Real Numbers: Overview and Definition
In the number system, real numbers are just the sum of rational and irrational numbers. All arithmetic operations may be done on these numbers in general and can also be represented on a number line. Simultaneously, imaginary numbers are unreal numbers that cannot be stated on a number line and are typically employed to represent complex numbers. Examples of actual numbers are 23, -12, 6.99, 5/2, and so on.
The combination of rational and irrational numbers yields real numbers. They can be either positive or negative and are represented by the letter "R". This category includes all-natural integers, decimals, and fractions.
Real Numbers: Real Numbers on Number Line
A number line assists us in displaying numbers by portraying each one with a distinct point on the line. Every point on the number line represents a separate real number. One can refer to the procedures below to portray actual numbers on a number line-
Step 1: Draw a horizontal line with arrows on both ends and a 0 in the center. The number 0 is known as the origin.
Step 2: Draw a straight line parallel to the origin and identify it with a specified scale.
Step 3: It should be noticed that positive numbers are on the right side of the origin and negative numbers are on the left.
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Real Numbers: Types of Real Numbers
Real numbers, as we already know, encompass both rational and irrational numbers. The following are some of the types of real numbers-
- Rational Numbers: Any integer that can be written as a fraction p/q is a rational number. The fraction's numerator is written as 'p,' while the denominator is represented as 'q,' where 'q' ≠ 0. A rational number can be a natural number, a whole number, a decimal number, or an integer. For Example:1/2, -2/3, 0.5, and 0.333 are all rational numbers.
- Irrational Numbers: Irrational numbers are real numbers that cannot be represented as a fraction p/q, where 'p' and 'q' are integers and the denominator 'q' > 0. For example: (pi) is an irrational number. π = 3.14159265...The decimal number in this scenario never stops at any point.
Read more about the Signum Functions in Mathematics.
Real Numbers: Symbols
The sign R represents real numbers. Here is a list of the different sorts of number symbols-
- Natural numbers (N)
- Whole numbers (W)
- Integers (Z)
- Rational numbers (Q)
- Irrational numbers (Q)
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Real Numbers: Group of Actual Numbers
The set of real numbers is divided into several groups, including natural and whole numbers, integers, and rational and irrational numbers. All real numbers formulae (i.e., the representation of the categorization of real numbers) are specified with examples in the table below-
Particulars | Definition | Examples |
Integers | The sum of all whole numbers and the inverse of all natural numbers. | It includes: Infinity (∞), -4, -3, -2, -1, 0, 1, 2, 3, 4,......+infinity (∞). |
Irrational Numbers | Numbers that are not rational and cannot be expressed in the manner described p/q. | Irrational numbers, such as 2, are non-terminating and non-repeating. |
Natural Numbers | Include all counting numbers beginning with 1. N = {1, 2, 3, 4,……} | All numerals begin with 1, 2, 3, 4, 5, 6, etc. |
Rational Numbers | Numbers that may be expressed as p/q, where q = 0. | Examples of rational numbers are 12, 5/4, and 12/6, among others. |
Whole Numbers | A collection of natural and zero numbers. W = {0, 1, 2, 3,…..} | All numerals beginning with 0, such as 0, 1, 2, 3, 4, 5, 6, etc. |
Real Numbers: Properties
The four main properties of real numbers are as follows-
Consider the three real numbers "a, b, and c." Then, as shown below, the aforesaid qualities may be represented using a, b, and c as follows-
Closure Property of Real Numbers
According to the closure property, the sum and product of two real numbers is always a real number. For Example: If a,b and c are real numbers then, a+ b x R = a x b + R.
Associative Property of Real Numbers
If a, b, and c are integers,
- Addition: The usual form is m + (n + r) = (m + n) + r. 10 + (3 + 2) = (10 + 3) + 2 is an example of an additive associative characteristic.
- Multiplication: (ab) c = a(bc), (9 x 6) 7 = 9 (6 x 7) is an example of a multiplicative associative property.
Commutative Property of Real Numbers
If the numbers are a and b, the typical form is a + b = b + a for addition and a.b = b.a for multiplication.
- Addition: a + b = b + a For instance, 7 + 9 = 9 + 7.
- Multiplication: a x b = b x a. For instance, 7 x 9 = 9 x 7.
Distributive Property of Real Numbers
The distributive property is expressed as follows for three real numbers m, n, and r-
m(n+ r) = mn + mr and (m + n)r = mr + nr. For Example: 8(5 + 1) = 8 x 5 + 8 x 1. In this case, both sides will produce 48.
Identity Property of Real Numbers
Identity can be additive or multiplicative.
For Addition: m + 0 = m. (0 represents the additive identity).
For Multiplication: m x 1 = 1 x m = m. (1 represents the multiplicative identity).
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Real Numbers: What is a Real Numbers Chart?
The chart below shows a set of real numbers that includes all of the components of real numbers-
Real Numbers: Difference between Real Numbers and Integers
The primary distinction between real numbers and integers is that real numbers contain integers. In other words, integers fall within the realm of real numbers. The following table will help us grasp the distinction between real numbers and integers-
Real Numbers | Integers |
Real numbers are composed of rational, irrational, whole, and natural numbers. | Negative numbers, positive numbers, and zero are all examples of integers. |
Real number examples include 1/2, -2/3, 0.5, and 2. | Integer Examples: -4, -3, 0, 1, 2. |
Every point on the number line corresponds to a different real number. | On a number line, integers are represented by whole numbers and negative numbers. |
Decimals and fractions are regarded as real numbers. | Decimals and fractions are not included in integers. |
R is the symbol used to represent real numbers. | Integers are represented by the symbol Z. |
Read more about Rational Numbers.
Real Numbers: Important Real Numbers Tips
The following are some tips for real numbers-
- Every irrational number is the same as a real number.
- Real numbers are always rational numbers.
- All numbers are real, with the exception of complex numbers.
Also Read- What is the Area of a Parallelogram?
Real Numbers: Solved Examples
Example 1: Fill in the blanks with the decimal equivalents of the following-
- ⅙
- ⅝
Solution- a) ¼
1/4 = (1 × 25)/(4 × 25) = 25/100 = 0.25
b) 5/8 = (5 × 125)/(8 × 125) = 625/1000 = 0.625
Example 2: What should be multiplied by 1.25 to get a result of 1?
1.25 = 125/100
If we divide this by 100/125,
We get 125/100 x 100/125 = 1.
Example 3: Find five reasonable numbers ranging from 1/2 to 3/5.
Solution: We will use the same denominator for both the supplied rational numbers,
(1 x 5)/(2 x 5) = 5/10 and (3 x 2)/(5 x 2) = 6/10.
Now, multiply the numerator and denominator of each rational number by 6, and we get.
(5 x 6)/(10 x 6) = 30/60 and (6x 6)/(10 x 6) = 36/60.
31/60, 32/60, 33/60, 34/60, and 35/60 are five rational numbers between 1/2 = 30/60 and 3/5 = 36/60.
Real Numbers: Practice Questions
- Is it possible to express any positive integer as 6x + 9 (where x is an integer)?
- Which of the following is the lowest composite number?
- What is the product of a rational non-zero number with an irrational number?
- Demonstrate that each positive odd number has the formula 3x + 1, 3x + 8, or 3x + 6.
- Calculate 8 + 9 x 2 - 6.