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**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**

**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.

**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.

**Integers:**

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.

**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.

**TYPES OF FRACTIONS:**

**Proper fraction :**If numerator is less than its denominator, then it is a proper fraction.

**Improper fraction:**If numerator is greater than or equal to its denominator, then it is a improper fraction.

**Mixed fraction:**it consists of an integer and a proper fraction.

Equivalent fraction/Equal fractions: Fractions with same value.

**Like fractions:**Fractions with same denominators.

**Unlike fractions:**Fractions with different denominators.

**Simple fractions:**Numerator and denominator are integers.

**Complex fraction:**Numerator or denominator or both are fractional numbers.

**Decimal fraction:**Denominator with the powers of 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 RULES**

**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.

**DIVISION ALGORITHM:**

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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