Number System

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

Number

A mathematical symbol representing a number in a systematic manner is called a numeral represented by a set of digits.there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits.

Face Value

Face value of a digit in a numeral is value of the digit itself. For example in 321, face value of 1 is 1, face value of 2 is 2 and face value of 3 is 3.

Place Value

Place value of a digit in a numeral is value of the digit multiplied by 10ⁿ where n starts from 0. For example in 321:

• Place value of 1 = 1 x 10⁰ = 1 x 1 = 1
• Place value of 2 = 2 x 10¹ = 2 x 10 = 20
•  Place value of 3 = 3 x 10² = 3 x 100 = 300

Example:  In the numeral 70984 ,Find place value of each digit

 Soluion: we have  Place value of 4 = (4x1)=4

    Place value of 8 = (8x10)=80

    Place value of 9 = (9x100)=900

    Place value of 7 = (7x10000)=70000

Note: Face value of 0 in any given number is 0 , at whichever place it may be.

Example:  What is the difference between the place value of 2 in the numeral 7229?
A.) 20         B.) 200          C.) 180               D.) 18
Solution:     Place value of 2 = (2x10)=20
                     Place value of 2 = (2x100)=200
                     Difference=200-20=180
Hence option c (180) is the correct answer

Types of Numbers

1. Natural Numbers - n > 0 where n is counting number; [1,2,3...]

2. Whole Numbers - n ≥ 0 where n is counting number; [0,1,2,3...]
Note: 0 is the only whole number which is not a natural number.
Every natural number is a whole number.

3. Integers - n ≥ 0 or n ≤ 0 where n is counting number;...,-3,-2,-1,0,1,2,3... are integers.

• Positive Integers - n > 0; [1,2,3...]
• Negative Integers - n < 0; [-1,-2,-3...]
• Non-Positive Integers - n ≤ 0; [0,-1,-2,-3...]
• Non-Negative Integers - n ≥ 0; [0,1,2,3...]

           0 is neither positive nor negative integer.
4. Even Numbers - n / 2 = 0 where n is counting number; [0,2,4,...]

 5. Odd Numbers - n / 2 ≠ 0 where n is counting number; [1,3,5,...]

 6. Prime Numbers - Numbers which is divisible by themselves only apart from 1.

1 is not a prime number.

To test a number p to be prime, find a whole number k such that k > √p. Get all prime numbers less than or equal to k and divide p with each of these prime numbers. If no number divides p exactly then p is a prime number otherwise it is not a prime number.

Example  191 is prime number or not?

Solution:

Step 1 - 14 > √191

Step 2 - Prime numbers less than 14 are 2,3,5,7,11 and 13.

Step 3 - 191 is not divisible by any above prime number.

Result - 191 is a prime number.

Example: 187 is prime number or not?

Solution:

Step 1 - 14 > √187

Step 2 - Prime numbers less than 14 are 2,3,5,7,11 and 13.

Step 3 - 187 is divisible by 11.

Result - 187 is not a prime number

7. Composite Numbers - Non-prime numbers > 1. For example, 4,6,8,9 etc.

1 is neither a prime number nor a composite number.

2 is the only even prime number.

8. Co-Primes Numbers - Two natural numbers are co-primes if their H.C.F. is 1. For example, (2,3), (4,5) are co-primes.

Note:  Coprime numbers may or may not be prime.

8. Rational Numbers:A number that can be expressed as p/q is called a rational number, where p and q are integers

    eg. 2/3,4/5 etc

9. Irrational Numbers:  The numbers that cannot be expressed in the form of p/q are called irrational numbers, where p and q are integers  eg   , ,  etc

When irrational number is expressed in decimal form, it goes forever without repeating.

10. Real Numbers:Real numbers include rational and irrational numbers both,
 Real numbers are denoted by R.

Divisibility:

Following are tips to check divisibility of numbers.

1.  Divisibility by 2 - A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.

Example: 64578 is divisible by 2 or not?

Solution:

Step 1 - Unit digit is 8.

Result - 64578 is divisible by 2.

Example: 64575 is divisible by 2 or not?

Solution:

Step 1 - Unit digit is 5.

Result - 64575 is not divisible by 2.

2.  Divisibility by 3 - A number is divisible by 3 if sum of its digits is completely divisible by 3.

Example: 64578 is divisible by 3 or not?

