Number System
A Number is an abstract entity that represents a count or measurement.
All numbers fall in 2 categories Real Number, Complex Number
A Real Number can either be a Rational number or an irrational number.
Rational number can either be Natural numbers or Negative integers or Fractions
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 add zero to our above list then we have the set of whole numbers.
i.e Whole numbers 0,1,2,3...
Negative numbers
Negative numbers are numbers which are less than zero. They are used to indicate a number that is opposite to the corresponding positive number (the absolute value), but equal in magnitude.
Exampe: -1, -2, -3, . . .
Remember -(n + 1)is always smaller than -n where n is a positive number.
Integer
Integers are the whole numbers, negative whole numbers, and zero.
One of the numbers ..., -2, -1, 0, 1, 2, . . .
But numbers like 1/2, 4.00032, 2.5, Pi, and -9.90 are not integers.
Note that zero is neither positive nor negative.
It may help you to think of numbers as occurring along a line that stretches infinitely in both directions. Numbers to the left of the 0 point are negative, numbers to the right are positive.
Along this line are a series of dots that correspond to whole numbers (integers). The spaces between the whole numbers are occupied by the numbers that are not whole (they contain fractions, and are called real numbers ex.1/2, 4.00032, 2.5, Pi, and -9.90).
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.
Again, this lets us talk about whether 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.
Representation of rational numbers in decimal form
Any positive rational number p/q, after actual division, if necessary can be expressed as,
p / q = m + r/q where m is non-negative integer and 0 ≤ r < 5 =" 6" 5 ="">
There are few fractions for which the right most digit(or set of right most digit)recurrs endlessly. For example 1/3 =0.33333. . . .
and 5/11 = 0.45454 . . . .
Note that the dots ........ represent endless recurrence of digits.
The above examples are decimal numbers of the "non-terminating type".
In case of "non-terminating type" we have decimal fractions having an infinite number of digits. Some decimal fractions from this group have digits repeating infinitely. They are called "repeating or recurring " decimals.
In "endless recurring or infinite repeating" decimal fractions we can see that when p is actually divided by q the possible remainders are 1, 2, 3, ..... , q -1. So one of them has to repeat itself in q steps. Thereafter the earlier numeral or group of numerals must repeat itself.
All the rational numbers thus can be represented as a finite decimal (terminating type) or as a recurring decimal.
Irrational Numbers
In mathematics, 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.
Prime number
A prime number is a whole number that is not the product of two smaller numbers.
Note that the definition of a prime number doesn't allow 1 to be a prime number : 1 only has one factor, namely 1.
Prime numbers have exactly two factors, not "at most two" or anything like that. When a number has more than two factors it is called a composite number.
Here are the first few prime numbers:
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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.
PRIME FACTORS
Suppose n is a natural number. Then there exists a unique sequence of prime
numbers p1, p2, p3, . . . , pm, such that both of the following statements are
true:
p1 ≤ p2 ≤ p3 ≤ . . . ≤ pm
p1 x p2 x p3 . . . x pm
The numbers p1, p2, p3, . . . , pm are called the prime factors of the natural number.
Every natural number n has one, but only one, set of prime factors.
This is an important principle known as the Fundamental Theorem of
Arithmetic.
Number of Prime Factors
A number N of the form
am x bn x cp
where a, b, c are all prime factors of number N
has (m + 1)(n + 1)(p + 1)no. of prime factors
What is the fastest way to determine if a number is Prime?
The easiest & simplest method is to divide the number up to the closet square root of that number.
Ex. Lets consider 53. Number close to 53 having a perfect square is 64 and its square root is 8. Now start dividing 53 from 2 to 8. There is no such number between 2 to 8 which divides 53 so 53 is a prime number. Related Links :Click to view the largest known prime number
Composite Numbers
A composite number is a positive integer which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called "composites" for short) are 4, 6, 8, 9, 10, 12, 14, 15, 16, . . .
