Number Systems:Face & Place Value Naturals, Integers, Rationals, Irrationals, Reals, and Beyond

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What is Number System?

A system in which we study different types of numbers, their relationship and rules govern in them is called a number system.

Face value and Place value in a number

Face value:

In a numeral, the face value of a digit is the value of the digit itself irrespective of its place in the numeral.

For example,75963 the face value of 7 is 7, the face value of 5 is 5, the face value of 6 is 6.

Place value:

In a numeral, the place value of a digit changes according to the change of its place.

Number System: Types of Numbers

Natural Numbers:

Natural numbers are set of counting numbers. It is denoted by N.

Where,N = {1, 2, 3, ……¥} . 0 is not a natural number

Whole Numbers:

Whole numbers are those numbers when 0 is included in the set of natural numbers.

It is denoted by W. where, W = {0, 1, 2, 3, …..¥}

Integers:

Integers are those numbers when in the set of whole numbers, natural numbers with negative sign are included .It is denoted by I or Z. I : [– ¥, …………………………. –4, –3, –2, –1, 0, 1, 2, 3, 4, ……….¥]

Integers are two types

Positive Integers:Z + = {1, 2, 3, ……………. ¥}

Negative Integers:Z – = {– ¥, ………………. –3, – 2, –1}

Even Numbers :

The set of all natural numbers which are divisible by 2 are called even numbers. It is denoted by E. Where, E = {2, 4, 6, 8, 10, ……¥}

Odd Numbers :

The set of all natural numbers which are not divisible by 2 are called odd numbers. In other words, the natural numbers which are not even numbers, are odd numbers. i.e., O = {1, 3, 5, 7, ……..¥}

Prime Numbers:

A counting number is called a prime number when it is exactly divisible by 1 and itself.

For example, 2,3,5,7,11,11….

Important facts:

* A prime number is always greater than 1

*2 is the only even number which is prime.

* The lowest odd prime number is 3

How to test a prime number?

If P= given number, then

i) find the whole number x such that x> rootP

ii) take all the prime numbers less than or equal to X

iii) if none of these divide P exactly, then P is prime otherwise 20 non Prime

*All prime numbers other than 2 are odd numbers but all odd numbers are not prime numbers.

Co-Prime Numbers :

Two numbers which have no common factor except 1, are called Co–Prime numbers. Such as, 9 and 16, 4 and 17, 80 and 81 etc.

Composite Numbers:

Composite numbers are non Prime natural numbers. They must have at least one factor apart from 1 and itself. As, 4, 6, 9, 15, ……..

* 1 is neither Prime number nor composite number.

*Composite numbers may be even or odd.

Rational Numbers :

A number that can be expressed as P/Q is called rational number, where P and Q are integers and Q not equal to 0

search as 4/ 5, 5/ 1, 1/ 2 etc are rational numbers.

Irrational Numbers :

The numbers that cannot be expressed in the form of P/Q are called irrational numbers,where P and Q are integers and Q not equal to 0,Such as

Root2 = 1.414213562……….

pi = 3.141592653 ………..

Real Numbers:

Set of all rational numbers as well as irrational numbers is called Real numbers. The square of all of them is positive.

Complex Numbers : Z = a + ib is called complex number, where a and b are real numbers, b not equal to 0 and i = root-1 . Such as,root -2 ,root -3 etc. So, a + ib or 4 + 5i are complex numbers.

Number System: Divisibility Tests

Divisibility by 2: If the last digit of a number is 0 or an even number then that number is divisible by 2. Such as 242, 540, etc.

Divisibility by 3: If the sum of all digits of a number is divisible by 3, then that number will be divisible by 3. Such as. 432 : 4 + 3 + 2 = 9 which is divisible by 3. So, 432 is divisible by 3. Divisibility by 4: If in any number last two digits are divisible by 4, the whole number will be divisible by 4. Such as 48424. In this number 24 is divisible by 4. So, 48424 will be divisible by 4.

Divisibility by 5: If the last digit of a number is 5 or 0, then that number is divisible by 5. Such as 200, 225 etc.

Divisibility by 6: If a number is divisible by both 2 and 3, then that number is divisible by 6 also, such as 216, 25614 etc.

Divisibility by 7: Here the concept of osculator should be applied. The meaning of negative osculator is – there increases or decreases 1 from the factor of 10 of the number. As, 21 : 2 × 10 + 1 = 21 49 : 5 × 10 – 1 = 50 – 1 = 49 To check the divisibility of 7, we use osculator ‘2’, as , 112 : 11 – 2 × 2 = 7 which is divisible by 7 Again, 343 : 34 – 2 × 3 = 28 which is divisible by 7. Then 343 will be divisible by 7.

 Divisibility by 8: If in any number last three digits are divisible by 8, then whole number is divisible by 8, such as, 247864 since 864 is divisible by 8. So, 247864 is divisible by 8. Similarly, 289000 is divisible by 8.

Divisibility by 9: If the sum of all digits of a number is divisible by 9, then that whole number will be divisible by 9. As, 243243 : 2 + 4 + 3 + 2 + 4 + 3 = 18 is divisible by 9. So, 243243 is divisible by 9. Divisibility by 10 : The number whose last digit is ‘0’, is divisible by 10, such as, 10, 20, 200, 300 etc.

Divisibility by 11: If the difference between “Sum of digits at even place” and “Sum of digits at odd place” is divisible by 11, then the whole number is divisible by 11.

Divisibility by 12: If a number is divisible by 3 and 4 both. Then the number is divisible by 12. Such as, 19044 etc.

Divisibility by 13: For 13 we use osculator 4, but our osculator is not negative here. It is one-more osculator (4). 143 : 14 + 3 × 4 = 26 and 26 is divisible by 13, So, 143 is divisible by 13. Similarly for 325 : 32 + 5 × 4 = 52 52 is divisible by 13 Hence, 325 will also be divisible by 13.

Divisibility by 14: If a number is divisible by 2 and 7 both then that number is divisible by 14 i.e. number is even and osculator 2 is applicable.

Divisibility by 15: If a number is divisible by 3 and 5 both, then that number is divisible by 15.

Divisibility by 16: If last 4 digits of a number are divisible by 16, then whole number is divisible by 16. Such as 341920.

Divisibility by 17: For 17, there is a negative ‘osculator 5’. This process is same as the process of 7. As. 1904 : 190 – 5 × 4 = 170. Q 170 is divisible by 17. So 1904 will be divisible by 17.

Divisibility by 18: If a number is divisible by 2 and 9 both, then that number is divisible by 18. Divisibility by 19 : For 19, there is one–more (positive) osculator 2, which is same processed as 13. As, 361 = 36 + 1 × 2 = 38 Q 38 is divisible by 19. So 361 is also divisible by 19.

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