Number System
INTRODUCTIONThe chapter of number system is amongst the most important chapters in the whole of mathematics syllabus for virtually all the entrance exams inspite of the fact that is the most elementary concept in mathematics. The fundamentals of numbers would be applied across arithmetic in different forms in algebra and also as an extended feature in fundamental principal of counting.
NUMBER SYSTEM:
Number System presents the simplest mathematical structure. This system includes the real numbers, complex numbers, rational number, irrational numbers, fractions, integers, whole number and natural numbers.
Decimal Number System
The system of number we use in general is Decimal System. There are ten digits in this system 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are also the face value of the digits. The place value of a digit will change according to its position in unit’s place, ten’s place, hundred’s place and so on.
e.g.: In 2010, Face value of “2” is 2 and Place value of “2” is thousand.
Real Number
Set of all number that can be represented on the number line are called real numbers.
For example 6, -7, 0, 6.3777, 7, 7+, 3, 311, etc
A Number Line is a straight line with an arbitrary point called origin. To the right of this point are positive numbers and to the left are negative numbers.
Imaginary Numbers
In real number system square root of negative numbers does not exist. Solution of certain equations like x2 + 3 = 0 leads to the concepts of imaginary number. Numbers which are not equal. i is the imaginary unit whose value is √(-1) and it is written as i and known as iota.
Natural Numbers
N = {1, 2, 3, 4, 5, 6, ………….. } is the set of all natural numbers. These are counting numbers.
1 + 2 + 3 + 4 + 5 + _____________ + n = n (n+1)2
Can also have sum of first “n” odd and “n” even numbers together. One tends to remember all sum related formula together.
12 + 22 + 32 + 42 + 52 + ____________ + n2 = n(n + 1)(2n + 1)6
13 + 23 + 33 + 43 + 53 + ____________ + n3 = nn +122
Sum of first “n” odd numbers = n2, Sum of first “n” even numbers = n x (n+1)
Whole Numbers
W = {0, 1, 2, 3, 4, 5, 6, ………….} is the set of all whole numbers.
Whole numbers are natural numbers together with zero.
It is denoted as W = N + {0}
Integers
1 = {-3, -2, -1, 0, 1, 2, 3, ....} is the set of all integers. It is a set of all numbers, 0 and negative of natural numbers.
Positive Integers
These are all natural numbers, I + =- {1, 2, 3, 4, …………….}
Negative Integers
I = {-1, -2, -3, ---},
0 is neither positive nor negative.
Rational Numbers
The numbers of the form p/q where p & q are integers and q ≠ 0 are rational numbers, Every integer is a rational number.
Irrational Numbers
The numbers which when expressed in decimal form are in non-terminating and non repeating forms are called irrational number.
e.g. √3, √11, π, π is not 22/7. It is the approximate value of π
Even Numbers
These are denoted by the expression 2I where I is an integer.
2 + 4 + 6 + 8 + …………………. + 2n terms = n(n + 1) [Sum of even numbers = n(n + 1)]
Odd Numbers
These are denoted by the expression 2 I + 1 where I is an integer.
1 + 3 + 5 + 7 + 9 + …………………. + n terms = n2 [Sum of odd numbers = n2]
Prime Numbers
A number which has exactly two factors, 1 and the number itself, is called prime number.
e.g. 2, 3, 5, 7, 11,.....
The Pratham Edge: A Prime Number ≥ 3 is always expressed in the form 6k ± 1. However it is not a test to check whether its prime or not.
Composite Numbers
A number which has more than 2 factors is called a composite number.
e.g. 4, 6, 8, 9, 12, …………
Co-Primes
Two natural numbers a and b are said to be co-prime if their HCF is I.
Ex. (2, 3), (4, 5), (7, 9), (8, 11) etc. are pairs of co-primes.
TEST OF DIVISIBILITY
1. Divisibility By 2: A number is Divisible by 2, if its unit’s digit is either 0, 2, 4, 6 or 8.
Ex. 84932 is divisible by 2, while 65935 is not.
2. Divisibility By 3: A number is divisible by 3, if the sum of its digits is divisible by 3.
Ex. 592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3.
But, 864329 is not divisible by 3, since sum of its digits (8+6+4+3+2+9) = 32, which is not divisible by 3.
3. Divisibility By 4 : A number is divisible by 4, if the number formed by the last two digits of the number is divisible by 4.
Ex. 982648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4.
But, 749282 is not divisible by 4, since the number formed by the last two digits is 82, which is not divisible by 4.
4. Divisibility By 5 : A number is divisible by 5, if its unit’s digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.
5. Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3.
Ex. The number 35356 is clearly divisible by 2.
Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3.
Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.
6. Divisibility By 7: Subtract 2 times the last digit from the rest.
Ex: 483 is divisible by 7.
48-(3x2) = 42, which is divisible by 7.
7. Divisibility by 8 : A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
Ex. 953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8.
But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.
8. Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible by 9.
Ex. 60732 is divisible by 9, since sum of digits = (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9.
But 68956 is not divisible by 9, since sum of digits (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by 9.
9. Divisibility By 10 : A number is divisible by 10, if it ends with 0.
Ex. 96410. 10480 are divisible by 10, while 96375 is not.
10. Divisibility By 11 :
This post first appeared on CTET DSSSB Teacher Recruitment Exam, please read the originial post: here
