Properties of Real Numbers

By the properties of real numbers, we basically mean the how various algebraic operations (eg., +, – , ×, ÷, <, > etc) work on real numbers. Some of the basic properties of real numbers are given as follows:

1. Closure property
2. Associative property
3. Distributive property
4. Identity property
5. Inverse property
6. Commutative property

We now provide the complete list and let us understand these properties of real numbers one by one with examples.

Closure Property

The closure property of real numbers says that if you add or multiply two real numbers then the outcome will also be a real number. For two real numbers x and y, we have that

x+y, xy ∈ℝ.

For example the sum and the product of 2 and 3 are 5 and 6 respectively, which are real numbers.

Associative Property

The associative property of real numbers states that the sum or the product of any three or more real numbers in any order will be the same.

For x, y and z ∈ ℝ, we have that

• x+(y+z) = (x+y)+z
• x(yz) = (xy)z

Distributive Property

The distributive property is used to multiply the sum of two real numbers by an another real number. This property states that for any x, y and z ∈ ℝ, we have that

• x(y+z) = xy+xz
• (x+y)z = xz+yz

Distributive property of real numbers:

2⋅(3+4) = 2⋅3+2⋅4	Both equal 14
(2+3)⋅4 = 2⋅4+3⋅4	Both equal 20

Read Also:

• Completeness Axiom of Real Numbers
• Supremum and Infimum of a Set

Identity Property

For addition, the identity element is such an element when we add any number m to this, we get the same number m. As m+0 = m for all m ∈ ℝ, we say that 0 is the identity element of real numbers for addition.

For multiplication, as m × 1 = m for all m ∈ ℝ, we say that 1 is the identity element of real numbers.

Note: 0 is the additive identity of real numbers where as 1 is the multiplicative identity of real numbers.

Inverse Property

Note that a+ (-a) = 0, the additive identity. So a and -a are the additive inverses of each other. They are also called opposite numbers of each other.

For the multiplication, $a \times \dfrac{1}{a}=1$, the multiplicative identity (when a≠0). Here a and 1/a are multiplicative inverses of each other.

Commutative Property

For two real numbers x and y, the commutative property says that the addition or the multiplication of x and y does not follow the order of the operation. In other words, we always have the following.

• x+y = y+x
• xy = yx

Examples of commutative property:

2+3 = 3+2	Both equal 5
2×3 = 3×2	Both equal 6
2+7 = 7+2	Both equal 9
2×7 = 7×2	Both equal 14

Order Property

Let us define a linear order relation < on the set ℝ of real numbers as follows: x < y if x is less than y and x, y ∈ ℝ. This order relation on ℝ satisfies the following conditions.

1. If x, y ∈ ℝ then exactly one of the following facts always hold: x < y, or x=y, or y < x. This is called the law of trichotomy of real numbers.
2. The law of transitivity of real numbers states that if x < y and y < z, then x < z where x, y, z ∈ ℝ.
3. For x, y, z ∈ ℝ, x < y implies that x+z < y+z.
4. For x, y, z ∈ ℝ with z > 0, we have that xz < yz.

Density Property

The density property of real numbers says that between any two real numbers x and y there always exist an another real number (in fact, infinitely many real numbers exist between x and y). For more details, we refer to the page “Density Property of Real Numbers“.

Archimedean Property

If x and y are two positive real numbers, then the Archimedean property says that there exists a positive integer n such that nx > y. For more details, see the page “Archimedean Property of Real Numbers“.

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