Pre-Calculus: Foundations of Algebra

Comprehensive vocabulary flashcards covering the core algebraic topics from Unit 1: Foundations of Algebra & Graphs.

Natural Numbers

The set of counting numbers denoted by $N = \{1, 2, 3, \dots\}$.

Whole Numbers

The set of numbers denoted by $W = \{0, 1, 2, 3, \dots\}$, comprising natural numbers along with $0$.

Integers

The set of numbers denoted by $I = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$, consisting of negative integers, zero, and positive integers.

Rational Number

Any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$, having a decimal representation that is either terminating or non-terminating but repeating.

Irrational Number

A number that cannot be expressed as a fraction of two integers $\frac{p}{q}$ ($q \neq 0$), characterized by a decimal form that is non-terminating and non-repeating.

Real Number

Any number that can be found on the number line, representing the union of rational numbers ($\mathbb{Q}$) and irrational numbers.

Commutative Property

The property stating that the order of terms does not affect the result: $a + b = b + a$ for addition and $a \times b = b \times a$ for multiplication.

Associative Property

The property stating that the grouping of terms does not affect the result: $(a + b) + c = a + (b + c)$ for addition and $(a \times b) \times c = a \times (b \times c)$ for multiplication.

Distributive Property

The property linking multiplication and addition: $a \times (b + c) = a \times b + a \times c$.

Additive Identity

The number $0$, which satisfies the condition $a + 0 = a$ for any real number $a$.

Multiplicative Identity

The number $1$, which satisfies the condition $a \times 1 = a$ for any real number $a$.

Additive Inverse

For any real number $a$, the number $-a$ such that $a + (-a) = 0$.

Multiplicative Inverse

For any non-zero real number $a \neq 0$, the reciprocal $\frac{1}{a}$ such that $a \times \frac{1}{a} = 1$.

Closure Property

The property stating that the sum or product of any two real numbers is always a real number ($a + b \in \mathbb{R}$, $a \times b \in \mathbb{R}$).

Zero Property of Multiplication

The property stating that multiplying any real number by $0$ results in $0$ ($a \times 0 = 0$).

Trichotomy Property

For any two real numbers $a$ and $b$, exactly one of the following relations is true: $a < b$, $a = b$, or $a > b$.

Absolute Value

The distance of a real number $a$ from $0$ on the number line, defined piecewise as $a$ if $a \ge 0$ and $-a$ if $a < 0$.

Exponential Notation

The expression of the $n\text{th}$ power of a number as $a^n = a \cdot a \cdots \cdot a$ ($n$ factors), where $a$ is the base and $n$ is a positive integer exponent.

Scientific Notation

The representation of a positive number $x$ in the form $x = a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

Principal $n\text{th}$ Root

For a positive integer $n$, the number $b$ such that $\sqrt[n]{a} = b$ means $b^n = a$, with the requirement that $a \ge 0$ and $b \ge 0$ if $n$ is even.

Rational Exponent

An exponent of the form $\frac{m}{n}$ in lowest terms ($n > 0$), defined such that $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.

Variable

A letter that represents any number from a given set of numbers.

Algebraic Expression

A mathematical phrase formed by combining variables and real numbers using operations such as addition, subtraction, multiplication, division, powers, and roots.

Monomial

An algebraic expression of the form $ax^k$, where $a$ is a real number and $k$ is a nonnegative integer.

Binomial

An algebraic expression that is the sum of two monomials.

Trinomial

An algebraic expression that is the sum of three monomials.

Polynomial

An expression in the variable $x$ of the form $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $a_0, a_1, \dots, a_n$ are real numbers and $n$ is a nonnegative integer.

Degree of a Polynomial

The highest power $n$ of the variable $x$ appearing in a polynomial with a nonzero coefficient ($a_n \neq 0$).

FOIL Method

A mnemonic method for multiplying two binomials $(a + b)(c + d)$ by adding the products of the First terms ($ac$), Outside terms ($ad$), Inside terms ($bc$), and Last terms ($bd$).

Domain of an Algebraic Expression

The set of real numbers that a variable is permitted to have such that the algebraic expression is defined.

Linear Equation in One Variable

A first-degree equation equivalent to the form $ax + b = 0$, where $a$ and $b$ are real numbers ($a \neq 0$) and $x$ is the variable.

Quadratic Equation

A second-degree polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers with $a \neq 0$.

Quadratic Formula

The algebraic formula used to find the roots of any quadratic equation $ax^2 + bx + c = 0$ ($a \neq 0$), given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Complex Number

An expression of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^2 = -1$.

Real Part

The real value $a$ in a complex number expressed in standard form $a + bi$.

Imaginary Part

The real coefficient $b$ of the imaginary unit $i$ in a complex number expressed in standard form $a + bi$.

Complex Conjugate

For a complex number $a + bi$, the number $a - bi$, obtained by changing the sign of the imaginary part.

Principal Square Root of a Negative Number

For any negative number $-r$ (where $r > 0$), the value defined by $\sqrt{-r} = i\sqrt{r}$.

Inequality

A mathematical statement indicating that two quantities are not equal, relating them using the symbols $<$, $\le$, $>$, or $\ge$.