Number System & Simplification for rrb

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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

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.

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.

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.

TYPES OF FRACTIONS:

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 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 r1and 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