4-8 The Real Numbers: Rational and Irrational Numbers

Vocabulary and classification practice flashcards based on the lecture on real numbers, rational numbers, and irrational numbers.

Real Number

All numbers that can be written on a number line, consisting of the set of rational numbers and the set of irrational numbers.

Rational Number

A number that can be written as the quotient of two integers (a fraction) or as either a terminating or repeating decimal.

Irrational Number

A number that can be written only as decimals that do not terminate or repeat, and cannot be written as the quotient of two integers.

Square Root of a Non-Perfect Square

An irrational number formed when taking the square root of a whole number that is not a perfect square, such as $\sqrt{2}$.

Calculator Display Caution for Decimals

A repeating decimal may not appear to repeat on a calculator because calculators show a finite number of digits.

Natural Numbers

The set of counting numbers starting from $1$ ($1, 2, 3\dots$).

Whole Numbers

The set containing zero and all natural numbers ($0, 1, 2, 3\dots$).

Integers

The set containing whole numbers and their negative opposites, such as $-3$ and $-19$.

Undefined Division

A fraction with a denominator of $0$, which is not a real number because division by zero is undefined.

Classification of $\sqrt{5}$

Irrational, real (because $5$ is a whole number that is not a perfect square).

Classification of $-12.75$

Rational, real (because $-12.75$ is a terminating decimal).

Classification of $\frac{\sqrt{16}}{2}$

Whole, integer, rational, real (since $\frac{\sqrt{16}}{2} = \frac{4}{2} = 2$).

Classification of $\sqrt{21}$

Irrational (because $21$ is a whole number that is not a perfect square).

Classification of $\sqrt{\frac{64}{81}}$

Rational (since $\sqrt{\frac{64}{81}} = \frac{8}{9}$).