Table of Contents
Real Numbers Class 10 Notes
R= Real Numbers
Real numbers include both irrational and rational numbers.
I= Integers
Integers are all numbers starting with (…-3, -2, -1, 0, 1, 2, 3…).
Q= Rational Numbers
Rational numbers are real numbers with the pattern p/q, q 0, and p, q I.
- Rational expressions can represent all integers, such as 5 = 5/1.
- Decimal expansion of recurring or non-recurring rational numbers.
Q’ = Irrational Numbers
Real numbers cannot be expressed in the form p/q, whose decimal expansions are non-terminating and non-recurring.
- Roots of primes like √2, √3, √5 etc., are irrational.
N= Natural Numbers
Natural numbers are those that can be counted. N = {1, 2, 3, …}
W= Whole Numbers
Whole numbers are the collective term for zero and all natural numbers. {0, 1, 2, 3,…}
Even Numbers
Even numbers are natural numbers with the form 2n. (2, 4, 6, …}
Odd Numbers
Odd numbers are natural numbers with the form 2n – 1; examples are 1, 3, 5, etc.
- Why can’t the form be expressed as 2n+1?
Remember!
*Natural numbers are exclusively whole numbers.
*Every Whole Number is an integer.
*Rational Numbers are the same as Integers.
*All Rational Numbers are Real Numbers.
Prime Numbers
Prime numbers are all natural numbers bigger than one that can be divided by 1 and the number itself. Examples of prime numbers are 2, 3, 5, 7, and 11.
- Since it only has one factor, 1, it is not a prime number.
Composite Numbers
Composite numbers are any natural numbers that may be divided by one, including the number one itself. Four, six, eight, nine, ten, etc.
- 1 is not a prime number nor a composite number.
Real Numbers Class 10 Notes- Methods
- Euclid’s Division Lemma
There are distinct integers q and r that satisfy the equation a = bq + r, 0 r b when given two positive integers a and b.
Observe this: Every time, “r” is lower than “b.” Each “q” and “r” is distinctive.
- Application Of Lemma
The HCF of two positive numbers is determined using Euclid’s Division Lemma.
Example: How can I find the HCF of 56 and 72?
Steps:-
- Lemma applied to 56 and 72.
- Find “b” and “r” by using a larger number. 72 = 56 × 1 + 16
- Consider 56 as the new dividend and 16 as the new divisor since 16 s not equal to 0. 56 = 16 × 3 + 8
- Once more, if 8 is not equal to 0, apply 16 as the new dividend and 8 as the new divisor. 16 = 8 × 2 + 0
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The remainder being zero, (8) is HCF’s divisor
Euclid’s Division Lemma can be extended to all integers except zero, i.e., b 0. This is true even if it is only stated for positive integers.
III. Constructing A Factor Tree
Steps for building a factor tree:
- Put the amount in the form of a prime number plus a composite number.
Example: 48 factorise
- Continue until all primes have been achieved.
∴ Prime factorization of 48 = 24 x 3
- Fundamental Theorem Of Arithmetic
With the exception of the sequence in which they appear, every composite number can be written as a product of primes, and this expression is singular.
Applications:
- In order to find the HCF and LCM of two or more positive integers.
- Demonstrating the irrationality of numbers
- To identify the type of rational number’s decimal expansion.
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