Key Points

A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and the denominator $q \neq 0$. Examples include $\frac{3}{5}$, $-7$ (as $\frac{-7}{1}$), and $0.5$ (as $\frac{1}{2}$).

An irrational number cannot be expressed as a fraction $\frac{p}{q}$. Their decimal representations are non-terminating and non-repeating. Famous examples are $\sqrt{2}$, $\sqrt{3}$, and $\pi$.

Real numbers ($\mathbb{R}$) are the set of all rational and irrational numbers combined. They represent every point on the number line.

The decimal expansion of a rational number is either terminating (e.g., $\frac{1}{8} = 0.125$) or non-terminating and repeating (e.g., $\frac{1}{3} = 0.333... = 0.\overline{3}$).

A rational number $\frac{p}{q}$ in its simplest form has a terminating decimal expansion if and only if the prime factorization of the denominator $q$ contains only powers of 2 and/or 5. For example, $\frac{7}{20}$ terminates because $20 = 2^2 \times 5^1$.

To convert a repeating decimal like $x = 0.\overline{45}$ to a fraction, multiply by a power of 10 to shift the decimal. Here, $100x = 45.\overline{45}$, so subtracting gives $99x = 45$, which means $x = \frac{45}{99} = \frac{5}{11}$.

To prove $\sqrt{2}$ is irrational, we use proof by contradiction. Assume $\sqrt{2} = \frac{p}{q}$ in simplest form. This leads to $p^2 = 2q^2$, implying both $p$ and $q$ must be even, which contradicts that the fraction was in simplest form.

Irrational numbers like $\sqrt{2}$ can be located on the number line using the Pythagoras theorem. Construct a right triangle with sides of length 1 and 1; the hypotenuse will have length $\sqrt{1^2+1^2} = \sqrt{2}$.

Between any two distinct rational numbers, there are infinitely many other rational numbers. A simple way to find one between $a$ and $b$ is to calculate their average, $\frac{a+b}{2}$.

To find $n$ rational numbers between $\frac{a}{b}$ and $\frac{c}{d}$, first find a common denominator $m$. Then find equivalent fractions $\frac{k_1}{m}$ and $\frac{k_2}{m}$ such that $k_2 - k_1 > n$, and pick integer numerators between $k_1$ and $k_2$.

Integers ($\mathbb{Z}$) include positive numbers, negative numbers, and zero. Key arithmetic rules include: a negative times a negative is a positive, e.g., $(-3) \times (-4) = 12$.

The absolute value of a number $x$, written as $|x|$, is its distance from 0 on the number line and is always non-negative. For example, $|5| = 5$ and $|-5| = 5$.

The repeating decimal $0.\overline{9}$ is exactly equal to 1. This can be shown algebraically: if $x = 0.999...$, then $10x = 9.999...$. Subtracting the first equation from the second gives $9x=9$, so $x=1$.

The number systems are nested: Natural Numbers $\mathbb{N} \subset$ Integers $\mathbb{Z} \subset$ Rational Numbers $\mathbb{Q}$. Real Numbers $\mathbb{R}$ are the union of Rational Numbers $\mathbb{Q}$ and Irrational Numbers.

The sum, difference, or product of a non-zero rational number and an irrational number is always irrational. The sum or product of two irrational numbers can be either rational or irrational (e.g., $\sqrt{2} \times \sqrt{2} = 2$, which is rational).