Properties of Real Numbers, Linear Equations, Exponents, and Polynomials

Comprehensive vocabulary flashcards covering sets of numbers, algebraic properties, linear equations, inequalities, exponent rules, and polynomial terminology.

Set

A collection of objects, each of which is called a member or element of the set, enclosed using curly braces ${}$.

Natural Numbers

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

Whole Numbers

The set of numbers consisting of zero and the natural numbers, denoted by $\{0, 1, 2, 3, \dots\}$.

Integer Numbers

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

Rational Numbers

The set of numbers that can be expressed in the form $\{\frac{a}{b} \mid a \text{ and } b \text{ are integers and } b \neq 0\}$.

Irrational Numbers

The set of decimal numbers that never end and never repeat, such as $\sqrt{2}$, $\sqrt{7}$, and $\pi$.

Real Numbers

The entire system of numbers that encompasses all rational numbers (including natural numbers, whole numbers, and integers) as well as all irrational numbers.

Commutative Property

A property stating that changing the order of the numbers will not change the result: $a + b = b + a$ for addition and $a \cdot b = b \cdot a$ for multiplication.

Associative Property

A property stating that changing the grouping of the numbers will not change the result: $a + (b + c) = (a + b) + c$ for addition and $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for multiplication.

Identity Property

A property stating that $0$ and $1$ preserve identities under addition and multiplication respectively: $a + 0 = a$ and $a \cdot 1 = a$.

Inverse Property

A property stating that for each real number $a$, there exists a unique inverse number whose sum is $0$ ($a + (-a) = 0$) and whose product is $1$ ($b \cdot \frac{1}{b} = 1$).

Distributive Property

A property stating that multiplication distributes over addition: $a \cdot (b + c) = ab + ac$.

Algebraic Expression

A mathematical phrase that combines numbers, variables, and operations, such as $2x + y$, $4y$, or $3x - y + 7$.

Term

A component of an algebraic expression that can be a number, a variable, or the product of a number and variables.

Variable Term

A term in an algebraic expression that contains one or more variables.

Constant Term

A term in an algebraic expression that consists only of a number.

Coefficient

The numerical factor of a term in an algebraic expression.

Like Terms

Terms that have the same variables raised to the same exponents.

Formula

An equation that describes a known relationship between quantities, such as $d = r \cdot t$, $A = \pi r^2$, or $P = 2l + 2w$.

Inequality

A mathematical statement comparing two quantities that contains an inequality symbol ($>$, $<$, $\ge$, or $\le$).

Product Rule for Exponents

An exponent rule stating that when multiplying powers with the same base, you add the exponents: $a^m \cdot a^n = a^{m+n}$.

Power Rule for Exponents

An exponent rule stating that when raising a power to another power, you multiply the exponents: $(a^m)^n = a^{m \cdot n}$.

Power of a Product Rule

An exponent rule stating that raising a product to a power applies that power to every factor inside: $(ab)^n = a^n b^n$.

Power of a Quotient Rule

An exponent rule stating that raising a quotient to a power applies that power to both the numerator and the denominator: $\left(\frac{a}{c}\right)^n = \frac{a^n}{c^n}$.

Quotient Rule for Exponents

An exponent rule stating that when dividing powers with the same base, you subtract the bottom exponent from the top exponent: $\frac{a^m}{a^n} = a^{m-n}$.

Zero Exponent Rule

An exponent rule stating that any non-zero base raised to the power of zero equals $1$: $a^0 = 1$.

Negative Exponent Rule

An exponent rule stating that a base with a negative exponent is equal to its reciprocal with a positive exponent: $a^{-n} = \frac{1}{a^n}$.

Scientific Notation

A method of writing a number as a value between $1$ and $10$ multiplied by $10$ raised to an integer exponent (e.g., $a \times 10^n$).

Polynomial

A specific type of algebraic expression composed of one or more terms consisting of constants and variables raised to non-negative integer powers.

Monomial

A polynomial that consists of exactly one term (e.g., $3xy^2$).

Binomial

A polynomial that consists of exactly two terms (e.g., $17 + 2x$).

Trinomial

A polynomial that consists of exactly three terms (e.g., $7x + 4y - 5xy$).

Degree of a Term

The sum of the exponents of all the variables contained in that individual term.

Degree of a Polynomial

The greatest degree among all individual terms in the polynomial.

FOIL Method

A mnemonic method used to multiply two binomials by summing the products of their First, Outer, Inner, and Last terms.

Product of the Sum and Difference of Two Terms

A special product pattern where multiplying the sum and difference of the same two terms results in the difference of their squares: $(a + b)(a - b) = a^2 - b^2$.