Number System

number system

In this topic of Number system, we will study the following-

Natural Numbers (N):  Simple counting numbers starting from 1 are called Natural Numbers. Set of natural numbers is denoted by 'N', then we write N = {1, 2, 3, 4, 5, 6,…}
The smallest natural number is 1 and the greatest or biggest natural Number is not possible to find or we can say the greatest or biggest natural Number doesn't exist.

Whole Numbers (W):,If we add Zero (0) in the set of Natural Numbers then we get the set of whole Numbers, then we write W = {0, 1, 2, 3, 4, 5,…}
The smallest whole number is 0 and greatest Whole number is not possible to find.

Integers (I): All the whole numbers and their negative numbers (Except zero because zero is a unique number. It is neither negative nor positive) are called Integers, then we write I = {– 3, –2, –1, 0, 1, 2, 3, …}

Integers play important role in Number system.

Rational Numbers: Any number which can be expressed in the form of p/q, where p and q are both integers and q # 0 are called rational numbers.

e.g. 3/2, 7/9, 5, 2, 3.2 = 32/10

There exists infinite number of rational numbers between any two rational numbers.

Decimal Expansion of rational Numbers are either Terminating (eg. 5.5, 3.65, 8.9) or Non-Terminating but Repeating (eg. 1.3333.., 2.777).

Irrational Numbers: Any number which can't be expressed in the form of p/q, where p and q are both integers and q # 0 are called irrational numbers.Decimal Expansion of irrational Numbers are Non-Terminating and Not-Repeating.

e.g. √3, √5,√29

Real Numbers: All rational and irrational numbers are called real number. If square of a number is positive then it is a real number otherwisw imaginary.

Eg. √-5 is not real because   (√-5)= -5.

Basic Rules on Natural Numbers

  1. One digit numbers are from 1 to 9. There are 9 one digit numbers. ie, 9 × 100.
  2.  Two digit numbers are from 10 to 99. There, are 90 two digit numbers. ie, 9 × 10.
  3.  Three digit numbers are from 100 to 999. There are 900 three digit numbers ie, 9 × 102.
    In general the number of n digit numbers are 9 × 10(n–1)

Sum of the first n, natural numbers ie, 1 + 2 + 3 + 4 + … + n = n (n+ 1) / 2

Sum of the squares of the first n natural numbers ie. 12 + 23 + 32 + 42 + …+ n2 =  n(n+1)(2n+1)/ 6

Different Types of Numbers

Even Numbers: Numbers which are exactly divisible by 2 are called even numbers.
eg, – 4, – 2, 0, 2, 4…
Sum of first n even numbers = n (n + 1)

Odd Numbers: Numbers which are not exactly divisible by 2 are called odd numbers.
eg, – 5, –3, –1, 0, 1, 3, 5…
Sum of first n odd numbers = n2

Learn about even and odd numbers in more detail

Prime Numbers: Definition-1)Numbers which are divisible by one and itself only are called prime numbers.(except 1), Definition-2)- Any number having exactly two factors is called Prime Number.
Eg. 2, 3, 5, 7, 11…

  • 2 is the only even prime number.
  • 1 is not a prime number because it has only one factor.
  • Every prime number greater than 3 can be written in the form of (6K + 1) or (6K – 1) where K is an integer.
  • There are 15 prime numbers between 1 and 50 and l0 prime numbers between 50 and 100.

Relative Prime Numbers: Two numbers are said to be relatively prime if they do not have any common factor other than 1.
eg, (3, 5), (4, 7), (11, 15), (15, 4)…
Twin Primes: Two prime numbers which differ by 2 are called twin primes.
eg, (3, 5), (5, 7), (11, 13),…
Composite Numbers: Numbers which are not prime arc called composite numbers
eg, 4, 6, 9, 15,…
1 is neither prime nor composite.
Perfect Number: A number is said to be a perfect number, if the sum of all its factors excluding itself is
equal to the number itself. eg, Factors of 6 are 1, 2, 3 and 6.
Sum of factors excluding 6 = 1 + 2 + 3 = 6.
6 is a perfect number.
Other examples of perfect numbers are 28, 496, 8128 etc.

Rules for Divisibility

In Number system, This topic of divisibility is very important.

Divisibility by 2: If any number has its last digit 0 or 2 or 4 or 6 or 8 then the number will be completely divisible by 2.
eg, 3582, 4800, 345673829874566, 65756878434.

Divisibility by 3: If sum of all digits of a number is divisible by 3 then number is divisible by 3.
eg, 453 => Sum of the digits = 4 + 5 + 3 = 12, 12 is divisible by 3 so the number 453 is divisible by 3.

Divisibility by 4: A number is divisible by 4, if the number formed with its last two digits is divisible by 4. eg, if we take the number 45024, the last two digits form 24. Since, the number 24 is divisible by 4, the number 45024 is also divisible by 4.

Divisibility by 5: A number having its last digit 0 or 5 is divisible by 5.
eg, 10, 25, 60

Divisibility by 6: A number divisible by 2 and 3 is also divisible by 6.
eg, 48, 24, 108

Divisibility by 7:
eg, 658
Divisibility by 8: A number is divisible by 8, if the number formed with its last two digits is divisible by 8. Eg.- if we take the number 45168, the last three digits form 168. Since, the number 168 is divisible by 8, the number 45168 is divisible by 8.

Divisibility by 9: If the sum of all digits of a number is divisible by 9 then number is divisible by 9.
Eg. 684 = 6 + 8 + 4 = 18.
18 is divisible by 9 so, 684 is divisible by 9.

Divisibility by 10: A number having its last digit 0 is divisible by 10. Eg, 20, 180, 350 etc.

Divisibility by 11:

eg, 30426

Division on Numbers

When we perform a division then we have four quantities.

(1) Dividend, (2) Divisor, (3) Quotient and (4) Remainder.

These quantities make a mathematical relation. Thios is=>> Dividend = Divisor × Quotient + Remainder.

If we want to divide a number 'a' by 'b' (a ÷ b) then we get the quotient(q) and remainder (r) and this relation becomes

a = bq + r, where r <b  (r is always less then b)

Eg. Let if we divide 32 by 10 then

Dividend (a) = 32, Divisor (b) = 10, Quotient (q) = 3 and Remainder (r) = 2

Relation a = bq + r for above division is

32 = 10 X 3 + 2

Factors and Multiples

Factor: A number which divides a given number exactly is called a factor of the given number,
eg, Q- Write all the factors of 24.

Soln - Since, 24 is divisible by 1, 2, 3, 4, 6, 8, 12 and 24, then

Factor of 24 = 1, 2, 3,  4, 6, 8, 12, 24

Some facts about Factors-
• 1 is a factor of every number
• A number is a factor of itself
• The smallest factor of a given number is 1 and the greatest factor is the number itself.
• If a number is divided by any of its factors, the remainder is always zero.
• Every factor of a number is either less than or at the most equal to the given number.
• Number of factors of a number are finite.

Multiples: A multiple is the result of multiplying a number by an integer (not a fraction).

so, if we want to find the multiples of 7 then multiples of 7 are 0, 7, 14, 21, 28, 35 ...... and negative multiples are -7, -14, -21, -28, ....

Some facts about Multiples-

  • Every number is a multiple of 1.
  • Every number is the multiple of itself.
  • Zero (0) is a multiple of every number.
  • The product of two or more factors is the multiple of each factor.
  • There is no end to multiples of a number. Eg 5, 10, 15, 20, 25, …………….., 100, 105, 110, …………………., are the multiples of 5.

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