Solution:

Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30

which is divisible by 3.

Result - 64578 is divisible by 3.

Example: 64576 is divisible by 3 or not?

Solution:

Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28

which is not divisible by 3.

Result - 64576 is not divisible by 3.

3. Divisibility by 4 - A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.

Example: 64578 is divisible by 4 or not?

Solution:

Step 1 - number formed using its last two digits is 78

which is not divisible by 4.

Result - 64578 is not divisible by 4.

Example: 64580 is divisible by 4 or not?

Solution:

Step 1 - number formed using its last two digits is 80

which is divisible by 4.

Result - 64580 is divisible by 4.

4.  Divisibility by 5 - A number is divisible by 5 if its unit digit is 0 or 5.

Example: 64578 is divisible by 5 or not?

Solution:

Step 1 - Unit digit is 8.

Result - 64578 is not divisible by 5.

Example: 64575 is divisible by 5 or not?

Solution:

Step 1 - Unit digit is 5.

Result - 64575 is divisible by 5.

5.   Divisibility by 6 - A number is divisible by 6 if the number is divisible by both 2 and 3.

Example: 64578 is divisible by 6 or not?

Solution:

Step 1 - Unit digit is 8. Number is divisible by 2.

Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30

which is divisible by 3.

Result - 64578 is divisible by 6.

Example: 64576 is divisible by 6 or not?

Solution:

Step 1 - Unit digit is 8. Number is divisible by 2.

Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28

which is not divisible by 3.

Result - 64576 is not divisible by 6.
 

6.  Divisibility by 7 - Double the last digit of the number and then subtract it from the remaining number. If the result is divisible by 7, then the original number will also be divisible by 7.

Example: 595 is divisible by 7 or not?

Solution:

Step 1 – double of the last digit 10 and 59-10=49 which is divisible by 7  

Result - 595 is  divisible by 7.

Example: 672 is divisible by 7 or not?

Solution:

Step 1 – double of the last digit 4 and 67-4=63 which is divisible by 7  

Result - 672 is  divisible by 7.

7.  Divisibility by 8 - A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.

Example: 64578 is divisible by 8 or not?

Solution:

Step 1 - number formed using its last three digits is 578

which is not divisible by 8.

Result - 64578 is not divisible by 8.

Example: 64576 is divisible by 8 or not?

Solution:

Step 1 - number formed using its last three digits is 576

which is divisible by 8.

Result - 64576 is divisible by 8.

8.  Divisibility by 9 - A number is divisible by 9 if sum of its digits is completely divisible by 9.

Example: 64579 is divisible by 9 or not?

Solution:

Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 9 = 31

which is not divisible by 9.

Result - 64579 is not divisible by 9.

Example: 64575 is divisible by 9 or not?

Solution:

Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 5 = 27

which is divisible by 9.

Result - 64575 is divisible by 9.

9. Divisibility by 10 - A number is divisible by 10 if its unit digit is 0.

Example: 64575 is divisible by 10 or not?

Solution:

Step 1 - Unit digit is 5.

Result - 64578 is not divisible by 10.

Example: 64570 is divisible by 10 or not?

Solution:

Step 1 - Unit digit is 0.

Result - 64570 is divisible by 10.

10.  Divisibility by 11 - A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.

Example: 64575 is divisible by 11 or not?

Solution:

Step 1 - difference between sum of digits at odd places

and sum of digits at even places = (6+5+5) - (4+7) = 5

which is not divisible by 11.

Result - 64575 is not divisible by 11.

Example: 64075 is divisible by 11 or not?

Solution:

Step 1 - difference between sum of digits at odd places

and sum of digits at even places = (6+0+5) - (4+7) = 0.

Result - 64075 is divisible by 11.
 

Division Algorithm

When a number is divided by another number then

Dividend = (Divisor x Quotient) + Reminder

1. The minimum number on adding of which the given number is completely divisible  =  Divisor - Remainder

2. The minimum number on subtraction of which the given number is completely divisible = Remainder

Example: Find the number, which on (1) addition (2) subtraction from the number 5029 is completely divisible by 17.

Solution:  Dividing 5029 by 17 we find.∴Remainder=141. 

The minimum number on adding of which the given number is completely divisible by 17=Divisor−Remainder=17-14 =.2.

The minimum number on subtraction of which the given number iscompletely divisible by 17=Remainder=14