Number System
Number system :
A study about the characteristics of different numbers is called a number system.
Different types of numbers
1) Natural Numbers
2) Whole Numbers
3) Integers
(a) Positive Integers b) Negative Integers c) Non-Positive Integers d) Non Negative Integers )
4) Rational Numbers
5) Irrational Numbers
6) Real Numbers
7) Imaginary Numbers
8) Even Numbers
9) Odd Numbers
10) Prime Numbers
11) Composite Numbers
12) Co - Prime Number
(1) Natural Numbers :
Counting numbers are called natural numbers. Set of Natural numbers denoted by N
Example
N=`{1,2,3,4,5,........ }`
Note -
Zero `(0)` is not a natural number
(2) Whole Numbers
All counting numbers with zero (0) are called whole numbers, set of Whole numbers denoted by W
Example
W=`{0,1,2,3,4.........}`
Note -
Zero `(0)` is the whole number.
(3) Integers:
All-natural numbers, `0` and negatives of counting numbers are called Integers. set of Integers is denoted by `I`.
Example
`I`= `{ 0,1,2,3,4,5,6,....-1,-2,-3,-4,-5,....}`
(a)Positive Integers :
Set of all positive integers.
Example
`I+` = `{1,2,3,4,5........}`
Note -
Positive integers and natural numbers are synonyms.
(b)Negative Integers:
Set of all negative integers
Example
`I-` = `{-1,-2,-3,-4,-5,..}`
(c) Non-positive Integers :
set of all non-positive integers.
Example
`{0,-1,-2,-3,-4 .....}`
(d)Non negative Integers:
set of all non negative integers.
Example
`{0,-1,-2,-3,-4,....}`
Note -
Zero is neither positive nor negative integer.
(4) Rational numbers ;
The numbers of the form `p/q` where `p` and `q` are integers and `q` are not equal to zero are known as rational numbers.
Example
`3/4, 56/75, 0/23.`
Terminating & Repeating Decimals:
Every rational number has peculiar characteristics that it when expressed in decimal form is expressible either in terminating decimals or in repeating decimals.
Example:
`1/2 = 0.5`
`1/3 =0.333`..
`8/45 = 0.177`...
(5)Irrational Numbers:
The numbers which of the not form `p/q` are known as irrational numbers.
All numbers which when expressed in decimal form are in non - terminating and non - repeating form, are known as irrational numbers.
Example:
`sqrt2, sqrt5, sqrt7`
Note -
The exact value of π is not `22/7` as `22/7` is rational while π is irrational. `22/7` is the approximate value of `π`. Similarly, `3.14` is not an exact value of `pi`.
Also `0.101001000100001`.... is a non -terminating and non-repeating decimal, so it is an example of an irrational number.
(6)Real Numbers :
The group of all rational and irrational numbers is called real numbers.
The set of real numbers is denoted by `R`
Note:-
a) The sum or difference of rational and irrational numbers is irrational numbers.
Example:-
`(2+sqrt3), (7-sqrt2), (2/3+sqrt3)` etc. are all irrational.
b) The Product of a rational and an irrational number is irrational, e. g.
`3sqrt2, -3sqrt3` etc. are all irrational.
(7) Imaginary Numbers:
Numbers that are not real numbers mean these numbers are in imagination, known as Imaginary Numbers.
Example:-
`sqrt(-7), sqrt(-3)`
Note:-
a) we cannot find the square root of negative numbers.
b) we can write Imaginary Numbers
as `( a + i b ), ( a - i b ).`
(8) Even Numbers :
Integers divisible by `2` are known as even integers.
Example:-
`-4,-2,0,2,4`.... etc are even integers.
(9) Odd Numbers:-
Integers not divisible by `2` are known as odd integers.
Example:-
`-5,-3,-1,1,3,5` .... etc are odd integers.
(10)Prime Numbers:-
A number is greater than `1` and has exactly two factors, `1` and itself.
Example:-
a) `2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 & 97.`
There are `25` prime numbers between `1` to `100`.
To test a number greater than `100,` whether it is prime or not :
let `x` be a given number and let `k` be an integer very near to `sqrtx` such that `kgtsqrtx`
If `x` is not divisible by any prime number less than `k`, then `x` is prime; otherwise, it is not prime.
Example:-
b) which of the following numbers are prime?
1)`165` 2) `247` 3) `571`
Sol.
1) The nearest value of `sqrt(165)` is `13`, i. e. `13`
`gt,sqrt165`. So, we test the divisibility of `165` by each of the prime numbers less than `13`.
We find that `165` is divisible by `11`.
So,`165` is not a prime number.
2) Clearly, `16gtsqrt247`.
So, we divide `247` by each number less than `16` which are `2,3,5,7,9,11,13`.
we find that `247` is divisible by `13`.
Hence `247` is not a prime number.
3)Clearly, `24gtsqrt571`
So, we divide `571` by each prime number less than `24` which are `2, 3,5,7,9,11,13,17,19,23.`
We find that `571` is not divisible by any of them.
So, `571` is a prime number.
(11) Composite Numbers:-
Numbers greater than `1` which are not prime, are known as composite numbers.
Example:-
`4,6,8,9,10,12` etc. are all composite numbers.
Note:-
1) `1` is neither prime nor composite.
2) `2` is the only even number that is prime.
3)There are `25` prime numbers between `1` and `100.`
(12) Co- Prime Numbers:-
HCF of two natural numbers is `1`, known as Co- prime numbers.
Example:-
`9` & `16` are Co- prime numbers.
Exercise
1) If n is a natural number, then `sqrtn` is :
a) Always a natural number
b) Always a rational number
c) Always an irrational number
d) Sometimes a natural number and sometimes an irrational number
2) Which of the following is irrational.
a) `sqrt(4/9)` b) ` 4/5` c) `sqrt7` d) `sqrt81`
3) The number `0.318564318564318564`..... is:
a) A natural number b) An integer
c) A rational number
d) An irrational number
4) In between two fractions, there are:
a) precisely two fractions
b) even number of fractions
c) a finite number of fractions
d) Infinitely many numbers of fractions
5) A prime number greater than `11` will never end with
a) `5` b) `7` c) `9` d) `1`
6) The numbers `p`, `p+2`, `p+4` are all prime if `p` is equal to :
a) `3` b) `5` c) `29` d) `191`
7) The product of two numbers is `-14/27.` If one of the numbers be `7/9,` then other number is :
a) `-3/2` b) `-2/3` c) `2/3` d) `3/2`
8) If `p` is a prime number and `p` divides `ab` (written as `p/(ab)` ), where a and b are integers, then
a) `p/a` or `p/b` b) `p/a` & `p/b`
c) `p/(a+b)` d) None
9) which one of the following numbers can be represented as non - terminating, repeating decimals?
a) `39/24` b) `3/16` c) `3/11` d) `137/25`
10) Which of the following is a correct statement?
a) exact value of `pi` is `3.14`
b) exact value of `pi` is `22/7`
c) `pi` is irrational
d) None of these.
Answer - 1) `d` 2) `c` 3) `c` 4) `d` 5) `a` 6) `a` 7) `b`
8) `a` 9) `c` 10) `c`
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