Class 10 Math Revision Notes: Real Numbers
Introduction
➥ eal numbers include both rational and irrational numbers. Rational numbers can be expressed as the quotient of two integers, while irrational numbers cannot.
Fundamentals
- A non-negative integer 'p' is divisible by an integer 'q' if there exists an integer 'd' such that p = qd.
- +1 divides every non-zero integer.
- 0 does not divide any integer.
Lemma
➥ A lemma is a statement which is already proved and is used for proving other statements.
Euclid’s Division Lemma
➥ Let 'a' and 'b' be any two positive integers, then there exist unique integers 'q' and 'r' such that a = bq + r, where 0 < r < b.
Natural Numbers
➥ All counting numbers are called Natural Numbers.
Examples: 1, 2, 3, 4, 5
Whole Numbers
➥ All natural numbers with Zero (0), are called Whole Numbers .
Examples: 0, 1, 2, 3, 4, 5
Integers
➥ All negative and non-negative numbers including zero.
Examples: -2, -1, 0, 1, 2, 3, 4, 5
Rational Numbers
➥ A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Example 1
Express 0.75 as a fraction.
Solution: 0.75 = 75/100 = 3/4
Irrational Numbers
➥ A number is irrational if it cannot be expressed n the form p/q, where p and q are integers.
They have non-terminating, non-repeating decimal expansions.
Example 2
Show that √2 is an irrational number.
Proof: Assume √2 is rational Number.
i.e., it can be written as √2 = p/q in its lowest terms, where p and q are integers and q ≠ 0.
Then, 2 = (p/q)^2 or p^2 = 2q^2, which implies p^2 is even.
Thus, p must be even.
Let p = 2k.
Then (2k)^2 = 2q^2 or 4k^2 = 2q^2, giving 2k^2 = q^2, which implies q^2 is even and hence q is even.
Thus, p and q are both even, contradicting the assumption that p/q is in its lowest terms.
Therefore, √2 is irrational.
Decimal Expansions
➥ The decimal expansion of rational numbers is either terminating or non-terminating recurring. For example, 1/2 = 0.5 (terminating) and 1/3 = 0.333... (non-terminating recurring).
Example 3
Convert the recurring decimal 0.666... to a fraction.
Solution: Let x = 0.666....
Both sides Multiplying by 10.
10x = 6.666....
Subtracting these, we get 10x - x = 6.666... - 0.666...
which simplifies to 9x = 6 or x = 6/9 = 2/3.
Thus, 0.666... = 2/3.
Terminating Decimals: -
If decimal expansion of rational numberp/q, where p and q are co-prime numbers (matlab aise numbers, jinka common factor 1 hai) and q
≠ 0 known as a rational number.
𝑒𝑔:- 2.25, 3.13, 2.3 𝑒𝑡𝑐.
Non-Terminating Decimals: - the decimal expansion obtained from p/q, repeats
periodically , then it is called non- terminating repeating (or recurring) decimal. A
bar is put over the repeating digits to show the repetition.
𝑒𝑔:-(i) 0.333333……………… = 0.3
(ii) 1/9 = 0.1111111....... = 0.1 𝑒𝑡𝑐.
Example 4
1. 64/455 = 64/(5×7×13)
Here, denominator is 5×7×13, which is not in the form 2𝑛5𝑚. Hence, it’s non-terminating.
2. 15/1600 = (3×5)/(2×2×2×2×2×2×5)
Here, denominator is 26×51, which n=6 and m=1. Hence, it’s terminating.
