Real Numbers

Number types and symbols

The numbers you use to count are known as whole numbers and are part of rational numbers. Rational numbers and whole numbers compose also the real numbers, but there are many more, and the list can be found below.

• Natural Numbers, with the symbol (N).

• Whole numbers, with the symbol (W).

• Integers with the symbol (Z).

• Rational numbers with the symbol (Q).

• Irrational numbers with the symbol (Q ').

Venn diagram of numbers

Types of real numbers

It is important to know that for any real Number picked, it is either a rational number or an irrational number which are the two main groups of real numbers.

Rational numbers

Rational numbers are a type of real numbers that can be written as the Ratio of two integers. They are expressed in the form p / q, where p and q are integers and not equal to 0. Examples of rational numbers are$\frac{1}{2},\frac{10}{12},\frac{3}{10}$ . The set of rational numbers is always denoted by Q.

Types of rational numbers

There are different types of rational numbers and these are

• Integers, for example, -3, 5, and 4.

• Fractions in the form p / q where p and q are integers, for example, ½.

• Numbers that do not have infinite decimals, for example, ¼ of 0.25.

• Numbers that have infinite decimals, for example, ⅓ of 0.333….

Irrational numbers

Irrational numbers are a type of real numbers that cannot be written as the Ratio of two integers. They are numbers that cannot be expressed in the form p / q, where p and q are integers.

As mentioned earlier, real numbers consist of two groups – the rational and irrational numbers, $(R-Q)$expresses that irrational numbers can be obtained by subtracting rational numbers group (Q) from real numbers group (R). That leaves us with the irrational numbers group denoted by Q '.

Examples of irrational numbers

• A common example of an irrational Number is 𝜋 (pi). Pi is expressed as 3.14159265….

The decimal value never stops and does not have a repetitive pattern. The fractional value closest to pi is 22/7, so most often we take pi to be 22/7.

• Another example of an irrational number is $\sqrt{2}$. the value of this is also 1.414213 ..., $\sqrt{2}$ is another number with an infinite decimal.

Properties of real numbers

Just as it is with integers and natural numbers, the set of real numbers also has the closure property, commutative property, the associative property, and the distributive property.

• Closure property

The product and sum of two real numbers is always a real number. The closure property is stated as; for all a, b ∈ R, a + b ∈ R, and ab ∈ R.

• Commutative property

The product and sum of two real numbers remain the same even after interchanging the order of the numbers. The commutative property is stated as; for all a, b ∈ R, a + b = b + a and a × b = b × a.

• Associative property

The product or sum of any three real numbers remains the same even when the grouping of numbers is changed.

The associative property is stated as; for all a, b, c ∈ R, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

• Distributive property

The distributive property of multiplication over addition is expressed as a × (b + c) = (a × b) + (a × c) and the distributive property of multiplication over subtraction is expressed as a × (b - c) = (a × b) - (a × c).