Examples of number systems include: natural numbers, integers, rational numbers, irrational numbers, real numbers, complex numbers.
All numbers are either real or complex numbers. The real numbers can be either rational or irrational numbers.
Natural Numbers
The natural numbers start off as follows: 1, 2, 3, 4, and 5 ... The "..." means that the list goes on forever. We give this set the name N.
If a number is in N, then its successor is also in N. Thus, there is no greatest number, because we can always add one to get a larger one. N is an infinite set . Since it is infinite, N can never be exhausted by removing its members one at a time.
Whole Numbers
If we include 0 among the natural numbers then the numbers 0,1,2,3,4,5 ........... are called whole numbers.
The set of whole numbers can be represented by W = {0,1,2,3,4,5, ........... .} Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number
Integers
All counting numbers and their negatives including zero are known as integers. The set of integers can be represented by z or I = {.........-4,-3,-2,-1,0,1,2,3,4..}. Every natural number is an integer but not every integer is natural number.
Positive Integers
The set I + = {1,2,3,4........} is the set of all positive integers. Positive integers and natural numbers are synonyms.
Negative Integers
The set I - = {-1,-2,-3,.......} is the set of all negative integers 0 is neither positive nor negative.
Non Negative Integers
The set {0,1,2,3........} is the set of all non negative integers.
Even and Odd
The terms even and odd only apply to integers. A number is said to be an even number if it is divisible by 2 or else it is an odd number.
Even numbers are: 2, 4, 6, 8, 10. . . . .40, 42, 44,. . . 312, 314, .... 1008,1010, . . . .686860....
Odd numbers are: . . 5, 7, 9. . . . .41, 43, 45,. . . 311, 313, .... 1007,1009, . . . .686861....
2.5 is neither even nor odd.
Zero, on the other hand, is even since it is 2 times some integer: it's 2 times 0. To check whether a number is odd, see whether it's one more than some even number: 7 is odd since it's one more than 6, which is even. Another way to say this is that zero is even since it can be written in the form 2*n, where n is an integer.Odd numbers can be written in the form 2*n + 1.
Negative numbers are even and odd: -9 is odd since it's one more than -10, which is even.
Every positive integer can be factored into the product of prime numbers, and there's only one way to do it for every number . For instance, 280 = 2x2x2x5x7, and there's only one way to factor 280 into prime numbers
Rational Number
A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0 i.e Rational numbers are simply defined as ratios of integers. 1/2 is a rational number. 2/3 is also a rational number.
Note that all Of the integers are rational numbers, because you can think of them as the ratio of themselves to 1, as in 2 = 2/1 which is certainly the ratio of two integers, and so 2 is a rational number. The decimal form of a rational number is either a terminating or repeating decimal.
Irrational Numbers
An irrational number is any real number that is not a rational number i.e., one that cannot be written as a ratio of two integers, i.e., it is not of the form a/b where a and b are integers and b is not zero.
It can readily be shown that the irrational numbers are precisely those numbers whose expansion in any given base (decimal, binary, etc) never ends and never enters a periodic pattern.
The square root of 2 is a classic example of an irrational number: you cannot write it as the ratio of ANY two integers.
Examples
1. If a number 774958A96B is to be divisible by 8 and 9, the values of A and B, respectively, will be?
2. The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is ?
3. The solution set (x, y) for the system of equations log2 xy = 5 and log1/2 (x/y) = 1, is
4. Find the remainder when 51138 is divided by 7.
5. The highest power of 2 in 10! + 11! + 12! + 13! + ...+ 1000! is ?
6. If ‘x’ is an odd number, what will be the remainder if x3– x + 1 is divided by 24?
7. How many consecutive zeros would be there at the end of 626! - 625!?
8. A certain number when divided by 899 leaves the remainder 63. Find the remainder when the same number is divided by 29.
9. Five digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?
10. If p is any three digit number and q is any number obtained with any type of permutations of the digits of p, then p – q is always divisible by?
Dividing by 2
A number can be divided by 2 if the last digit is even.
Dividing by 3
A number is divisible by 3 if the sum of the digits is 3, 6 or 9.
Examples: 111111: the digits add to 6 so the whole number is divisible by three. 87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three.
Dividing by 4
A number is divisible by 4 if the number made by the last two digits can be divided by 4.
Examples: 100 is divisible by 4. 1732782989264864826421834612 is divisible by four also, because 12 is divisible by four.
Dividing by 5
A number is divisible by 5 if the last digit is a 5 or a 0.
Dividing by 6
A number can be divided by 6 if the last digit is even and the sum of all the digits is 3, 6 or 9.
Dividing by 7
To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number.
Example: If you had 203, you would double the last digit to get six, and subtract that from 20 to get 14. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again.
Dividing by 8
A number is divisible by 8 if the number made by the last three digits will be divisible by 8.
Example: 33333888 is divisible by 8; 33333886 isn't.
Dividing by 9
A number is divisible by 9 if the sum of all the digits will add to 9.
Example: 12348 is divisible by 9; as the sum is 18
Dividing by 10
A number can be divided by 10 if the last digit is a 0.
Dividing by 11
A number is divisible by 11 if the difference of the Sum of the digits at odd places and sum of the digits at the even places is either zero or divisible by 11.
Example: In the number 9823, the sum of the digits at odd places is 9 + 2 = 11 and the sum of the digits at even places is 8 + 3= 11.
The difference between it is 11 - 11 = 0 therefore given number is divisible by 11.
Dividing by 12
A number is divisible by 12 if it is divisible by 3 and 4.
Example: The number 1644 is divisible by 12 as it is divisible by 3 and 4.
Dividing by 18
An even number satisfying the divisibility test by 9 is divisible by 18.
Example: The number 80388 is divisible by 18 as it satisfies the divisibility test of 9.
Dividing by 25
A number is divisible by 25 if the number formed by the last two digits is divisible by 25 or the last two digits are zero.
Example: The number 7975 is divisible by 25 as the last two digits are divisible by 25.
