**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 Q
^{c}.

**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 r
_{1}and r_{2}When the sum of them is divided by the same divisor, the remainder is r_{3}. Then, divisor D = r_{1}+ r_{2}– r_{3} - 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 = x^{a}× y^{b}× z^{c}**STEP 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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