Real Numbers

Real numbers include all the numbers that can be found on the number line. This includes both rational and irrational numbers. In other words, any number that can be expressed as a decimal, whether that decimal is terminating, repeating, or non-repeating, is a real number.

Definition

The set of real numbers, commonly denoted as ℝ, can be defined as follows:

ℝ = {x | x is a rational or an irrational number}

This means that any number that can be written as a fraction (rational number) or as a non-repeating, non-terminating decimal (irrational number) is a real number.

Properties of Real Numbers

Real numbers have several important properties:

  1. Closure Property: If you add, subtract, or multiply two real numbers, or divide one real number by another (except for division by zero), the result is always a real number.

  2. Associative Property: For all real numbers a, b, and c, (a + b) + c = a + (b + c), (a - b) - c = a - (b + c), and (a * b) * c = a * (b * c).

  3. Commutative Property: For all real numbers a and b, a + b = b + a and a * b = b * a.

  4. Distributive Property: For all real numbers a, b, and c, a * (b + c) = a * b + a * c.

  5. Identity Property: The number 0 is the additive identity in the set of real numbers, and the number 1 is the multiplicative identity.

  6. Additive and Multiplicative Inverses: Every real number a has an additive inverse -a, such that a + (-a) = 0, and a multiplicative inverse 1/a (if a ≠ 0), such that a * 1/a = 1.

  7. Density Property: Between any two real numbers, there exists another real number.

  8. Ordering Property: For any two real numbers, one and only one of the following is true: the first number is greater than the second, the first number is less than the second, or the two numbers are equal.

Relationship to Rational and Irrational Numbers

The real number system is an extension of the rational numbers, which include fractions and integers, by the irrational numbers, which cannot be expressed as fractions. The real number system is used to represent quantities that can be either continuous or discrete.

Applications

Real numbers are used in virtually every field of mathematics, including algebra, calculus, statistics, and more. They're essential for measuring quantities in the physical world, such as distance, time, and temperature. Real numbers are also used in the sciences, engineering, economics, and many other fields.