Today we are going to study about the number system and how we can really play with the numbers.

## NUMBER SYSTEM

The system of numeration employing then ten digits 0, 1,2,3,4,.....,8 ,9 is known as

**number system .**
A group of digits, denoting a number is called a numeral.

A number may be a natural number , a whole number , an integer , a rational number , a real number etc.It is important to mention here that the above classification of numbers is not mutually exclusive, e.g. . An integer can also be a whole number or a natural number.

### VARIOUS TYPES OF NUMBERS

#### 1. NATURAL NUMBERS

Numbers which are used for counting i.e. 1,2,3,4.... are called natural numbers. The set of natural numbers is denoted by 'N' . Smallest natural number is 1 but we cannot find the largest natural number as successor of every natural number is again a natural number.

#### 2. WHOLE NUMBERS

Counting numbers including zero are known as whole numbers . The set of whole numbers is denoted by W. Thus, W = {0,1,2,3,4,5,....} is the set of whole numbers.

* Every natural number is a whole number.

* Zero is the smallest whole number but there is no largest whole number.

* Zero is the only whole number which is not a natural number.

#### 3. EVEN NUMBERS

The numbers which are divisible by 2 are called as even numbers . Such as 2,4,6,8,10,..... . In general , these are represented by 2m, where m belongs to N.

#### 4. ODD NUMBERS

The number which are not divisible by 2 are called as odd numbers . Such as 1,3,5,7,9,.... . In general , these are represented by (2m ± 1), where m belongs to N.

#### 5. PRIME NUMBERS

Those numbers which are divisible by 1 and the number itself are known as primes numbers, e.g. 2,3,5,7,....,etc, are prime numbers.

* If a number is not divisible by nay of the prime numbers upto square root of that number then it is prime number.

* 2 is the only even number which is prime.

* The prime number upto 100 are : 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 and 97 i.e. there are 25 prime numbers upto 100.

#### 6. COPRIME NUMBERS

Two natural numbers x and y are said to be copirme , if their HCF is ! i.e. they do not have a common factor other than 1.

e.g. (9,2), (5,6), are the pairs of caprice numbers.

* If x and y are any two caprices, a number p is divisible by x as well as by y , then the number is also divisible by xy.

* Coprime are also called as relatively prime numbers.

#### 7. COMPOSITE NUMBERS

A composite number is any integer greater than one that is not a prime number, e.g. 4,6,8,9,.... all are composite numbers.

* '1' is neither prime nor composite.

INTEGERS

The collection of all whole numbers i.e. positive , zero and negative numbers, are called integers.

The set of integers is denoted by Z or I.

Thus , Z or I = {.....,-4,-3,-2,-1,0,1,2,3,4,...} is the set of integers.

* Here every natural number and whole number is a part of integer.

### SOME SUBSETS OF INTEGERS

#### 1. POSITIVE INTEGERS

The set of all positive integers is denoted by I⁺ = {1,2,3,4,5,....} is a set of all positive integers. Here we can see that N and I⁺ are the same.

#### 2. NEGATIVE INTEGERS

The set of all negative integers is denoted by I⁻={-1,-2,-3,-4....} is a set of all negative integers.

* Here it is notable that 0 is neither positive now negative.

#### 3. NON-NEGATIVE INTEGERS

The set {0,1,2,3,4,....} is called the set of non negative integers.

#### 4. NON-POSITIVE INTEGERS

The set {0,-1,-2,-3,....} is the set of non positive integers.

#### 5. RATIONAL NUMBERS

The numbers which are expressed in the form of p/q , where p,q belongs to I , q≠0 are called rational numbers, where p and q are coprimes. The set of rational numbers is represented by 'Q' .

* Every natural number say x, can be written as x/1, so every natural number is a rational number.

* Zero is a rational number since we can write 0 = 0 /1

* Every natural number , whole number and integer is a rational number.

* Every non zero integer can be written as k/ 1, so every integer is a rational number.

EQUIVALENT RATIONAL NUMBERS

* Two rational numbers are said to be equivalent , if both the numerator and denominators are in proportion or they are reducible to be equal.

* There are infinitely many rational numbers between anyhow given rational numbers.

