This concept note is designed for CBSE 10th class maths students, This concept notes to do help students for the CBSE board examination and other competitive exams also…
Rational number: The number, which is written in the form of is called a rational number. It is denoted by Q.
Irrational number: – the number, which is not rational is called an irrational number. It is denoted by Q’ or S.
Prime number: – The number which has only two factors 1 and itself is called a prime number. (2, 3, 5, 7 …. Etc.)
Composite number: – the number which has more than two factors is called a composite number. (4, 6, 8, 9, 10… etc.)
Co-prime numbers: – Two numbers are said to be co-prime numbers, if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7) …etc.]
Euclid division lemma: – For any positive integers a and b, then q, rare integers exist uniquely satisfying the rules a = bq + r, 0 ≤ r < b.
To find H.C.F by using Euclid division lemma:
- For any two integers a and b (a > b). Apply Euclid division lemma, to a and b, we find whole numbers q and r such that a = bq + r, 0≤r<b.
- If r = 0, b is the H.C.F of a and b. If r≠ 0, apply Euclid division lemma, to b and r.
- Continue the process till the remainder is zero. The divisor at this stage is the required H.C.F.
- Euclid division lemma also called a division algorithm.
- Euclid division lemma is stated for only positive integers, it can be extended for all integers except 0.
The fundamental theorem of arithmetic:
Every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which prime factors occur.
Ex: – 12 = 2 ×2× 3, 15 = 3× 5 and so on.
To find LCM and HCF by using prime factorization method:
H.C.F = product of the smallest power of each common prime factors of given numbers.
L.C.M = product of the greatest power of each prime factor of given numbers.
- ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a.
- Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non- terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non- terminating repeating decimals.
Decimal expansion of rational numbers:
Decimal expansion of rational numbers is either terminating or non-terminating repeating (recurring)decimals.
Ex: – 1.34, 2.345, 1.2222… and so on.
In p/q, if prime factorization of q is in the form 2m 5n, then p/q is terminating decimal. Otherwise non-terminating repeating decimal.
Decimal expansion of irrational numbers:
Decimal expansion of irrational numbers is non-terminating decimals.
Ex: – 1.414…., 1.314….