Real Numbers
are used to measure continuous quantities like distance, time, and temperature. They include both rational and irrational numbers and can be represented as points on an infinite number line.
Each real number has a unique decimal expansion, which can go on forever. The real number system plays an important role in Mathematics, especially in calculus, where it helps define concepts such as limits, continuity, and derivatives.
The term “real” was introduced by René Descartes to distinguish these numbers from imaginary ones. In this blog, we will discuss real numbers in detail.
What are Real Numbers?
Real numbers
are all the numbers that can be plotted on a number line. They include both
rational
and
irrational numbers
. Rational numbers can be expressed as fractions, such as 7, -3, 0.5, or 4/3, while irrational numbers cannot be expressed as exact fractions, like √2 and π.
Real numbers cover whole numbers, integers, fractions, and decimals, representing a continuous and complete set of values. Some examples are 3, -1, 1/2, √2, π, and 2.5.
So, what types of numbers are not real numbers?
-
Imaginary numbers
, such as √−1 (the square root of -1), are not real numbers.
-
Infinity
is also not a real number.
These special types of numbers are used in advanced mathematics but fall outside the category of real numbers.
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The Real N
umber Line
The real number line is a visual representation of all real numbers on a continuous, unbroken line. A specific point on this line is marked as the "origin" (0). Numbers to the right of the origin are positive, and numbers to the left are negative.
A fixed distance is chosen to represent "1"; from there, whole numbers like 1, 2, 3, etc., are marked off to the right, while negative numbers like -1, -2, -3, etc. are marked to the left.
Any point on this line represents a real number. These can include:
-
Whole Numbers:
5, 10, -3
-
Rational Numbers:
3/4, -2.5
-
Irrational Numbers:
√3, π
However, you will
not
find numbers like
infinity
or
imaginary numbers
(such as √−1) on the real number line, as these do not belong to the set of real numbers.
Real Numbers
Symbol
Real numbers are represented by the symbol R or ℝ. Here are symbols for other types of numbers:
-
Natural Numbers (N):
1, 2, 3,...
-
Whole Numbers (W):
0, 1, 2, 3,...
-
Integers (Z):
-2, -1, 0, 1, 2,...
-
Rational Numbers (Q):
Numbers that can be written as fractions, like 1/2 or -3/4
-
Irrational Numbers (Q’):
Numbers that can't be written as fractions, like √2 or π
Properties of Real Numbers
Real numbers follow a set of basic rules for
addition
and
multiplication
. These rules help make calculations easier. Here’s a simple explanation of each property with examples.
Commutative Property
The order of the numbers doesn't matter.
For Addition:
a+b=b+a
Example
: 3+5=5+3 (Both equal 8)
For Multiplication:
a×b=b×a
Example:
4×2=2×4 (Both equal 8)
Associative Property
The grouping of numbers doesn’t matter.
For Addition:
(a+b)+c = a+(b+c)
Example:
(2+3)+4=2+(3+4) (Both equal 9)
For Multiplication:
(a×b)×c=a×(b×c)
Example:
(2×3)×4 = 2×(3×4)
(Both equal 24)
Distributive Property
You can distribute multiplication over addition.
Rule
a×(b+c) = a × b + a × c
Example
: 3×(4+2)=3 × 4 + 3 × 2
3 × 6 = 12 + 6 (Both equal 18)
Identity Property
Some numbers do not change their value when added or multiplied.
For Addition (Additive Identity):
a+ 0 = a
Example:
7
+ 0 = 7
For Multiplication (Multiplicative Identity):
a × 1 = a
Example:
5×1=5
Inverse Property
Combining a number with its inverse returns a neutral value
.
For Addition (Additive Inverse):
a+(−a)= a + (-a) = 0
Example:
6+(−6)=0
For Multiplication (Multiplicative Inverse):
a×(1/a)=1 (if a≠0)
Example:
4×1/4=1
Real Numbers Solved
Examples
Here are a few solved examples based on real numbers to help you better understand their application:
Example 1: Which of the following are real numbers?
333, −2.5-2.5−2.5,
√4
Solution:
-
333 is a whole number, which is a real number.
-
−2.5 is a decimal number, which is also a real number.
-
√4 equals 2, a whole number, making it a real number
-
π is an irrational number, but it is still a real number.
-
√-1 is an imaginary number, so it is
not
a real number.
Answer:
33
3
,
−2.5-2.5
−
2.5
,
√4 and
π
are real numbers.
Example 2: Find which of the variables
a,
b
,
c
, and d represent rational numbers and which irrational numbers:
(i)
a
2
=7
(ii)
b
2
=16
(iii)
c
2
=0.25
(iv)
d
2
=20/9
Solution:
(i)
a
2
= 7a
On simplifying, we get:
a = ±
√7
Therefore,
a
is an irrational number
2. b
2
=16
On simplifying, we get:
b = ±
√4
Therefore,
b
is a rational number.
3
. c
2
= 0.25
On simplifying, we get:
c=±0.5
Therefore,
c
is a rational number.
4.
d
2
= 20/9
On simplifying, we get:
d = ±20/3
Since
√20 is irrational,
Therefore, d
is an irrational number
Example 3: Place 7/4fraction on the number line.
Solution:
7/4=1.75
This lies between
1
and
2
on the number line.
Answer:
7/4 is located between 1 and 2.
Example 4: Simplify
12√7/4√7
Solution:
12√7/4√7
12/4
×
√7/√7
= 3
Example 5:
What are the first 10 real numbers?
Solution:
Real numbers include all numbers on the number line, such as integers, fractions, decimals, and irrational numbers. However, if we focus on the first 10 real numbers, considering integers, they are:
−5,−4,−3,−2,−1, 0, 1, 2, 3, 4
Real numbers form the backbone of mathematics, allowing us to measure, calculate, and solve real-world problems. Understanding real numbers helps unlock concepts like limits, continuity, and calculus. They are essential for both practical applications and advanced mathematical theories.
