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Number system.
Numbers: we have ten digits , namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
A number is denoted by a group of digits called numerals.
Face value &place value of a digit in a numeral:
1. The face value of a digit in a numeral is its own value, at whatever place it may be.
For example, in the numeral 9368, the face value of 8 is 8, the face value of 6 is 6, the face value of 3 is 3, the face value of 9 is 9.
2. In a given numeral:
The place value of unit digit: (unit digit)*1.
The place value of tens digit: (tens digit)*10.
The place value of hundred’s digit: (hundred’s digit)*100.
So on….
For example, in the numeral 9368:
The place value of 8 = 8*1 =8
The place value of 3 =3*10 =30
The place value of 6 = 6*100 = 600
The place value of 9 =9*1000 =9000
Types of numbers:
1. Natural numbers: the counting numbers are called natural numbers.
Therefore 1, 2, 3, 4, 5, 6, 7 … are all natural numbers.
2. Whole numbers: all counting numbers starting from 0 are called the whole numbers.
Therefore 0, 1, 2, 3, 4, 5, 6, 7 … are all whole numbers.
Every natural number is a whole number but not every whole number is natural number. 0 is a whole number but not a natural number.
3. Integers: all counting numbers, zeros and negatives of counting numbers from the set of integers.
Therefore, ….,5,4,3,2,1,0,1,2,3,4,5,… are all integers.
4. Even numbers: a counting number divisible by 2 is called an even number.
Therefore 0, 2, 4, 6, 8, 10, 12, 14, …etc. are all even numbers.
5. Odd numbers: a counting number not divisible by 2 are called odd numbers.
Therefore 1, 3, 5, 7,9 ,11, 13, 15, ….etc. are all odd numbers.
6. Prime numbers: a counting number, which have only two factors, namely the number itself and 1 is called prime number.
For example… 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, etc…
A Test for a number to be prime: Let x be the given number and let n be smallest counting number such that n^{2}=x. if x is divisible by the prime number less than or equal to n then x is not a prime number, else x is prime.
For example: let us check whether 123 is prime or not.
We know that (12)^{2}>123.
Prime numbers less than 12 are 2, 3, 5, 7, and 11
Clearly out of the above prime numbers, 3 divide 123.
Thus 123 is not a prime number.
7. Composite numbers: The natural numbers which are not prime, are called composite numbers.
Composite numbers can also be defined as the numbers which have more than two factors.
8. Coprime numbers: Two natural numbers x and y are said to be coprime if their HCF is 1.
For example: (2,3), (4, 5), (6, 7),(11,13) etc. are pairs of coprimes.
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