## Number System

In this system we study different types of numbers .

In the Hindu-Arabic system, we use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 .

### Face Value :-

The face value of a digit in any number is the value of the digit itself.

For example- 48763

Face value of 7 = 7

Face value of 8 = 8

Face value of 4 = 4

Face value of 6 = 6

Face value of 3 = 3

### Place Value :-

The Place value of a digit in any number is changes according to the change of its place .

For example- 56348

Place value of 8 = 8 × 1 = 8

Place value of 4 = 4 × 10 = 40

Place value of 3 = 3 × 100 = 300

Place value of 6 = 6 × 1000 = 6000

Place value of 5 = 5 × 10000 = 50000

### Types of Numbers :-

#### Natural Numbers :-

A number which start from 1 and end to infinity is called natural numbers.

Natural numbers are also called counting numbers. All natural numbers are positive.

Natural numbers denoted by N. Smallest natural number is 1.

For example- N = { 1, 2, 3, 4, ……… }

#### Whole Numbers :-

When 0 is included in natural numbers then they are called whole numbers.

All whole numbers are positive.

Whole numbers are denoted by W.

Smallest whole number is 0.

For example- W = { 0, 1, 2, 3, ……..} **Note**- All natural numbers are whole numbers, but all whole numbers are not a natural numbers.

#### Integers :-

When negative numbers included in whole numbers, then they are called integers.

Integers are denoted by I.

For example- I = { …..,-3, -2, -1, 0, 1, 2, …..}.

Integers are two types**Positive Integers **

Natural numbers are called positive integers.

They are denoted by I^{+}.

For example- I^{+} = { 1, 2, 3, ……} **Negative Integers **

Negative natural numbers are called negative integers.

They are denoted by I^{-}.

For example- I^{-} = { -1, -2, -3, ……..}.

Zero in neither positive nor negative integer.

#### Even Numbers :-

A number which is multiple of 2, is called an even number.

Or, A number whose last digit's 0, 2, 4, 6 or 8 is called an even number.

For example- 2, 4, 6, 8, 10, 12,……. etc.

#### Odd Numbers :-

A number which in not multiple of 2, is called an odd number.

A number whose last digit's 1, 3, 5, 7 or 9 is called an odd numbers.

For example- 1, 3, 5, 7, 9, 11, 13, ……. etc.

#### Prime Numbers :-

A number which have only two factors, factors are 1 and number itself is called prime numbers.

For example- 2, 3, 5, 7, 11,….. etc.

Smallest prime number is 2.

Even prime number is 2. 2 is only one even prime.

#### Composite Numbers :-

A number which have more than two factors, is called composite number.

Smallest composite number is 4.

For example- 4, 6, 8, 9, 10, 12,……… etc.

Note- 1 is neither a prime nor composite number.

#### Co-Prime :-

Pair of numbers whose common factor 1, is called co-prime.

or, Pairs of numbers whose HCF 1, is called co-primes.

For example- ( 1, 3), ( 16, 17 ) etc.

#### Twin-Prime :-

Pair of prime numbers whose difference 2, is called twin-primes.

For example- ( 3, 5 ), ( 5, 7 ), ( 11, 13 ), ( 17, 19 ) etc.

#### Rational Numbers :-

A number which is in the form of | is called rational numbers. | |

q |

For example- | , | , | etc. | |||

3 | 5 | 13 |

#### Irrational Numbers :-

A number whose decimal expression is non terminating and non repeating is called irrational numbers.

or, A number which in not in the form of | is called irrational number. | |

q |

#### Real Numbers :-

A number which include rational and irrational numbers both is called real numbers.

For example- | , √7 etc. | |

3 |

### Test of Divisibility :-

#### Divisibility by 2 :-

If the unit's digit of a number is 0, 2, 4, 6 or 8, then the number is divisible by 2.

For example- 1234, 242, 3230 etc.

#### Divisibility by 3 :-

If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3.

For example- 3252 - 3 + 2 + 5 + 2 = 12 which is divisible by 3.

∴ 3252 is divisible by 3.

For example- 4653 - 4 + 6 + 5 + 3 = 18, which is divisible by 3.

∴ 4652 is divisible by 3.

