Real Numbers and their Decimal Expansions
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.
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 | 10 | 3.333 | 1 | ||
| 1 | 1 | 1 |
What have you noticed? You should have noticed at least three things:
(i) The remainders either become
(ii) The number of entries in the repeating string of remainders is
(iii) If the remainders repeat, then we get a repeating block of digits in the quotient (for
Although we have noticed this pattern using only the examples above, it is true for all rationals of the form
On division of p by q, two main things happen – either the remainder becomes zero or never becomes zero and we get a repeating string of
Let us look at each case separately.
Case (i) : The remainder becomes zero
In the example of
In all these cases, the decimal expansion terminates or ends after a finite number of steps.
We call the decimal expansion of such numbers
Case (ii) : The remainder never becomes zero
In the examples of
For example,
The usual way of showing that 3 repeats in the quotient of
Similarly, since the block of digits 142857 repeats in the quotient of
Thus, we see that the decimal expansion of rational numbers have only two choices: either they are terminating or non-terminating.
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 non-terminating, can you conclude that it is a rational number?
The answer is Yes!
We will not prove it but illustrate this fact with a few examples. The terminating cases are easy.
Example 6: Show that 3.142678 is a rational number. In other words, express 3.142678 in the form
We have 3.142678 =
Now, let us consider the case when the decimal expansion is non-terminating recurring.
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 =
× = ... - Now, 3.3333... = 3 + x, since x = 0.3333... Therefore, 10 x =
+ x - Solving for x, we get
x = - 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
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 =
233 10 - You can also check the reverse that
233 990
So, every number with a non-terminating recurring decimal expansion can be expressed in the form
Let us summarise our results in the following 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
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
Interestingly, in the Sulbasutras (rules of chord), a mathematical treatise of the Vedic period (800 BC - 500 BC), you find an approximation 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!
Archimedes (287 BCE – 212 BCE)
Example 10: Now, let us see how to obtain irrational numbers.
Find an irrational number between