Real Numbers

The Real Number Line

The real number line is a visual representation of all real numbers arranged in order on a straight line. Every point on the line corresponds to a real number, and every real number corresponds to a point.

Ordering of Real Numbers

For any two real numbers \(a\) and \(b\):

• \(a > b\) means \(a\) lies to the right of \(b\) on the number line
• \(a < b\) means \(a\) lies to the left of \(b\) on the number line

The number line extends infinitely in both directions, with \(0\) at the center, positive numbers to the right, and negative numbers to the left.

Rational vs Irrational Numbers

All real numbers are either rational or irrational.

Rational Numbers

A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).

Examples: \(3\), \(-5\), \(\frac{2}{7}\), \(4.5\), \(-2.\overline{3}\)

Irrational Numbers

An irrational number cannot be expressed as a ratio of two integers. Their decimal expansions neither terminate nor repeat.

Key distinction: \(\sqrt{4} = 2\) (rational) but \(\sqrt{2}\) is irrational because 2 is not a perfect square.

Properties of the Real Numbers (\(\mathbb{R}\))

Closure Properties

For all \(a, b \in \mathbb{R}\):

Commutative Properties

Associative Properties

Distributive Property

Identity Elements

• Additive identity: \(a + 0 = a\)
• Multiplicative identity: \(a \times 1 = a\)

Inverse Elements

• Additive inverse: \(a + (-a) = 0\)
• Multiplicative inverse: \(a \times \frac{1}{a} = 1\) (for \(a \neq 0\))

Density Property

Between any two distinct real numbers, there exists infinitely many other real numbers.