Classification of Real Numbers
In real life, we use Hindu Arabic numerals  a system which consists of the symbols 0 to 9.
This system of reading and writing numerals is called, “Base ten system” or “Decimal number system”.
Natural Numbers
Counting numbers are called natural numbers. These numbers start with the smallest number 1 and go on without end. The set of all natural numbers is denoted by the symbol ‘N’.
N = { 1, 2, 3, 4, 5, .......} is the set of all natural numbers.
Whole Numbers
Natural numbers together with zero (0) are called whole numbers. These numbers start with the smallest number 0 and go on without end.
The set of all whole numbers is denoted by the symbol ‘W’.
W = { 0, 1, 2, 3, 4, 5, .......} is the set of all whole numbers.
Integers
The whole numbers and negative numbers together are called integers.
The set of all integers is denoted by Z.
Z = {...  2,  1, 0, 1, 2, ...,} is the set of all integers
Fractions
A fraction is a part or parts of a whole.
In a fraction, the number above the line is called the numerator and the number below the line is called the denominator.
Decimal Numbers
A number in which we have "point" is called a decimal number.
A decimal number has two parts namely an integral part and a decimal part.
Examples:
1) Let us consider the decimal number 0.6
0.6 can be written as 0 + 0.6
Here, integral part = 0 and decimal part = 6
2) Let us consider the decimal number 7.2
7.2 can be written as 7 + 0.2
Here, integral part = 7 and decimal part = 2
In a decimal number, the digits to the left of the decimal point is the integral part.
The digits to the right of the decimal point are the decimal part.
The value of all the decimal parts is less than 1.
Rational Numbers
Both "p" and "q" must be integers and q≠ 0
So, any number in the form of a fraction can be treated as a rational number.
Examples:
5, 2.3, 0.02, 5/6
Because all these numbers can be written as fractions.
5 = 5/1
2.3 = 23/10
0.02 = 2/100 = 1/50
5/6 (This is already a fraction)
Apart from the above examples, sometimes we will have recurring decimals like 1.262626..........
1.262626........ is a nonterminating recurring decimal.
All these recurring decimals can be converted into fractions and they are also rational numbers.
Important note:
All the fractions and decimal numbers will come under this category.
Hence, all the fractions and decimal numbers to be considered as rational numbers.
Irrational Numbers
A number that can not be converted into a fraction is called irrational numbers.
Examples:
All the above nonterminating numbers can not be converted into fractions because they do not have repeated patterns.
When we are trying to find the square root of a number that is not a perfect square, we get this nonrepeating and nonterminating decimal.
And these nonrecurring decimals can never be converted into fractions and they are called irrational numbers.
There are three types of decimal expansions of real numbers.
1. Terminating
2. NonTerminating but Repeating
3. NonTerminating and NonRepeating
1. Terminating: The remainder becomes zero.
Let us take examples to know it.
Example 1:
Expansion of 7/4
After performing some steps, we get the remainder as 0.
The remainder is 0, and the decimal expansion ends at 5. So it means the expansion is terminating.
Example 2:
Expansion of 32/5
Again we found that after some steps the remainder becomes zero with the decimal expansion as 6.4.
Similarly, the expansion of 32/5 is terminating.
Example 3:
Expansion of 578/25
Decimal expansion of 578/25 = 23.12
Therefore, the expansion of 578/52 is terminating.
2. The remainder never becomes zero: There are two cases:
(a) NonTerminating but Repeating.
(b) NonTerminating and NonRepeating
Example: Expansion of 10/3
Decimal expansion of 10/3 = 3.333333...
The expansion of 10/3 does not end that is not terminating, and number 3 is repeating, so it is a nonterminating but recurring expansion.
Example: Expansion of 1/7
Decimal expansion of 1/7 = 1.142857...
The expansion of 1/7 does not end that is not terminating, and numbers 142857 are repeating, so it is a non terminating but recurring expansion.
1/7 =
Example:
π is a nonterminating, nonrepeating decimal. π = 3.141 592 653 589 793 238 462 643 383 279 ...
e is a nonterminating, nonrepeating decimal. e = 2.718 281 828 459 045 235 360 287 471 352 ...
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