Real numbers are values that can be written as an endless decimal. Real numbers include common numbers like pi, fractions like 1/4, and decimals like 0.2. Real numbers can be visualized as an infinitely long number line that extends infinitely in both the positive and negative directions.

## Types of Numbers and Their Symbols

The numbers we use to count are called whole numbers, which fall under the broader category of rational numbers. Rational and whole numbers both make up real numbers, but there are many more types of real numbers too, as listed below.

- Natural Numbers (N)
- Whole Numbers (W)
- Integers (Z)
- Rational Numbers (Q)
- Irrational Numbers (Q')

## Categories of Real Numbers

It's important to recognize that any real number falls into one of two main groups: rational numbers or irrational numbers.

### Rational Numbers

Rational numbers are a type of real number that can be written as a fraction of two integers. They are expressed as p/q, where p and q are integers not equal to zero. Examples include 1/2, 10/12, and 3/10.

The set of rational numbers is denoted by Q.

#### Types of Rational Numbers

There are several types of rational numbers:

- Integers like -3, 5, and 4.
- Fractions like 1/2 where the numerator and denominator are integers.
- Numbers that terminate, like 1/4 or 0.25.
- Numbers that repeat endlessly like 1/3 or 0.333...

### Irrational Numbers

Another type of real numbers are irrational numbers. These numbers cannot be written as a fraction of two integers. They cannot be expressed as p/q where p and q are integers.

As mentioned, real numbers consist of rational and irrational numbers. The irrational numbers group, denoted Q', is what remains when we subtract the rational numbers group Q from the real numbers group R.

#### Examples of Irrational Numbers

- A common irrational number is π (pi). Pi is roughly 3.14159265358979... and never terminates or repeats.
- Another is the square root of 2. Its value is approximately 1.41421356237309... and also goes on forever without repeating.

## Properties of Real Numbers

Just like integers and natural numbers, real numbers exhibit closure, commutativity, associativity, and distributivity properties.

#### Closure Property

The sum or product of two real numbers is always another real number. For example, 13 + 23 = 36, which is real, and 13 × 23 = 299, also real.

#### Commutative Property

The order of real numbers doesn't change the result of addition or multiplication. For example, 0.25 + 6 = 6 + 0.25 = 6.25, and 0.25 × 6 = 6 × 0.25 = 1.5.

#### Associative Property

The grouping of real numbers in addition or multiplication doesn't change the result. For example, 0.5 + (2 + 0) = (0.5 + 2) + 0 = 2.5.

#### Distributive Property

The distributive property holds for real numbers. For example, 19 × (8.11 + 2) = (19 × 8.11) + (19 × 2) = 192.09.

## Key Takeaways about Real Numbers

- They can be written as endless decimals.
- The main types are rational and irrational.
- R is the symbol for real numbers.
- Whole, natural, rational, and irrational numbers are all real numbers.