#### 6. IRRATIONAL NUMBERS

The number which cannot be expressed in the form p/q, where p and q both are integers and q≠0 are known as irrational numbers.

The irrational numbers when expressed in decimal form are ion nonterminating and non repeating form .

* Here it tis notable that exact number value of 𝝿 is not 22/7 or 3.14 , as 22/7 is a rational number while 𝝿 is irrational.

* Numbers like 0.101005001 .... is a non terminating and non repeating decimal. So it is an irrational number.

* if a + √b = x + √y , where a and x are rational and √b and √y are irrational , then a = x and b = y .

* The sum or difference of a rational and irrational number is also an irrational number.

*If we add, multiply or divide two irrational numbers, we may get an irrational number or rational number.

Have you ever heard about what are real numbers in

*number system*?

### REAL NUMBERS

The collection of all rational and all rational and all irrational numbers together forms, the set of real numbers and is denoted by 'R'. Thus , all natural numbers, whole numbers, integers, rational and irrational numbers are real numbers.

#### PROPERTIES OF REAL NUMBERS

GENERAL PROPERTIES OF R

1. If x and y are two real numbers, then either x>y, y>x or x=y.

2. If x and y are two real numbers, then

* x > y => 1 / x < 1 / y

* x > y => -x < -y

* x > y => xa > ya, when a > 0

* x > y => x + a > y + a

3. If xy = 0 => x = 0 or y = 0

#### PROPERTIES OF ADDITION ON R

1. CLOSURE PROPERTY

The sum of two real number is always a real number . iu.e. a belongs to R , b belongs to R

=> a + b belongs to R

Hence, R is closed for addition.

2. ASSOCIATIVE LAW

If a, b , c belongs to R then

(a + b) + c = a + (b + c)

3. COMMUTATIVE LAW

a + b = b + a for all a, b belongs to R

4. ADDITIVE IDENTITY

Additive identity is the number which when added to any number of the set , then the number remains unaltered. Such as zero is a real number such that 0 + a = a + 0 = a for all a, 0 belongs to R.

So, '0' is the additive identity in R.

5. ADDITIVE INVERSE

It is a number in the set which when added to any number then the result is additive identity.

If a belongs to R, -a belongs to R then a + (-a) = 0 , then -a is called as additive inverse of a.

#### PROPERTIES OF MULTIPLICATION ON R

1. CLOSURE PROPERTY

R is closed for multiplication. i.e. a, b belongs to R then ab or ba belongs to R , such that a,b belongs to R

2. ASSOCIATIVE LAW

If a, b, c belongs R , then (ab)c= a(bc)

3. COMMUTATIVE LAW

If a,b belongs to R then

a(b + c) = ab + ac

4. DISTRIBUTIVE LAW

If a, b ,c belongs to R then,

a(b + c) = ab + ac

5. MULTIPLICATIVE IDENTITY IN R

It is a number which when multiplied by a real number , the number remains unaltered , i.e. if a belongs to R then 1 . a = a = a . 1, here 1 is the real number .

So , '1' is the multiplicative identity in R.

6. MULTIPLICATIVE INVERSE

It is a number which when multiplied by the number the result is 1. If a belongs R, then a . (1 / a)= 1 , so 1 / a is multiplicative inverse of a or vice versa.

* Zero has no reciprocal, thus has no multiplicative inverse.

#### PROPERTIES OF SUBTRACTION AND DIVISION ON R

1. CLOSURE PROPERTY

R is closed for subtraction, if a,b belongs R, then a - b belongs to R.

2. Subtraction on R does not satisfies the commutative and associative laws.

3. Division of real number does not holds closure property , since 3 belongs to R , 0 belongs to R but 3/0 does not belongs to R.

#### ABSOLUTE VALUE OF A REAL NUMBER

The absolute value of a real number x is denoted by | x | .

Thus | 3 | = 3 and | -4 | = 4

If x is any real number, then | x | = { x, when x > 0

{ 0, when x = 0

{ -x, when x < 0

For example,

if x = 5 , | 5 | = 5

if x = 0 , | 0 | = 0

SOME PROPERTIES OF ABSOLUTE VALUES

1. | x | ≥ 0 for all real x.

