( 2x \)

• Five more: \( 2x + 5 \)

Example: "Six times a number plus one"

• \( 6x + 1 \)

Quick Reference Table: Number Sets

Super Important Facts to Memorize

• Natural Numbers: \( 1, 2, 3, \ldots \)

• Whole Numbers: \( 0, 1, 2, 3, \ldots \)

• Integers: \( \ldots, -2, -1, 0, 1, 2, \ldots \)

• Rational Numbers: Can be written as a fraction.

• Irrational Numbers: Cannot be written as a fraction; decimal never ends or repeats.

• Real Numbers: Rational + Irrational.

• Absolute Value: \( |-7| = 7 \)

• Opposite: Opposite of \( -7 \) is \( 7 \)

• Evaluate: If \( x = 4 \), then \( 3x + 2 = 3(4) + 2 = 12 + 2 = 14 \)

Test Tip

A number can belong to more than one set. For example:

• 5 is natural, whole, integer, rational, and real.

• 0 is whole, integer, rational, and real (but not natural or irrational).

• -3 is integer, rational, and real.

• \( \sqrt{2} \) is irrational and real.