Number System

Aug 08, 2022, 16:45 IST

About Number System

The Integers

Integers are classified into negative integers, zero and positive integers, integers can be classified as prime numbers and composite numbers. Read formulas of number system .

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number.

Real Numbers

The totality of rational numbers and irrational numbers is called the real numbers. i.e. rational numbers and irrational numbers taken together are called Real Numbers.

Prime Number

A natural number is said to be prime, if it has only two different(distinct) factors namely one and itself. We say 4 and 9 are not prime, whereas 2, 3, 5, 7, 11 are prime numbers.

1 is not a prime number.

The only even prime number is 2.

The smallest prime number is 2.

All prime numbers are odd except 2.

Twin-Prime

  A pair of prime numbers is said to be twin-prime if they differ by 2.

            For example {3, 5}, {11, 13}, {17, 19}, {29, 31}, {29, 31}, {41, 43}, {71, 73} are all twin-prime.

Composite Number

A natural number is said to be composite, if it has atleast three different factors. 4, 6, 12, are all composite numbers.

1 is not a composite number.

4 is the smallest composite number.

1 is neither prime nor composite number.

Co-prime Number

A pair of numbers is said to be co-prime, if the numbers have no common factor other than one.

Two numbers are said to be co-prime if and only if their highest common factor is 1.

Two numbers are co-prime or relatively prime, if there is no factor common to them except 1.

For example: The pairs of number (2, 3), (5, 9), (7, 13), (12, 17), (12, 35), (63, 26), (162, 35), etc are co-prime and the pairs of numbers (4, 6), (45, 65), (25, 125), (60, 75) are not co-prime.

Perfect Number

A number is said to be perfect, if it is equal to the sum of its factors other than itself.

            For example: 6 = (1 + 2 + 3); 28 = (1 + 2 + 4 + 7 + 14) therefore 6 and 28 are perfect numbers.

Even Numbers

A natural number is said to be even if it is multiple of 2 or it is divisible by 2

            2, 4, 6, 8, 10, 12 ……. are examples of even numbers.

Odd Numbers

A natural number is said to be odd, if it is not even or if it is not divisible by 2.

Prime Factors

A factor of a given number is called a prime factor, if this factor is a prime number.

Example: The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. Out of these 2, 3 and 7 are prime numbers. therefore, 2, 3 and 7 are the prime factors of 42.

 

Common Factors

A number which divides each one of the given numbers exactly, is called a common factor of each of the given numbers.

Example: 4 divides each one of 212 and 356 exactly. Therefore, 4 is a common factor of 212 and 356.

HCF is the lowest power of common prime and LCM is the highest power of primes.

Find below pdf consist of examples of Number system