Number System

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Natural numbers :- These are the numbers (1, 2, 3 ...... etc) that are used for counting. in other words, all positive integers are natural numbers.

⇒ The least natural number is 1 but there is no largest natural number.

⇒ The set of natural number is denoted by N.

Thus N = {1, 2, 3 ......}

Whole numbers :- The set of numbers that includes all natural numbers and zero, are called whole numbers.

⇒ The set of whole numbers is denoted by W.

⇒ Whole numbers are also called as "Non-negative" integers.

Integers :- All the natural numbers, zero and the negatives of natural numbers called integers.

I = {...... -3, -2, -1, 0, 1, 2, 3 ......}

(i) set of negative integers = {-1, -2, -3, ......}

(ii) set of non-negative integers = {0, 1, 2, 3 ......}

(iii) set of positive integers = {1, 2, 3 ......}

(iv) set of non-positive integers = {0, -1, -2, -3 ......}

Note :- '0' is definitely a non-negative integer as well as a non-positive integer.

Rational numbers ⇒ The numbers which can be expressed in the form of (p/q), where p and q are integers and q ≠ 0, are called rational numbers and their set is denoted by θ.

e.q. 1 over 6 comma space 1 over 8 comma space fraction numerator negative 3 over denominator 7 end fraction etc.. open parentheses as space 6 space equals space 6 over 1 close parentheses etc. are rational numbers.

⇒ The set of rational numbers encloses the set of integers and fractions.

Representation of rational numbers as decimals :- The decimal form of a rational number is either terminating or non-terminating.

E.g. 17 over 4 space equals space 4.25 comma space 21 over 5 space equals space 4.2 → terminating (or finite) decimal.

16 over 3 space equals space 5.3333.... space space comma space 2 over 3 space equals 0.666666.... space space equals space space 0. top enclose 6 → non-terminating (or recurring) decimal.

⇒ Irrational numbers :- The numbers which when expressed in decimal form , neither terminating nor repeating decimals are called "Irrational numbers".

e.g. square root of 2 comma space square root of 3 comma space square root of 50 comma space square root of 7 comma space straight pi etc.

 

Real numbers :- All rational and irrational numbers together form the set of real numbers. denoted by R.

⇒ Thus every natural number, every whole number every integer, every rational number and every irrational number is a real number.

Note (i) :- The sum (or difference) of a rational and an irrational number is irrational

e.g. open parentheses 4 space plus space square root of 3 close parentheses comma space open parentheses 2 space minus space square root of 5 close parentheses comma space open parentheses 3 over 2 space minus space square root of 2 close parentheses comma space open parentheses 7 space plus space straight pi close parentheses etc. are all irrational.

Note (ii) :- The product of a rational and an irrational number is irrational.

e.g. 4 square root of 3 comma space minus 2 square root of 5 etc. are all irrational.

 Even and odd numbers :- Integers divisible by 2 are called even numbers, while those which are not divisible by 2 are known as odd integers. Thus ......6, -4, -2, 0, 2, 4, 6....... etc are even integers and .......-5, -3, -1, 1, 3, 5....... etc are odd integers.

Prime numbers :- A number greater than 1 is called a prime number, If it has exactly two factors, namely 1 and itself.

E.g. 2, 3, 5, 7, 17, 19, 23, 29, 31, 37, 41 ......

⇒ 2 is only even number which is prime.

Composite numbers :- Composite numbers are the numbers greater than 1 which are not prime.

E.g. 4, 6, 9, 14, 15 etc.

Note :- 1 is neither prime nor composite. There are 25 prime numbers between numbers 1 and 100.

⇒ Test for prime numbers:- Let x be a given number and let 'k' be an integer, such that k

If x is not divisible by any prime number less than k then x is prime, otherwise. it is not prime.

 Co-prime numbers :- Two numbers are Co-prime, if their H.C.F. (Highest common factor) is 1.

⇒ (2, 3), (3, 13), (5, 7) etc. are Co- Prime numbers.

Test of Divisibility

(i) Divisibility by 2 :- A number is divisible by 2 if its unit digit is any of 0, 2, 4, 6, 8.

(ii) Divisibility by 3 :- A number is Divisible by 3 only when the sum of its digits is divisible by 3.

(iii) Divisibility by 9 :- A number is divisible by 9 only when the sum of its digits is divisible by 9.

(iv) Divisibility by 4 :- A number is divisible by 4 is the sum of its last two digits is divisible by 4.

(v) Divisibility by 8 :- A number is divisible by 8 if the last three digits  of the given number is divisible by 8.

(vi) Divisibility by 10 :- A number is divisible by 10 only when its unit digit is 0.

(vii) Divisibility by 5 :- A number is divisibility by 5 only when its unit digit is 0 or 5.

(viii) Divisibility by 11 :- A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

(ix) Divisibility by 12 :- A number is divisible by 12 only when it is divisible by 4 and 3 both at the same time.

(x) Divisibility by 15 :- A number is divisible by 15 only when it is divisible by 3 and 5 both in same time.

⇒ Remainder rules :-

Rule 1 :- If a number N is divisible by D then the product of N with any other integral number k, is also divisible by D.

i.e. if  straight N over straight D → Remainder 0 ⇒ then  NK over straight D → Remainder => 0

Rule 2 :- If N1 and N2 two different numbers are divisible by D individually then their sum (i.e. N1 + N2) and their difference (N1 - N2) is also divisible by D.

Rule 3 :- If a number N1 is divisible by N2 and N2 is divisible by N3 then N1 must be divisible by N3.

Rule 4 :- If two numbers N1 and N2 are such that they divide mutually each other it means they are same i.e. N1 = N2 only.

⇒ In general we use four terms for the divisibility

or Dividend = (Divisor × Quotient) + Remainder.

(2) The remainder concepts :-

a) Remainder of the expression apostrophe fraction numerator straight a space cross times space straight b space cross times space straight c over denominator straight n end fraction apostrophe is equal to remainder of expression

Where ar is remainder of apostrophe straight a over straight n apostrophe

br is remainder of apostrophe straight b over straight n apostrophe

cr is remainder of apostrophe straight c over straight n apostrophe

* Remainder of  straight N to the power of straight P minus 1 end exponent over straight P is always 1 where 'P' is a prime number.

(3) Counting the number of zeroes in a number or in a factorial :-

a) When we are asked to count number of zeroes in a number, here basically we have to count number of fives [5].

Example → Find number of zero in term 5 × 4 × 3 × 2 × 1

Here no. of five (5) are 1. So no. of zeroes will be 1.

b) When we are asked to count no. of zeroes in a factorial, just follow a simple formula.

no. of zeroes = straight n over 5 space plus space straight n over 5 squared space plus space straight n over 5 cubed space plus space.......

Example → Count number of zeroes in 10!

Using formula → 10 over 5 space plus space 10 over 5 squared space plus space.......

⇒ 2 + 0

Here second term is taken as zero, because after division we will not get a whole number.

So number of zeroes are = 2

(i) System of division →

Terminology : Divisor → 7, Dividend → 43, Quotient → 6, Remainder → 1

Dividend = (divisor × quotient) + remainder.

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