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NUMBER SYSTEM math capsule

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.

So lets start to learn a bit more about 'number system'.

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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