1.Real Numbers

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                                      1.Real Numbers

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N = Natural Numbers:

Counting numbers are called natural numbers. N = {1, 2, 3, …}

W = Whole Numbers:

Zero along with all natural numbers are together called whole numbers. {0, 1, 2, 3,…}

Even Numbers:

Natural numbers of the form 2n are called even numbers. (2, 4, 6, …}

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.

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

• List of Prime Number till 1000

• 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101
  103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197
  199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311
  313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431
  433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557
  569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673
  677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811
  821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941
  947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039

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.

Example


$16=2\times 2\times 2\times 2$


This is unique, We cannot express it in any other prime numbers

I. Euclid’s Division lemma Click Here to Get the App

Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r ≤ b.

Exercise 1.1

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IV. 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.

Prime Factorisation Tree Form
HCF and LCM by prime factorization method

We can find HCF and LCM between two number by the prime factorization method also
HCF (Highest Common factor) = product of the smallest power of each common factor in the numbers


Example: Suppose the number are 14,24
1. Lets do prime factorization method for both the numbers
$14=2\times 7$


$24=2\times 2\times 2\times 3=2^3\times 3$


2. Now as per method 2 is the only factor and lowest power is 1
HCF =2


LCM( Lowest Common Multiple) = Product of the greatest power of each prime factor involved in the number
Example: Suppose the number are 14,24
1. Lets do prime factorization method for both the numbers
$14=2\times 7$


$24=2\times 2\times 2\times 3=2^3\times 3$