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**The real number system**

**Fig:Real Numbers Classification**

**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 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 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.

Fig: Fractions

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 as 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 is 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 fraction can be treated as rational number.

**Examples of rational number: **

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 non terminating 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 which can not be converted into fraction is called as irrational numbers.**Examples of irrational number: **

All the above non terminating numbers can not be converted into fractions.

Because, they do not have repeated patterns.

When we are trying to find square of a number which is not a perfect square, we get this non repeating non terminating decimal.

And these non recurring decimals can never be converted in to fractions and they are called as irrational numbers. **Decimal Expansions of Real Numbers **

There are three types of decimal expansions of real numbers.

1. Terminating

2. Non Terminating, Repeating

3. Non Terminating, Non Repeating**1.** **Terminating:** The remainder becomes zero.

Let us take examples to know it.**Example 1:**

Expansion of 7/4

In the example of 7/4,

We found that after some steps the remainder becomes zero and the decimal expansion of 7/4 = 1.75

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 and the decimal expansion of 32/5 = 6.4

Similarly expansion of 32/5 is Terminating.**Example 3: **

Expansion of 578/25

Decimal expansion of 578/25 = 23.12

Therefore, Expansion of 578/52 is Terminating.

**2. The remainder never becomes zero. **There are two cases:**(a)** **Non Terminating, Repeating.****(b)** **Non Terminating, Non Repeating**

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 non terminating, Repeating expansion.

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, Repeating expansion.

1/7 =

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