# Real Numbers and their Decimal Expansions

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In this section, we are going to study rational and irrational numbers from a different point of view. We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals. We will also explain how to visualise the representation of real numbers on the number line using their decimal expansions. Since rationals are more familiar to us, let us start with them. Let us take three examples :

Pay special attention to the remainders and see if you can find any pattern.

**Example 5 : Find the decimal expansions of **

We write the numbers as follows in a row :

**Solution:**

3 | 10 | Divident | Remainder | ||

3 | 9 | 3 | 1 | ||

3 | 10 | 3.3 | 1 | ||

3 | 9 | 3.33 | 1 | ||

3 | 9 | 3.333 | 1 | ||

1 | 1 | 1 |

**3**.so they are written as such in the next row.

What have you noticed? You should have noticed at least three things:

(i) The remainders either become **0** after a certain stage, or start **repeating** themselves.

(ii) The number of entries in the repeating string of remainders is less than the divisor (in **326451** in the repeating string of remainders and 7 is the divisor).

(iii) If the remainders repeat, then we get a repeating block of digits in the quotient (for **142857** in the quotient).

Although we have noticed this pattern using only the examples above, it is true for all rationals of the form

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#### Case (i) : The remainder becomes zero

In the example of

We call the decimal expansion of such numbers

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#### Case (ii) : The remainder never becomes zero

In the examples of **non-terminating recurring**. For example,

Similarly, since the block of digits **142857** repeats in the quotient of **0.142857** , where the bar above the digits indicates the block of digits that repeats.Also **3.57272...** can be written as **3.572**. So, all these examples give us **non-terminating recurring (repeating)** decimal expansions.

Thus, we see that the decimal expansion of rational numbers have only two choices: either they are

Now suppose, on the other hand, on your walk on the number line, you come across a number like **3.142678** whose decimal expansion is terminating or a number like **1.272727...** that is, **1.27** , whose decimal expansion is

We will not prove it but illustrate this fact with a few examples. The terminating cases are easy.

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**Example 6 : Show that 3.142678 is a rational number. In other words, express 3.142678 in the form **

**3.142678 =**3142678 1000000 , and hence is a rational number.

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**Example 7 : Show that 0.3333... = 0.3 can be expressed in the form **

- Since we do not know what 0.3... is , let us call it ‘x’ and so x =
- Now here is where the trick comes in. Look at 10 x =
× = 3.333... - Now, 3.3333... = 3 + x, since x = 0.3333... Therefore, 10 x =
+ x - Solving for x, we get 9x =
- i.e., x =

**Example 8 :Show that 1.272727... = 1.27 can be expressed in the form **

- Let x = 1.272727... Since two digits are repeating, we multiply x by 100 to get 100 x =
- So, 100 x = 126 + 1.272727... =
+ x - Therefore, 100 x – x =
, - i.e.,
x = - i.e., x =
=126 99 - You can check the reverse that
= 1.27.14 11

**Example 9 : Show that 0.2353535... = 0 235 . can be expressed in the form **

- Let x = 0.235. Over here, note that 2 does not repeat, but the block 35 repeats. Since two digits are repeating, we multiply x by 100 to get 100 x =
... - So, 100 x = 23.3 + 0.23535... =
+ - Therefore,
x = - i.e., 99 x =
, which gives x =233 10 - You can also check the reverse that
= 0.235.233 990

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So, every number with a non-terminating recurring decimal expansion can be expressed in the form

*The decimal expansion of a rational number is either terminating or nonterminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.*

So, now we know what the decimal expansion of a rational number can be. What about the decimal expansion of irrational numbers? Because of the property above, we can conclude that their decimal expansions are non-terminating non-recurring. So, the property for irrational numbers, similar to the property stated above for rational numbers, is

*The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.*

Recall s = 0.10110111011110... from the previous section. Notice that it is nonterminating and non-recurring. Therefore, from the property above, it is irrational. Moreover, notice that you can generate infinitely many irrationals similar to s. What about the famous irrationals

π = 3.14159265358979323846264338327950...

(Note that, we often take

Over the years, mathematicians have developed various techniques to produce more and more digits in the decimal expansions of irrational numbers. For example, you might have learnt to find digits in the decimal expansion of

Notice that it is the same as the one given above for the first five decimal places. The history of the hunt for digits in the decimal expansion of π is very interesting.

The Greek genius Archimedes was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 – 550 C.E.), the great Indian mathematician and astronomer, found the value of π correct to four decimal places (3.1416). Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places!

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Now, let us see how to obtain irrational numbers.

**Example 10 : Find an irrational number between **