The Real Number System and Basic Rules of Algebra

Vocabulary flashcards reviewing real number classifications, intervals, set operations, absolute value, and basic algebraic properties.

Irrational Numbers

Real numbers represented by decimals that never end and have no repeating pattern.

Rational Numbers

Real numbers including all fractions and decimals that end or have a repeated pattern.

Ordering of Real Numbers

The relation $a < b$ means $a$ is less than $b$ if and only if $a$ lies to the left of $b$ on the number line.

Bounded Intervals

Intervals of real numbers where $a$ and $b$ act as endpoints.

Union of Sets

For sets $S$ and $T$, the set $S \cup T$ consisting of all elements that are in $S$ or $T$ (or in both).

Intersection of Sets

For sets $S$ and $T$, the set $S \cap T$ consisting of all elements that are in both $S$ and $T$.

Absolute Value

The distance between the origin and a point on the real number line, defined as $|a| = a$ if $a \ge 0$, and $|a| = -a$ if $a < 0$.

Distance Between Two Points

The distance $d(a, b)$ between two real numbers $a$ and $b$ on the number line, given by $d(a, b) = |a - b| = |b - a|$.

Commutative Property of Addition

An algebraic property stating $a + b = b + a$ for real numbers $a$ and $b$.

Commutative Property of Multiplication

An algebraic property stating $ab = ba$ for real numbers $a$ and $b$.

Associative Property of Addition

An algebraic property stating $(a + b) + c = a + (b + c)$ for real numbers $a$, $b$, and $c$.

Associative Property of Multiplication

An algebraic property stating $(ab)c = a(bc)$ for real numbers $a$, $b$, and $c$.

Distributive Property

An algebraic property stating $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for real numbers $a$, $b$, and $c$.

Additive Identity Property

An algebraic property stating $a + 0 = a$ for any real number $a$.

Multiplicative Identity Property

An algebraic property stating $a \cdot 1 = a$ for any real number $a$.

Additive Inverse Property

An algebraic property stating $a + (-a) = 0$ for any real number $a$.

Multiplicative Inverse Property

An algebraic property stating $a \cdot \frac{1}{a} = 1$ for any real number $a \neq 0$.

Zero-Factor Property

An algebraic property stating that if $ab = 0$, then $a = 0$ or $b = 0$.