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

${1,2,3,4,5,........ }$

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

${0,1,2,3,4.........}$

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

$( a + i b ), ( a - i b ).$

(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:-

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 & 97.$

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

$kgtsqrtx$

$x$ 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?

$165$ 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:-

$1$ is neither prime nor composite.

$2$ 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.

$sqrt(4/9)$ 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

$5$ b)

$7$ c)

$9$ d)

$1$

6) The numbers $p$ , $p+2$ , $p+4$ are all prime if $p$ is equal to :

$3$ 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 :

$-3/2$ 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

$p/a$ or

$p/b$ b)

$p/a$ &

$p/b$

$p/(a+b)$ d) None

9) which one of the following numbers can be represented as non - terminating, repeating decimals?

$39/24$ 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

$22/7$

$pi$ is irrational

d) None of these.

Answer - 1)

$d$ 2)

$c$ 3)

$c$ 4)

$d$ 5)

$a$ 6)

$a$ 7)

$b$

$a$ 9)

$c$ 10)

$c$

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Number System

Number system :

(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 Numbers:-

Exercise

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