**NUMBER SYSTEM**

Number system comprises of 10 digits (0, 1, 2,3,4,5,6,7,8,9 ). When these digits are grouped together it becomes a number. The place at which the digits are placed is its **Place Value**. But the value of digit remains the same wherever it is placed hence we call it as Face Value

**Ex**. Let us take a number 395284 and place the digits in place value chart.

Place Value |
Lacs |
Ten Thousand |
Thousands |
Hundreds |
Tens |
Units |

Digit |
3 | 9 | 5 | 2 | 8 | 4 |

Power of 10 |
5 | 4 | 3 | 2 | 1 | 0 |

**Types of Numbers:**

**Natural Numbers:**Positive numbers (1,2,3,4,5,6……) beginning from 1 that can be counted are known as Natural Numbers. These numbers are also known as Positive Integers.**Whole Numbers:**All the numbers that can be counted including zero ( 0,1,2,3,4….) are known as Whole Numbers.**Integers:**All the numbers that can be counted (positive and negative) and 0 are Integers. …-3,-2,-1,0,1,2,3….

**Positive Integers:**Positive numbers including zero 0,1,2,….**Negative Integers:**Negative numbers ….-4,-3,-2,-1

**Rational Number:**Number that can be expressed in the format of (a/b) where b≠0 is a rational number. This includes all integers, zero or fraction.**Irrational Numbers:**Numbers that cannot be expressed in proper (a/b) format are irrational numbers.

For eg: π = 1415926535897932384626433832795028… And this value of pi never terminates and we use 3.14 or 22/7 (where 22/7 is an irrational number). That’s why irrational numbers are also known for their recurring property.**Even Numbers:**Integers (+ve & -ve) that can be divided by 2 are even numbers**Odd Numbers:**Integers that cannot be divided by 2 are odd numbers**Prime Numbers:**Positive Integers that have only two factors 1 and itself are Prime numbers.i.e., 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.**It is advisable to know all 25 prime numbers below 100**

**NOTE**: *2 is the only even prime number.*

**Composite Number:**All the numbers that are not prime are Composite Numbers.**Co- Prime Numbers:**Two numbers that have HCF as 1 i.e, no other common factor exists between them are called Co- Prime Numbers. Ex: 5 and 24**Real Number:**All the numbers discussed above comes under the umbrella of Real numbers. These numbers can be represented in the number line.

** ****Divisibility Tests**

Any number is considered divisible when it is divided by other number or digit without leaving a remainder.

Divisibility Rules

**for 2:**If the unit digit is 0,2,4,6,8 then that number is divisible by 2.**for 3:**A number is only divisible by 3 when the sum of all the digits of the number is divisible by 3.

**Eg:** 942, In the number 942351 sum of digits is 24 which is divisible by 3, therefore 942351 is also divisible by 3.

**for 4:**If the last digit of a number is 0 or the Last two digits are divisible by 4 then that number is divisible by 4.**for 5:**If the number has 0 or 5 at its units place that it is divisible by 5**for 6**: If a number is divisible by 2 and 3 both are also divisible by 6**for 7**: Let’s learn this using an example 161. The first step would be to double the number at units place i.e., 1*2 = 2. Now subtract this from remaining number 16-2=14. Check the reduced number is divisible by 7 or not. If Yes than number 161 is also divisible.**for 8**: A number is only divisible by 8 when its last 3 digits (i.e., digits at hundreds, tens and units place) are divisible by 8 or if the last 3 digits are zero.

**Eg**: Let’s check if 1111128 is divisible by 8 or not. The last 3 digits are 128 when we check the number after dividing it to 8 it leaves no remainder, therefore we can say 1111128 is divisible by 8.

**for 9:**A number is only divisible by 9 if the sum of all its digits is divisible by 9.

**Ex:** 652491, the sum of all the digits is 27, which is divisible by 9 therefore 652491 is also divisible by 9.

**for 11:**If the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or multiple of 11, then the number is divisible by 11.

**Ex:** 95667, sum of digits at odd places = 7+6+9 = 22;

sum of digits at even places = 5+6 = 11;

Difference = 22-11 = 11;

Number is divisible by 11 as the difference is a multiple of 11.

**for 25:**If the last two digits of a number are either of these: 00, 25, 50 or 75 (i.e., multiple of 25), then the number is divisible by 25.