Number Systems

Number Systems

A measurement carried out in the physical world leads directly to a number or some meaningful value which is known as Real Number.

Real Numbers can be divided into two main groups: (A) Rational Numbers: A rational number can always be represented by a fraction of the form (p/q) where p and q are integers and q is not = 0. Every integer and all the fractions are rational numbers including finite decimal numbers.

(B) Irrational Numbers: An irrational number cannot be expressed in the form p/q where q is not = O. e.g., under root of 7 gives an approximate answer in the form of a fraction or decimal but the digits after the decimal point are non-ending, hence ?7 is an irrational number. Similarly, ? = 3.14...is irrational. In other words, an infinite non-recurring decimal is an irrational number which is non-terminating.
Note: No number can be both rational and irrational.

Prime Numbers: A prime number is a number which has no other factors except itself and unity
e.g., 2, 3, 5, 7, 11 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107...173, 179, 181, 191, 193, 197, 199, etc., 2 is the only even number which is prime.

There are 25 prime numbers between 1 & 100.

To check if a particular number 'k' is prime or not, we should apply divisibility tests of all primes up to ?k

If a number has no factor equal to or less than its square root, then that number must be prime.
'1' is neither Prime nor composite.

Largest prime no. known so far is (22281 - 1) which is of about 700 digits.

Co-Prime Numbers: Two numbers are relatively prime to each other when their highest common factor is 1 or we can say that they have no common factor except 1.

For being co-prime the numbers themselves need not be prime
E.g. 35 has factors 1, 5, 7, 35 i.e., it is not a prime number
And 49 has factors 1, 7, 49 i.e., again it is not a prime number

BUT these are Co-prime numbers as no common factor except 1.

Composite Numbers: A composite number is a number which has more than 2 factors i.e., itself and unity, e.g., or in other words we can say any number that has atleast 3 factors. e.g., 14, 36, 345 etc., i.e., it must have 3 factors - (1, itself and one More)

Even numbers are divisible by 2 and end up with 0, 2, 4, 6 or 8

Odd numbers are not divisible by 2 and end up with 1, 3, 5, 7 or 9.

Some important results on Numbers:
If n is odd, n (n2 - 1) is divisible by 24 and if n is an odd prime greater than 3, n2 - 1 is divisible by 24 e.g., if n = 11 then 112 - 1 = 120 is divisible by 24.
If n is odd, 2n + 1 is divisible by 3, and if n is even, 2n - 1 is divisible by 3.
If n is prime to 5, then {n (n4 - 1}) is divisible by 30.
If n is odd, {22n + 1} is divisible by 5, and if n is even, {22n - 1} is divisible by 5.
If n is odd {52n + 1} is divisible by 13 and if n is even, {52n - 1} is divisible by 13.
If a > b, then a - b = 1 - b and if a < b, then a - b = b - a.
There are an infinite number of rational numbers between any two rational numbers.

Test for Divisibility:
(1) A number is divisible by 2, if it ends with 0, 2, 4, 6 or 8 e.g., 4172, 790, 714, etc.

(2) A number is divisible by 3, if the sum of its digits is divisible by 3. e.g., 6072, 4701063.

(3) A number is divisible by 4, the number formed by the last two right hand digits is divisible by 4, or if the last two digits are 00 e.g., 362164, 725500.

(4) A number is divisible by 5, when its unit's digit is 5 or 0. e.g., 2745, 5760.

(5) A number is divisible by 6, when it is divisible by 2 and 3 both, e.g., 654, 984.

(6) A number is divisible by 8, if the number formed by the last three right hand digits is divisible by 8, or when the last three digits are 000. e.g., 80800, 8844832.

(7) A number is divisible by 9, when the sum of its digits is divisible by 9. e.g., 4263567.

(8) A number is divisible by 10, when its units digit is 0. e.g., 120, 4350.

(9) A number is divisible by 11, when the difference between the sum of the digits in the odd and even places is 0 or a multiple of 11. e.g., 6172859. Here (9 + 7 + 8 + 6) = 30 and 1 + 2 + 5 = 8, 30 - 8 = 22 which is a multiple of 11.

Note: When any number with an even number of digits is added to its reverse, the sum is always a multiple of 11. e.g., abcd + dcba = pqrs...Where pqrs is divisible by 11.

(10) A number is divisible by 12, when it is divisible by 3 and 4 both. e.g., 81004572.

(11) A number is divisible by 25, when the number formed by the last two right hands digits is divisible by 25, e.g., 783425.

(12) A number is divisible by 125, when the number formed by the last three right hand digits is divisible by 125. e.g., 824625.