Real Numbers and Their Properties

Study Notes

Real numbers can be expressed in decimal form and include values such as -3/4, 1/3, and √2.
Infinite decimal expansions, such as 0.999..., represent the same real number as 1.000... .
Real numbers can be represented as points on a number line, known as the real line, denoted by the symbol R.
Properties of real numbers are categorized into algebraic properties, order properties, and completeness.

If a, b, and c are real numbers, certain rules for inequalities apply:
- a < b implies a + c < b + c.
- a < b implies a - c < b - c.
- a < b and c > 0 implies ac < bc.
- a < b and c < 0 implies bc < ac, with the special case that a < b implies -b < -a.
- If a > 0, then 1/a > 0.
- For both positive or negative a and b, a < b implies 1/a < 1/b.
Multiplying by a positive number preserves an inequality, while multiplying by a negative number reverses it.

The completeness property ensures there are no "holes" in the real numbers, which is crucial for the concept of limits.
This property is foundational for many calculus theorems.

Natural numbers include 1, 2, 3, 4, and so on.
Integers encompass 0, positive, and negative whole numbers (±1, ±2, ±3, ...).
Rational numbers are numbers expressible as a fraction m/n, where m and n are integers and n ≠ 0. Examples include 1/3, -9, and 4/200.