Types of Number

Rational Numbers are any values that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. This category includes all terminating decimals (like $0.75$) and recurring decimals (like $0.333...$).

Irrational Numbers cannot be written as simple fractions. Their decimal expansions are non-terminating and non-recurring, meaning they go on forever without a repeating pattern.

Common examples of irrational numbers include $\pi$ and the square roots of non-square integers, such as $\sqrt{2}$ or $\sqrt{7}$.

A Factor is a number that divides into another number exactly without leaving a remainder. For instance, the factors of $10$ are $1, 2, 5,$ and $10$.

A Multiple is the result of multiplying a number by an integer. The multiples of $5$ are $5, 10, 15, 20, \dots$, effectively forming the 'times table' for that number.

Prime Numbers are integers greater than $1$ that have exactly two distinct factors: $1$ and the number itself. The number $2$ is unique as the only even prime number, while $1$ is excluded from being prime because it only has one factor.

Square Numbers are produced by multiplying an integer by itself ($n^2$), while Cube Numbers result from multiplying an integer by itself twice ($n^3$). Recognizing the first fifteen squares and first five cubes is a vital skill for mental arithmetic.

Roots are the inverse operations of powers. A square root ($\sqrt{x}$) finds the number that, when squared, equals $x$, while a cube root ($\sqrt[3]{x}$) finds the number that, when cubed, equals $x$.

The Reciprocal of a number is found by dividing $1$ by that number. For a fraction $\frac{a}{b}$, the reciprocal is simply $\frac{b}{a}$, effectively 'flipping' the numerator and denominator.

Factors vs. Multiples: Factors are 'smaller' (or equal) divisors of a number, whereas multiples are 'larger' (or equal) products of that number.

Prime vs. Composite: Prime numbers have exactly two factors, while composite numbers have more than two factors.

The Number 1: Always remember that $1$ is neither prime nor composite. It is often used as a 'trick' in questions asking for the smallest prime number (which is $2$).

Negative Roots: Be cautious with square roots of negative numbers; in the set of real numbers, $\sqrt{-x}$ is undefined. However, cube roots of negative numbers (e.g., $\sqrt[3]{-8} = -2$) are perfectly valid.

Reciprocal of Zero: The number $0$ does not have a reciprocal because division by zero is undefined. Always check for this constraint in algebraic expressions.

Decimal Identification: If a decimal has a clear repeating pattern (like $0.121212...$), it is rational. If it seems random and never ends, it is likely irrational.