Classification of Numbers



Number: - A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral.

Real Number: - The real numbers include all of the measuring numbers.

Imaginary Number: - Number with square root of negative number is called imaginary number. These numbers are in the form of a + ib.

Rational Number: - A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator.

Irrational Number: - In mathematics, an irrational number is any real number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero and is therefore not a rational number.

Integers: - The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2 ...}.

Fractions: - A fraction is a number that can represent part of a whole.

Proper Fraction: Value of the fraction is less than 1. These are in the form of ½, 2/3 , ¾ ..

Improper Fraction: - Value of the fraction is greater than 1. These are in the form of 3/2 , 4/3 …

Mixed Fraction: - When an integer is given before proper fraction then number is called mixed fraction. Numbers of the form 2¾.

Whole Number: All non negative numbers including Zero. W ~ (0,1,2,3,4...)

Natural Numbers: - All the number starting from 1. N ~ (1,2,3,4...)

Natural Numbers = Whole Number - Zero

Basic Operations on Numbers :

Let a and b any two numbers then find the results of basic operations below.

Points to remember:-
1. Sum and Difference of any two odd or two even numbers is always even.
2. a^b always depends on a.
ie Even raise to any number is even and Odd raise to any number is odd.
eg 2^ 3 = 8 and 3^ 2 = 9.
3. If atleast one number is even - Product is always even.

Converting a recurring decimal to vulgar fraction :

A decimal with recurring value is called recurring decimal.
E.g: 2/9 will give 0.22222222...... where 2 is recurring number.

Method:
a) Separate the recurring number from the decimal fraction.
b) Annex denominator with "9" as many times as the length of the recurring number.
c) Reduce the fraction to its lowest terms.

Example: Consider 0.2323232323

Step a: The recurring number is 23
 Step b: 23/99 [the number 23 is of length 2 so we have added two nines] 
Step c: Reducing it to lowest terms : 23/99 [it can not be reduced further].

How to Convert a mixed-recurring decimal to vulgar fraction ? A decimal with both recurring and non-recurring value is called mixed recurring decimal.
E.g: 28/25 will give 1.1199999999...... where 11 is non-recurring number and 9 is recurring number.

Method:
a) Separate the recurring number, non recurring number from the decimal fraction.
b) Round the decimal after point to the first recurring value.
c) Result of step b - non recurring number.
d) Annex as many "0" as non-recurring number length and as many "9" as recurring number length.
e) Step c / Step d
f) Add the fraction with the number before decimal point.
Example: Consider 1.11999999...

Step b and c: 119-11 [rounded value of number after decimal point - non recurring value]Step d: 900
Step e: 108/900 [c/d]
Step f: 1+¹⁰⁸/₉₀₀ [adding with number before decimal point ]

Reducing it to lowest terms : 900+108 / 900 = 1008/900 = 28/25.Step a: The recurring number is 11, non-recurring number is 9

This is a very important topic as time is concern.