What is the Number System ?
The Number System is defined as a way of writing or representing numbers. The main two types of number system are :-
•Real Numbers
•Imaginary Numbers
There are many other types of numbers. They are as follows :
Natural Number
The counting numbers are called natural numbers, i. e., the numbers starting from 1 are called natural numbers. There is no last natural number.
The collection of all natural numbers is denoted by N and is written as
N = {1,2,3,4,5,......,99,100,...}
Whole Number
The numbers starting from 0 are called natural numbers. There is no last whole number.
This new collection of whole numbers is denoted by W and is written as
W = {0,1,2,3,4,5,......,99,100,...}
Integers
If we take the mirror images of natural numbers in the plane mirror placed at O and denote the images of 1, 2, 3, 4, …, as -1, - 2, -3, -4, …, we get new numbers. These numbers are called negative integers. These negative integers when included in the collection of whole numbers give us a new collection of numbers known as integers.
This new collection is called the collection of integers and is denoted by Z and is written as
Z = {....,-99,......,-5,-4,-3,-2,-1,0,1,2,3,4,5,......,99,...}
Rational Number
The numbers in the form of p/q, where q is not equal to 0 and p, q are integers.
The collection of all such new numbers are called collection of rational numbers and is denoted by Q and written as
Q = { 0/2, 1/4, 2/7, 3/2, 4/1,5/9,......,99/10,100/90,...}
Finding Rational Number Between Two Rational Numbers
If we have to find a rational number between two rational numbers x and y, we have to add x and y and then divide it by 2.
That is, (x+y)/2
EXAMPLE : Find a rational number between 2 and 3. SOLUTION : Here, x = 2, y = 3. So, (x+y)/2 = ( 2+3 )/2 = 5/2
If we have to find more than one rational number between two rational numbers x/a and y/a then here are the steps.
n = Total number of rational numbers to be found
m = n + 1
Let's take an example.
EXAMPLE : Find 5 rational numbers between 2 and 3.
SOLUTION : n = 5, m = n+1 = 5+1 = 6. Now, 2×6/6 = 12/6 and 3×6/6 = 18/6 Therefore, the five rational numbers between 2 and 3 are
12/6 < 13/6 < 14/6 < 15/6 < 16/6 < 17/6 < 18/6
Decimal Expansion of Rational Numbers
The way of converting the p/q form of a number to decimal is called the Decimal Expansion. In the rational number system, decimal expansions are of two types. They are :-
Finite or Terminating Decimal : The rational numbers with a finite decimal part or for which the long division terminates after a finite number of steps are called finite or Terminating Decimal. Example : ⅞ = 0.875, 12/5 = 2.4, etc.
Non-Terminating Recurring Decimal : These numbers when divided don't terminate itself but keep repeating a set of numbers. They are called non-terminating recurring decimals.
Example : ⅔ = 0.6666…., 2/11 = 0.181818…., etc.
These decimals are written by putting a bar on the first block of repeating decimal parts.
Example : ⅔ = 0.6666…. = 6̅ , 2/11 = 0.181818…. = ̅1̅8̅, etc.
Types of Non-Terminating Recurring Decimal
In a non-terminating repeating decimal, there are two types of decimal representationsPure Recurring Decimals : A decimal in which all the digits after decimal point are repeated is known as pure recurring decimal.
Example : ⅔ = 0.6666…. = 6̅ , 2/11 = 0.181818…. = ̅1̅8̅, etc.
Mixed Recurring Decimals : A decimal in which at least one of the digits after the decimal point is not repeated and then some digits are repeated is known as mixed recurring decimals.
Example : 2.16̅, 0.31̅, 8.531̅, etc.
Conversion of A Pure Recurring Decimal to the Form p/q
To convert a pure recurring decimal to the form p/q, follow the steps given below :
1. Obtain the repeating decimal and put it equal to x.
2. Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice.
3. Determine the number of digits having bar on their heads.
4. If the repeating decimal has one place of repetition, multiply by 10; two place of repetition, multiply by 100 and so on.
5. Subtract the number in step second from the number obtained in step fourth.
6. Divide both sides of the equation by the coefficient of x and write the rational number in its simplest form.
EXAMPLE : Express 0.3̅ in the form of p/q.
SOLUTION : Let x = 0.3̅.
Then, x = 0.3333….. (i)
10x = 3.3333…. (ii)
On subtracting (i) from (ii), we get
9x = 3 => x=3/9=⅓
Conversion of A Mixed Recurring Decimal to the Form p/q
To convert mixed recurring decimal in the form of p/q, follow the steps given below.
1. Obtain the mixed recurring decimal and write it equal to x.
2. Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point.
3. Multiply both sides of x by 10n so that only repeating decimal is on the right side of the decimal point.
4. Use the method of converting pure recurring decimal to the form p/q and obtain the value of x.
EXAMPLE : Express 0.32̅ in the form of p/q.
SOLUTION : Let x = 0.32̅
Since there is just one digit on the right side of the decimal point without bar, we will multiply both sides by 10 so that only the repeating decimal remain on the right side of the decimal point.
=> 10x = 3.2̅
=> 10x = 3 + 0.2̅
=> 10x = 3 + 2/9
=> 10x = (9 × 3 + 2)/9 => 10x = 29/9
=> x = 29/90
Irrational Number
The numbers having non-terminating and non-repeating decimal representation are called Irrational Numbers. An irrational number cannot be written in the form of p/q, where q is not equal to 0.
Theorems of Irrational Numbers
1. Negative of an irrational number is an irrational number.
2. The sum of rational number and an irrational number is an irrational number.
3. The product of a non-zero rational number and an irrational number is an irrational number.
4. The sum, difference, product and quotient of two irrational numbers need not to be an irrational number.
0 Comments