Real Numbers Class 10 Notes Maths Chapter 1

Some Important Definitions

1. Real Numbers(R): All rational and irrational numbers are called real numbers.
2. Integers(I): All numbers from (…-3, -2, -1, 0, 1, 2, 3…) are called integers.
3. Rational Numbers(Q):Real numbers of the form p/q, q ≠ 0, p, q ∈ I are rational numbers.
   • All integers can be expressed as rational, for example, 5 = 5/1.
   • Decimal expansion of rational numbers terminating or non-terminating recurring.
4. Irrational Numbers(Q’ ): Real numbers which cannot be expressed in the form p/q and whose decimal expansions are non-terminating and non-recurring.
   • Roots of primes like √2, √3, √5 etc. are irrational
5. Natural Numbers(N): Counting numbers are called natural numbers. N = {1, 2, 3, …}
6. Whole Numbers(W): Zero along with all natural numbers are together called whole numbers. {0, 1, 2, 3,…}
7. Even Numbers: Natural numbers of the form 2n are called even numbers. (2, 4, 6, …}
8. Odd Numbers: Natural numbers of the form 2n -1 are called odd numbers. {1, 3, 5, …}

Remember this!

• All Natural Numbers are whole numbers.
• All Whole Numbers are Integers.
• All Integers are Rational Numbers.
• All Rational Numbers are Real Numbers.

Prime Numbers

The natural numbers greater than 1 which are divisible by 1 and the number itself are called prime numbers, Prime numbers have two factors i.e., 1 and the number itself For example, 2, 3, 5, 7 & 11 etc.

Note: 1 is not a prime number as it has only one factor.

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Composite Numbers

The natural numbers which are divisible by 1, itself and any other number or numbers are called composite numbers. For example, 4, 6, 8, 9, 10 etc. 

Note: 1 is neither prime nor a composite number.

Constructing a factor tree

Step 1: Write the number as a product of prime number and a composite number.

Step 2: Repeat the process till all the primes are obtained.

Example : Factorize 8190

So we have factorised 8190 as 2 × 3 × 3 × 5 × 7 × 13 as a product of primes, i.e., 8190 = 2 × 3² × 5 × 7 × 13 as a product of powers of primes

Fundamental theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this expression is unique, apart from the order in which they appear.
Applications:

• To locate HCF and LCM of two or more positive integers.
• To prove irrationality of numbers.
• To determine the nature of the decimal expansion of rational numbers.

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Algorithm to locate HCF and LCM of two or more positive integers

Step 1: Factorize each of the given positive integers and express them as a product of powers of primes in ascending order of magnitude of primes.

Step 2: To find HCF, identify common prime factor and find the least powers and multiply them to get HCF.

Step 3: To find LCM, find the greatest exponent and then multiply them to get the LCM.

To prove Irrationality of numbers

• The sum or difference of a rational and an irrational number is irrational.
• The product or quotient of a non-zero rational number and an irrational number is irrational.

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To determine the nature of the decimal expansion of rational numbers

• Let x = p/q, p and q are co-primes, be a rational number whose decimal expansion terminates. Then the prime factorization of’q’ is of the form 2^m5ⁿ, m and n are non-negative integers.
• Let x = p/q be a rational number such that the prime factorization of ‘q’ is not of the form 2^m5ⁿ, ‘m’ and ‘n’ being non-negative integers, then x has a non-terminating repeating decimal expansion.