NUMBERS

 1. Numbers:

          In Hindu-Arabic system we have ten digits, namely 0,1,2,3,4,5,6,7,8,9 called zero, One, Two, Three, Four, Five, Six, Seven, Eight, Nine respectively. These number are the building blocks of anything. 

Hindu-Arabic  numerals are also called as Decimal (or) Base ten (or) Place value number system. 

A number is denoted by a group of digits, called "Numeral".

Ex: 0, 1, 2 is called numbers and 120 (or) One hundred and twenty is called Numeral.

For denoting a numeral, we use the Place-value chart, given below.

Ex: Write each of the following numerals in words.

	Crores	Ten lakhs	Lakhs	Ten Thousands	Thousands	Hundreds	Tens	One’s (or) Units
1)					5	4	8	9
2)			2	6	4	8	7	3
3)	7	2	8	4	6	9	5	2

Sol: The given numbers in words are:

1) Five thousand four hundred eighty nine.

2) Two lakhs sixty-four thousand eight hundred seventy three.

3) seven crore twenty- eight lakh forty-six thousand nine hundred fifty two.

Ex: Write each of the following numbers in figures.

1) six crore fifty-four lakhs eighteen thousand four hundred sixty nine.

2) Two lakh sixty-four thousand eight hundred seventy three.

3) Seventy two thousand eight hundred forty-six.

Sol:

	Crores	Ten lakhs	Lakhs	Ten Thousands	Thousands	Hundreds	Tens	One’s (or) Units
1)	6	5	4	1	8	4	6	9
2)			2	6	4	8	7	3
3)				7	2	8	4	6

 

2. Face value and Place value (Local value) of a digit in a numeral :

i) The face value of a digit in a numeral is its own value, at whatever place it may be.

Ex: In the numeral 5476, the face value of 4 is 4, the face value of 7 is 7, the face value of 6 is 6.

ii) The place value is the value denoted by a digit according to its position in the number.

Ex: In the numeral 5476, the Place value of 4 is 400 (4X100), the place value of 7 is 70 (7X10), the place value of 6 is 6 (6X1).

Note: 1) Place value of "0" in a given numeral is "0", at whatever place it may be.

2) Place valve and Face value of "0" is always same i.e., "0".

 3. Even and Odd Numbers:

i) Even Number: A counting number which is divisible by '2' is called as "Even Number" i.e., 0, 2, 4, 6, 8, 10, 12, 14 ..... etc., 

ii) Odd Number: A counting number which is not divisible by '2' is called as "Odd Number"  i.e., 1, 3, 5, 7, 9, 11, 13 ...... etc.,

 4. Various types of Numbers:

Natural Numbers: All Counting numbers are called "Natural Numbers", Thus 1, 2, 3, 4, 5 ......... are all natural numbers.

Whole Numbers: All counting numbers and 0 form the set of whole numbers. Thus 0, 1, 2, 3, 4, 5 ..... are all whole numbers.

Note: 1) Every natural number is whole number.

2) "0" is a whole number which is not a natural number.

Integers: All counting numbers, zero, and negatives of counting numbers are called Integers. 

Thus   -∞ ................. -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 .............. ∞

Note: 

1) Set of positive integers {1, 2, 3, 4, 5, ........  ∞ }

2) Set of negative integers {-1, -2, -3, -4, -5, .........  ∞ } 

3) Set of Non-negative integers {0, 1, 2, 3, 4, 5 ...... ∞ }

 5. Prime Numbers:

A counting number is called a prime number if it has exactly two factors namely itself and 1. 

Ex: All prime numbers below 100 are 

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

Note: 1) '2' is the only even prime number, rest of all is Odd numbers. 

2) Number of Prime numbers below 50 is '15'.

3) Number of Prime numbers below 100 is '25'.

4) Number of Prime numbers below 500 is '95'.

5) Number of Prime numbers below 1000 is '168'.

6) Even numbers placed in the unit’s position of a number cannot be prime number.

7) In a number, the sum of all digits in that number is divisible by 3 then the number is not a prime number.

8) 1 is neither a prime number nor a composite number.

9) No prime number greater than 5  ends with 5 as its unit digit.

6. Composite Numbers:

The natural numbers which are not prime numbers, are called Composite Numbers. 

Ex: 4, 6, 8, 9, 10, .......

Note:

1) 1 is neither a prime number nor a composite number.

2) Every composite number has more than 2 factors.

3) Zero (0) is not considered a prime or a compound number because it has no factors.

4) The smallest composite number is '4'.

5) Each composite number is a factor of itself.

7. Co-Primes:

 Two natural numbers 'a' and 'b' are said to be C0-Primes if their HCF is '1'.

 Ex: (2,1),  (2,3),  (3,5),  (7,9),  (5,11), .......

Note:

1) The common factor of co-primes is '1'.

2) Since all even numbers have the same common factor as 2, two even numbers can never be a co-primes.

3) Any two prime numbers are co-primes. ex: (2,3),  (3,7),  (5,11) etc., 

4) 

Series 

Note: 

1) Smallest natural number is '1'.

2) Natural numbers are whole numbers excluding '0'.

3) Natural Numbers are Non-Negative Integers (all positive integers).

4) The Sum of a 2 digit number and the number obtained by interchanging (swapping) the digits is always divisible by 11. 

Example:  Consider a two digit number is : 23

Number formed after swapping the digits is : 32

sum of the 2 digit number and the number obtained by interchanging is : 23 + 32 = 55.

The result (55) is divisible by 11.

5) The Difference of a 2 digit number and the number obtained by interchanging (swapping) the digits is always divisible by 9. 

Example:  Consider a two digit number is : 32

Number formed after swapping the digits is : 23

Difference of the 2 digit number and the number obtained by interchanging is : 32 - 23 = 9.

The result (9) is divisible by 9.