Note that the number 1 is a special case which is considered to be neither composite nor prime.
Numeric Operations:
1. A + 0 = A , A - 0= A , A x 0 = 0 , A/0= Value Does not exist
2. A x 1= 1 , A + (-A) = 0 , A x (1/A) = 1
3. A + B = B + A
4.A x B = B x A
5. A ( B + C) = AB + AC
Integer Roots
Suppose that a is a positive real number. Also suppose that n is a positive integer. Then the n th root of a can also be expressed as the 1/ n power of a . Thus, the second root (or square root) is the same thing as the 1/2 power; the third root (or cube root) is the same thing as the 1/3 power; the fourth root is the same thing as the 1/4 power; and so on.
Irrational-Number Powers
Suppose that a is a real number. Also suppose that b is a rational number such that b = m / n , where m and n are integers and ≠ 0. Then the following formula holds true:
ab = am/n =a m(1/n ) =a (1/n ) m and
(1/a)b = 1/(ab
In Case of a negative power a-b = (1/a)b = 1/(a b )
Important Formula A(b+c) = Ab A c and A(b-c) = A b / A c
Let A be a real number. Let b and c be rational numbers. Then the following formula holds good:
A bc = ( Ab) c = ( Ac) b
CAT TRICKS
Divisibility rules
1. xn + anis exactly divisible by (x + a)
if n is odd, but not if n is even
2. xn - an is divisible by (x + a)
if n is even but not if n is odd
3. xn - an is always divisible by (x - a)
Remainder theorems
If a rational integral functions f(x) is divided by ( x + a) the remainder is f(-a)
Ex1: What is the remainder when x 2 - 3x + 2 is divided by (x - 2)
Ans: der = f-(-2)= f(2) = 4 - 6 + 2 = 0
Thats true beause x 2 - 3x + 2 = (x - 2)(x - 1)
Ex2: Find a if (ax3 + 3x2 - 3) and (2x3 - 5x + a) when divided by (x-4) leave same rmainder.
Ans: remainders are,
R1 = f(4) = a(4)3+ 3(4)2-3= 64a+45
R2 = f(4) = 2(4)3- 5(4) + a= a +108
Since R1= R2
64a +45= a+108 => 63a = 63 => a =1
Factor theorem
A binomials of the form (x + a) or (x - a) will be a factors of rational integers function f(x) if it leaves no remainder when divided by (x-a) or (x+a)
This provides a test for divisibility of an expression f(x) by (x+a).
The reminder of the number N (= n1 + n2 i.e. sum of two differnt numbers n1 and n2) when divided by a is equal to the sum of the reminders obtained when a divides n1 and n2 individually.
For Example, when we divide 47 by 5 we get reminder as 2.
Now 47 can be written as 40 + 7 and when divide 40 by 5 the reminder is 0 and when 7 is divided by 5 the remainder is 2 so some of the reminders 2 + 0 =2 is same as intial reminder.
The reminder of the number N (= n1 x n2 i.e. product of two differnt numbers n1 and n2) when divided by a is equal to the product of the reminders obtained when a divides n1 and n2 individually.
Number of a's (a prime number) in N!
Number of a’s in N! = [N/a] + [N/a2] + [N/a3] + . . . . .
Where [] shows the floor or integer function
The floor function means round down to the nearest integer.
For example how many 2’s are in 12!
Ans: It ‘s [12/2] + [12/4] + [12/8] + [12/16]
= 6 + 3 + 1 + 0 = 10
While doing the calculation do it as follows first 12/2 = 6
Second number is 6/2= 3
Third number = 3/2 =1
So you should write directly as 6 + 3 + 1
Ex2: 45! Ends with _ zeros?
Number zeros depends upon number of 5’s and 2’s.
Number of 5’s : 9 + 1 = 10
Number of 2’s : 22 + 11 + 5 + 2 + 1 = 41
So number of zero’s = 10
AT is tough not because of its content, but because of certain misconceptions in the students' mindset.