2. | x | = a means x = a or x = -a

3. | x | > a means x > a or x < -a

4. √x² = | x | = + x , if x > 0 - x , if x < 0

5. | ab | = | a | | b |

6. | a / b | = | a | / | b | if b ≠ 0

7. | a + b | ≲ | a | + | b |

8. | a - b | ≳ | a | - | b |

#### TO FIND THE UNIT'S PLACE DIGIT OF A GIVEN EXPONENTIAL

Let the exponential number of the form be aⁿ and n belongs to I.

1. In case, if a is any of (0,1,5,6) then the units's place digit is 0,1,5 and 6 respectively.

2. In case , if a is any of (4 and 9)

(a) and if power is odd then the unit's place digit is 4 and 9 respectively

(b) and if power is even then the unit's place digit is 6 and 1 respectively

3. In case if a is any of (2,3,7,8) then see the following examples

TO FIND THE UNIT'S PLACE DIGIT OF (134647)^553

STEP 1

Divide 553 / 4 gives 1 as remainder , this remainder is taken as new power.

STEP 2

(134647)^553 = (134647)^1=7^1 = 7

The unit's place digit is 7.

or

STEP 3

If on dividing the remainder obtained is zero, take 4 as new power instead of zero.

e.g .

(134647)^552 = (134647)^0 = 7 ^ 0

= 7 ^ 4 = 2401

The unit's place digit is 1.

#### DIVISION ON NUMBERS

#### (Division Algorithm Lemma)

Let 'a ' and 'b' be two integers such that b ≠ 0, on dividing 'a' by 'b', 'q' will be the quotient and 'r' will be the remainder, then the relationship between a,b,q and r is

a = b q + r

or in general we can say

dividend = divisor * quotient + remainder

#### DIRECT DIVISIBLITY TEST

1. TEST FOR A NUMBER DIVISIBLE BY 2

If the number is an even number or has '0' in its unit place.

2. TEST FOR A NUMBER DIVISIBLE BY 3

If the same of the digits of the given number is divisible by 3, then the number is divisible by 3.

3. TEST FOR A NUMBER DIVISIBLE BY 4

If the number formed by the ten's place and unit place digits is divisible by 4 or last two digits are zero or divisible by 4.

4. TEST FOR A NUMBER DIVISIBLE BY 5

If the digit at unit place is 5 or 0 then the number is divisible by 5

5. TEST FOR A NUMBER DIVISIBLE BY 6

If a number is divisible by 2 and 3 , then it is also divisible by 6.

6. TEST FOR A NUMBER DIVISIBLE BY 7

If double of unit place digit of given number is subtracted from rest of digits and if the remainder is divisible by 7 , then that number is divisible by 7.

7. TEST FOR A NUMBER DIVISIBLE BY 8

If the last three digits of a number is completely divisible by 9 then the number is divisible by 9.

8. TEST FOR A NUMBER DIVISIBLE BY 9

If the sum of all the digits of a number is completely divisible by 9 , then the number is divisible by 9

9. TEST FOR A NUMBER DIVISIBLE BY 10

If zero exists at the unit place then the number is divisible by 10

10. TEST FOR A NUMBER DIVISIBLE BY 11

If the difference between the sum of digits at even places and sum digits at odd places is (0) then the number is divisible by 11

### THEOREM OF DIVISIBILITY

1. If N is a composite number of the form N = (a^p)(b^q)(c^r)..... , where a,b,c are primes then the number of divisors of N represented by m is given by

m = (p + 1)(q + 1)(r + 1).....

2. The sum of the divisors of N represented by S is given by

S =

__(a^(p+1) - 1)____(b^(q + 1) - 1)____(c^(r+1) - 1)__
(a - 1) (b - 1) (c - 1)

SOME IMPORTANT RESULTS ON DIVISION

* If p divides q and r then p divided their sum and difference also.

* For any natural number n , (n³ - n) is divisible by 6

* The product of three consecutive natural numbers is always divisible by 6

* (xⁿ - aⁿ) is divisible by (x + a) for even values of n.

* xⁿ + aⁿ is divisible by (x + a) for odd values of m

* xⁿ - aⁿ is divisible by (x - a) for all values of m

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__number system__?

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