Real Numbers Definition
Real numbers can be defined as the union of both rational and irrational numbers.
They can be both positive or negative and are denoted by the symbol “R”.
All the natural numbers, decimals and fractions come under this category.
Related Links-
Set of Real Numbers
The set of real numbers consists of different categories, such as natural and whole numbers, integers, rational and irrational numbers. In the table given below, all the real numbers formulas (i.e.) the representation of the classification of real numbers are defined with examples.
| Category | Definition | Example |
|---|---|---|
| Natural Numbers |
Contain all counting numbers which start from 1.
N = {1, 2, 3, 4,……} |
All numbers such as 1, 2, 3, 4, 5, 6,…..… |
| Whole Numbers |
Collection of zero and natural number.
W = {0, 1, 2, 3,…..} |
All numbers including 0 such as 0, 1, 2, 3, 4, 5, 6,…..… |
| Integers | The collective result of whole numbers and negative of all natural numbers. | Includes: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞) |
| Rational Numbers | Numbers that can be written in the form of p/q, where q≠0. | Examples of rational numbers are ½, 5/4 and 12/6 etc. |
| Irrational Numbers | The numbers which are not rational and cannot be written in the form of p/q. | Irrational numbers are non-terminating and non-repeating in nature like √2 |
You may also read-
Properties of Real Numbers
The following are the four main properties of real numbers:
- Commutative property
- Associative property
- Distributive property
- Identity property
Consider “m, n and r” are three real numbers.
Then the above properties can be described using m, n, and r as shown below:
Commutative Property
If m and n are the numbers, then the general form will be m + n = n + m for addition and m.n = n.m for multiplication.
- Addition: m + n = n + m. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2.
- Multiplication: m × n = n × m. For example, 5 × 3 = 3 × 5, 2 × 4 = 4 × 2.
Associative Property
If m, n and r are the numbers. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication.
- Addition: The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 10 + (3 + 2) = (10 + 3) + 2.
- Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).
Distributive Property
For three numbers m, n, and r, which are real in nature, the distributive property is represented as:
m (n + r) = mn + mr and (m + n) r = mr + nr.
- Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.
Identity Property
There are additive and multiplicative identities.
- For addition: m + 0 = m. (0 is the additive identity)
- For multiplication: m × 1 = 1 × m = m. (1 is the multiplicative identity)
Real Numbers Class 9 and 10
In real numbers Class 9, the common concepts introduced include representing real numbers on a number line, operations on real numbers, properties of real numbers, and the law of exponents for real numbers. In Class 10, some advanced concepts related to real numbers are included. Apart from what are real numbers, students will also learn about the real numbers formulas and concepts such as Euclid’s Division Lemma, Euclid’s Division Algorithm and the fundamental theorem of arithmetic in class 10.
What are Natural and Real Numbers?
Natural numbers are all the positive integers starting from 1 to infinity. All the natural numbers are integers but not all the integers are natural numbers. These are the set of all counting numbers such as 1, 2, 3, 4, 5, 6, 7, 8, 9, …….∞.
Real numbers are numbers that include both rational and irrational numbers. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5) and irrational numbers such as √3, π(22/7), etc., are all real numbers.
Is Zero a Real or an Imaginary Number?
Are there Real Numbers that are not Rational or Irrational?
Is the real number a subset of a complex number?
What are the properties of real numbers?
Commutative Property
Associative Property
Distributive Property
Identity Property