#### Divisibility by 4 :-

If the last 2 digit's of a number is divisible by 4, then the number is divisible by 4.

or, If the last 2 digits or more than 2 digits of a number is zero, then the number is divisible by 4.

For example- 2532 is divisible by 4, since 32 is divisible by 4.

2400, 163000 are divisible by 4 as they have two or more zeroes at end.

#### Divisibility by 5 :-

If the last digit of a number is 0 or 5, then the number is divisible by 5.

For example- 235, 24340, 645 etc.

#### Divisibility by 6 : -

If the number is divisible by 2 and 3 both then, the number is also divisible by 6.

For example- 528, 738 etc.

#### Divisibility by 7 :-

If the difference between twice the last digit and the number formed by remaining digits is either zero or a multiple of 7, then the number is divisible by 7.

For example- 826, 82 - ( 2 × 6 ) = 82 - 12 = 70, which is divisible by 7.

∴ 826 is divisible by 7.

#### Divisibility by 8 :-

If the last three digits of a number is divisible by 8, then the number is divisible by 8.

or, If the last three digits or more than three digits of a number is zero, then the number is divisible by 8.

For example- 2368 - Last three digits = 368, which is divisible by 8.

∴ 2368 is divisible by 8.

4354000, 54340000 are divisible by 8 as they have two or more then two zeroes at end.

#### Divisibility by 9 :-

If the sum of the digits of a number is divisible by 9, then the number is also divisible by 9.

For example- 24345- 2 + 4 + 3 + 4 + 5 = 18, which is divisible by 9.

∴ 24345 is divisible by 9.

#### Divisibility by 10 :-

If the last digit of a number is 0, then the number is divisible by 10.

For example- 74540, 12740, 75800 etc.

#### Divisibility by 11 :-

If the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11, then the number is divisible by 11.

For example- 529782 = ( 5 + 9 + 8 ) - ( 2 + 7 + 2 )

= 22 - 11

= 11, which is multiple of 11.

∴ 529782 is divisible by 11.

#### Divisibility by 12 :-

If the number is divisible by both 3 and 4, then the number is also divisible by 12.

For example- 5412 is divisible by both 3 and 4. Therefore, it is divisible by 12.

#### Divisibility by 14 :-

If the number is divisible by both 2 and 7, then the number is also divisible by 14.

For example- 826 is divisible by both 2 and 7. Therefore, it is divisible by 14.

#### Divisibility by 15 :-

If the number is divisible by both 3 and 5, then the number is also divisible by 15.

For example- 2415 is divisible by both 3 and 5. Therefore, it is divisible by 15.

#### Divisibility by 16 :-

If a number whose last four digits is divisible by 4, then the number is divisible by 16.

For example- 375248 - Last 4-digits = 5248, which is divisible by 16.

∴ 375248 is divisible by 16.

#### Divisibility by 18 :-

If a number is divisible by both 2 and 9, then the number is also divisible by 18.

For example- 23436 is divisible by both 2 and 9. Therefore, it is divisible by 18.

### Unit's Place of an Expression :-

**Method to find unit digit :-**

Unit's Place of an Expression

( 0 )^{n} = 0

( 1 )^{n} = 1

( 5 )^{n} = 5

( 6 )^{n} = 6

If the unit digit of any number are 0, 1, 5 or 6, then the unit's digit remains same.

