Operations on Real Numbers, Exponents, and Inequalities: Study Notes

Review of the Real Number System

Key Terms and Definitions

• Set: A collection of objects, often numbers, considered as a whole.

• Elements: The individual objects in a set.

• Finite Set: A set with a countable number of elements.

• Infinite Set: A set with uncountable or endless elements.

• Empty Set: A set with no elements, denoted as ∅ or {}.

• Variable: A symbol, usually a letter, that represents a number.

• Set-Builder Notation: A concise way of writing sets by stating the properties that its members must satisfy.

• Number Line: A line with a scale to indicate the set of real numbers.

• Coordinate Graph: A visual representation of pairs of numbers as points in a plane.

• Additive Inverse: The number that, when added to a given number, results in zero. For any number a, its additive inverse is -a.

• Signed Numbers: Numbers that include both positive and negative values.

• Absolute Value: The distance of a number from 0 on the number line, denoted as $|a|$.

• Equation: A mathematical statement that two expressions are equal.

• Inequality: A mathematical statement that compares two values, using symbols such as <, >, ≤, ≥, ≠.

Examples and Applications

• Example: The set of even numbers less than 10: {2, 4, 6, 8}

• Example: The absolute value of -5 is $|-5| = 5$.

• Example: The additive inverse of 7 is -7, since $7 + (-7) = 0$.

Exponents and Operations

Exponents and Exponential Expressions

• Exponent: Indicates how many times a number (the base) is multiplied by itself.

• Base: The number that is multiplied by itself.

• Exponential Expression: An expression of the form $a^n$, where a is the base and n is the exponent.

• Square Root: A number that, when multiplied by itself, gives the original number. The square root of a is written as $\sqrt{a}$.

Operations with Signed Numbers

• Addition and Subtraction: Combine like terms, paying attention to signs.

• Multiplication and Division: The product or quotient of two numbers with the same sign is positive; with different signs, it is negative.

Examples

• Evaluate: $-3^2 = -(3 \times 3) = -9$

• Evaluate: $(-3)^2 = (-3) \times (-3) = 9$

• Evaluate: $\frac{15}{12} \div \frac{5}{4} = \frac{15}{12} \times \frac{4}{5} = \frac{15 \times 4}{12 \times 5} = \frac{60}{60} = 1$

Inequalities and Their Properties

Understanding Inequality Symbols

• < : Less than

• ≤ : Less than or equal to

• > : Greater than

• ≥ : Greater than or equal to

• = : Equal to

• ≠ : Not equal to

Examples and Applications

• Example: $4 < 5$ is read as "4 is less than 5".

• Example: $8 ≤ 9$ is true because 8 is less than 9.

• Example: $4 < 4$ is false because 4 is not less than itself.

• Equivalent Statement: The inequality $5 < 7$ can be rewritten as $7 > 5$.

Operations with Fractions

Key Steps

• To add or subtract fractions, use a common denominator.

• To multiply fractions, multiply numerators and denominators.

• To divide fractions, multiply by the reciprocal of the divisor.

Example

• Evaluate: $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$

Summary Table: Key Algebraic Terms

Term	Definition	Example
Set	A collection of objects	{1, 2, 3}
Element	Member of a set	2 in {1, 2, 3}
Absolute Value	Distance from zero	$|-7| = 7$
Exponent	Indicates repeated multiplication	$2^3 = 2 \times 2 \times 2 = 8$
Inequality	Compares two values	$3 > 1$

Additional info:

• These notes cover foundational concepts in College Algebra, including sets, real numbers, exponents, and inequalities, which are essential for further study in algebra and related fields.