Section P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

Vocabulary flashcards covering algebraic expressions, order of operations, mathematical models, set concepts, set operations, and subsets of real numbers.

Variable

A letter, such as $x$ or $y$, used to represent various numbers.

Algebraic Expression

A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots.

Exponential Expression

An expression written in the form $b^n$, defined as the product of $n$ factors of $b$.

Base

In an exponential expression of the form $b^n$, the factor $b$ that is multiplied repeatedly $n$ times.

Exponent

In an exponential expression of the form $b^n$, the number $n$ (also called the power) that indicates how many times the base $b$ appears as a factor.

Evaluating an Algebraic Expression

The process of finding the value of an algebraic expression for a given value of the variable.

Order of Operations Agreement

A set of rules for evaluating algebraic expressions without a calculator: 1) Perform operations within innermost parentheses working outward (treating fraction numerators and denominators as if enclosed in parentheses); 2) Evaluate exponential expressions; 3) Perform multiplications and divisions working left to right; 4) Perform additions and subtractions working left to right.

Equation

A mathematical statement formed when an equal sign is placed between two algebraic expressions.

Formula

An equation that uses variables to express a relationship between two or more quantities.

Model Breakdown

A situation where a mathematical model gives an estimate that is not a good approximation or is extended to include values of the variable that do not make sense.

Set

A collection of objects whose contents can be clearly determined.

Element

An individual object contained within a set.

Roster Method

A form of set representation that uses braces {} and commas to separate the listed elements of the set.

Ellipsis

A symbol consisting of three dots ($\dots$) indicating that there is no final element and that the listing of a set goes on forever.

Set-Builder Notation

A notation for representing a set in which elements are described using a rule or property rather than listed, such as {x \mid x \text{ is a counting number less than } 6}.

Intersection of Sets

The set of elements common to both set $A$ and set $B$, written $A \cap B = \{x \mid x \text{ is an element of } A \text{ AND } x \text{ is an element of } B\}$.

Empty Set

A set that has no elements, also called the null set, represented by the symbol $\varnothing$.

Union of Sets

The set of elements that are members of set $A$, set $B$, or both sets, written $A \cup B = \{x \mid x \text{ is an element of } A \text{ OR } x \text{ is an element of } B\}$.

Natural Numbers

The set of counting numbers, denoted by $\mathbf{N}$ and represented as {1, 2, 3, 4, 5, \dots}.

Whole Numbers

The set of numbers including $0$ and the natural numbers, denoted by $\mathbf{W}$ and represented as {0, 1, 2, 3, 4, 5, \dots}.

Integers

The set of numbers including the whole numbers and the negatives of the natural numbers, denoted by $\mathbf{Z}$ and represented as {\dots, -3, -2, -1, 0, 1, 2, 3, \dots}.

Rational Numbers

The set of all numbers that can be expressed as a quotient of two integers $\frac{a}{b}$ with denominator $b \neq 0$, denoted by $\mathbf{Q}$, which can be written as terminating or repeating decimals.

Irrational Numbers

The set of all numbers whose decimal representations are neither terminating nor repeating, which cannot be expressed as a quotient of integers.

Real Numbers

The set of numbers formed by taking the union of the set of rational numbers and the set of irrational numbers, represented by $\mathbf{R} = \{x \mid x \text{ is rational}\} \cup \{x \mid x \text{ is irrational}\}$.

Subset

A set whose elements are all also elements of a larger specified set.