**Ex**- Find the unit digit of 2345^{127}.

**Solution**:- ( 2345 )^{127}

= 5

**Method**- If unit's place is 3, 7 or 9.

**Step1**- First divide the power of last 2-digits by 4 and find the remainder.

**Step2**- On unit digit put the remainder and simplify.

**Step3**- After simplifying write unit digit.

**Ex**- Find the unit digit of ( 1537 )^{1537}

**Solution**:- ( 1537 )^{1537}

= 7^{37}

= 7^{37 ÷ 4}

= 7^{1}

= 7

**Ex**- Find the unit digit of ( 143 )^{143}

**Solution**:- ( 143 )^{143}

= 3^{43}

= 3^{43 ÷ 4}

= 3^{3}

= 27

= 7

**Method**- If unit's place 2, 4 or 8.

**Step 1**- First divide the power of last 2-digits by 4 and find the remainder.

**Step 2**- On unit digit put the remainder and simplify.

** Step 3**- After simplifying write unit digit.

**Step 4**- If unit digit is 1 then the answer will be 6.

**Ex**- Find the unit digit of ( 1242 )^{1448}.

**Solution**:- ( 1242 )^{1448}

= 2^{48}

= 2^{0}

= 1

= 6

**Ex**- Find the unit digit of ( 1748 )^{1748}.

**Solution**:- ( 1748 )^{1748}

= 8^{48}

= 8^{0}

= 1

= 6

### Basic Number Theory :-

**1. Sum of first n natural numbers**1 + 2 + 3 + 4 + ……..+ n = | Where, n = numbers of terms | |

2 |

**Ex**- Find the first sum of first 50 natural numbers. **Solution**:- 1 + 2 + 3 + ……..+ 50

Here, n = 50

Sum = | = | |||

2 | 2 |

= | = 25 × 51 = 1275 | |

2 |

**2. Sum of first n odd numbers**

Where, n = number of terms

**1 + 3 + 5 + 7 +…….. = n**^{2}Where, n = number of terms

**Ex**- Find the sum of first 10 odd numbers.

**Solution**:- 1 + 3 + 5 + 7 + ………

Here, n = 10

Sum = n^{2}

= 10^{2}

= 100

Sum of first 10 odd numbers is 100.

**3. Sum of first n even numbers**

**2 + 4 + 6 + 8 + …….. = n ( n + 1 )****Ex**- Find the sum of first 20 even numbers. **Solution**:- 2 + 4 + 6 + 8 + …….

Here, n = 20

Sum = n ( n + 1 )

= 20 ( 20 + 1 )

= 20 × 21

∴ Sum= 420

**4. Sum of square of first n natural numbers**1^{2} + 2^{2} + 3^{2} + ……. = |
||

6 |

**Ex**- Find the sum of square of first 10 natural numbers. **Solution**:- 1^{2} + 2^{2} + 3^{2} + …….

Here, n = 10

Sum = | ||

6 |

= | = | = 5 × 11 × 7 = 385 | ||

6 | 6 |

**5. Sum of cubes of first n natural numbers** 1^{3} + 2^{3} + 3^{3} + ……. = |
^{2} |
|||

2 |

**Ex**- Find the sum of cubes of first 5 natural numbers.

**Solution :**- 1^{3} + 2^{3} + 3^{3} + …….

Here, n = 5

Sum = | ^{2} |
|||

2 |

= | ^{2} |
|||

2 |

^{2}

= 15

^{2}

= 225

### Comparison of Numbers :-

**1.** To compare two or more numbers, first of all the denominators of the power of given numbers are made equal to each other and then simplify and compared.

**Ex**- Which of the following number is largest?

√2 , ^{4}√3 , ^{3}√4 , ^{12}√131

**Solution**:- LCM of 2, 3, 4 and 12 = 12

= 2^{6}, 3^{3}, 4^{4}, 131^{1}

= 64, 27, 256, 131

∴ Largest = ^{3}√4

**2.** To compare two or more fractions, first find the gap between numerator and denominator, then divide the numerator by gap.

In this method **N < D**

gap |

Ex- |
, | , | , | , | , | . Find smallest and largest. | ||||||

67 | 8 | 19 | 34 | 45 | 88 |

Solution:- |
, | , | , | , | , | |||||||

67 | 8 | 19 | 34 | 45 | 88 |

= | , | , | , | , | , | |||||||

6 | 1 | 2 | 3 | 4 | 7 |

^{1}/

_{6}, 7,8

^{1}/

_{2}, 10

^{1}/

_{3}, 10

^{1}/

_{4}, 11

^{4}/

_{7}

∴ Largest = | and Smallest = | ||

88 | 8 |

**3.** In this method first find the HCF of power and then divide the power by HCF.

**Ex**- 2^{52}, 3^{78}, 4^{65}, 5^{91} Find the smallest and largest. **Solution**:-

HCF of 52, 78, 65 and 91 = 13

2^{52/13}, 3^{78/13}, 4^{65/13}, 5^{91/13}

= 2^{52/13}, 3^{78/13}, 4^{65/13}, 5^{91/13} = 2^{4}, 3^{6}, 4^{5}, 5^{7}

∴ Largest = 5^{91}

∴ Smallest = 2^{52}