Number System & Simplification PDF

Number System & Simplification 

The ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits, which can represent any number.

Natural Numbers:

  • These are the numbers (1, 2, 3, etc.) that are used for counting. It is denoted by N.
  • There are infinite natural numbers and the smallest natural number is one (1).

Even numbers:

  • Natural numbers which are divisible by 2 are even numbers. It is denoted by E.
  • E = 2, 4, 6, 8, ….
  • Smallest even number is 2. There is no largest even number.

Odd numbers:

  • Natural numbers which are not divisible by 2 are odd numbers.
  • It is denoted by O.
  • O = 1, 3, 5, 7, ….
  • Smallest odd number is 1.
  • There is no largest odd number.

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Based on divisibility, there could be two types of natural numbers:

Prime and Composite

  1. Prime Numbers: Natural numbers which have exactly two factors, i.e., 1 and the number itself are called prime numbers. The lowest prime number is 2. 2 is also the only even prime number.
  2. Composite Numbers: It is a natural number that has atleast one divisor different from unity and itself.

Every composite number can be factorized into its prime factors.

For example: 24 = 2 × 2 × 2 × 3. Hence, 24 is a composite number.

The smallest composite number is 4.

Whole Numbers:

  • The natural numbers along with zero (0), from the system of whole numbers.
  • It is denoted by W.
  • There is no largest whole number and
  • The smallest whole number is 0.


  • The number system consisting of natural numbers, their negative and zero is called integers.
  • It is denoted by Z or I.
  • The smallest and the largest integers cannot be determined.

Real Numbers:

  • All numbers that can be represented on the number line are called real numbers.
  • It is denoted by R.
  •  R+: Positive real numbers and R : Negative real numbers.

Real numbers = Rational numbers + Irrational numbers.

  1. Rational numbers:
  • Any number that can be put in the form of , where p and q are integers and q 0, is called a rational number.
  • It is denoted by Q.
  • Every integer is a rational number.
  • Zero (0) is also a  rational number. The smallest and largest rational numbers cannot be determined. Every fraction (and decimal fraction) is a rational number

2. Irrational numbers:

  • The numbers which are not rational or which cannot be put in the form of , where p and q are integers and q 0, is called irrational number.
  • It is denoted by Q’ or Qc.

Fraction:  A fraction is a quantity which expresses a part of the whole.


  1. Proper fraction : If numerator is less than its denominator, then it is a proper fraction.
  2. Improper fraction: If numerator is greater than or equal to its denominator, then it is a improper fraction.
  3. Mixed fraction: it consists of an integer and a proper fraction.
  4. Equivalent fraction/Equal fractions: Fractions with same value.
  5. Like fractions: Fractions with same denominators.
  6. Unlike fractions: Fractions with different denominators.
  7. Simple fractions: Numerator and denominator are integers.
  8. Complex fraction: Numerator or denominator or both are fractional numbers.
  9. Decimal fraction: Denominator with the powers of 10.
  10. Vulgar fraction: Denominators are not the power of 10.

Operations: The following operations of addition, subtraction, multiplication and division are valid for real numbers.

  • Commutative property of addition: a + b = b + a
  • Associative property of addition: (a + b) + c = a + (b + c)
  • Commutative property of multiplication:  a * b = b * a
  • Associative property of multiplication:   (a * b) * c = a *  (b * c)
  • Distributive property of multiplication with respect to addition (a + b) * c = a * c + b * c

Complex numbers:

  • A number of the form a + bi, where a and b are real number and i =  (imaginary number) is called a complex number.
  • It is denoted by C.
  • For Example: 5i (a = 0 and b = 5),  + 3i (a =  and b = 3)


  • Divisibility by 2: A number is divisible by 2 if it’s unit digit is even or 0.
  • Divisibility by 3: A number is divisible by 3 if the sum of it’s digit are divisible by 3.
  • Divisibility by 4: A number is divisible by 4 if the last 2 digits are divisible by 4, or if the last two digits are 0’s.
  • Divisibility by 5: A number is divisible by 5 if it’s unit digit is 5 or 0.
  • Divisibility by 6: A number is divisible by 6 if it is simultaneously divisible by 2 and 3.
  • Divisibility by 7: We use osculator (-2) for divisibility test.
  • Divisible by 11: In a number, if difference of sum of digit at even places and sum of digit at odd places is either 0 or multiple of 11, then no. is divisible by 11.
  • Divisible by 13: we use (+4) as osculator.
  • Divisible by 17: We use (-5) as osculator.
  • Divisible by 19: We use (+2) as osculator.
  • Divisibility by a Composite number: A number is divisible by a given composite number if it is divisible by all factors of composite number.


Dividend = (Divisor × Quotient) + Remainder where, Dividend = The number which is being divided Divisor = The number which performs the division process Quotient = Greatest possible integer as a result of division Remainder = Rest part of dividend which cannot be further divided by the divisor.

Complete remainder:

  • A complete remainder is the remainder obtained by a number by the method of successive division.
  • Complete reminder = [I divisor × II remainder] + I remainder
  • Two different numbers x and y when divided by a certain divisor D leave remainder r1 and r2 When the sum of them is divided by the same divisor, the remainder is r3. Then, divisor D = r1 + r2 – r3
  • Method to find the number of different divisors (or factors) (including 1 and itself) of any composite number N:STEP I: Express N as a product of prime numbers as  N = xa × yb × zcSTEP II: Number of different divisors (including l and itself) = (a + 1) (b + 1) (c +1) …..

Counting Number of Zeros

  •  Sometimes we come across problems in which           we have to count number of zeros at the end of             factorial of any numbers. for example-           Number of zeros at the end of 10!
  •  10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 ×1
  •  Here basically we have to count number of     fives, because multiplication of five by any    even number will result in 0 at the end of final    product. In 10! we have 2 fives thus total number of zeros are 2.

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