Irrational Numbers as Decimals

Introduction

In Rational and Irrational Numbers post, we have discussed that  \sqrt{2} is irrational.  Aside from its radical form,  using a calculator or a computer, we can approximate its value; for example, \sqrt{2} \approx 1.414213562. As we can see, irrational numbers can also be represented as decimals. The more powerful the computer, the more accurate we can approximate.

Some Definitions

Decimal numbers with finite number of digits are called terminating decimals, while decimals with infinite number of digits are called non-terminating decimals. The number 0.345 is a terminating decimal, while 0.999 \cdots is a non-terminating decimal. The \cdots symbol means that the 9s extend indefinitely.

Decimals with repeating digits; that is, the digits that repeat infinitely are called repeating decimals. The numbers 0.999 \cdots0.4545 \cdots,  and 5.144144144 \cdots are repeating decimals.

Behavior of Decimals

We can easily show that terminating decimals can be expressed as fractions and therefore are not irrational numbers.  For instance, 0.3 = \frac{3}{10}, 0.45 = \frac{45}{10^2}, 0.231 = \frac{231}{10^3} and so on. From here, it follows that (1) all irrational numbers are non-terminating decimals. However, we also have discussed that the non-terminating, repeating decimal  0.999 \cdots = 1 (Why?), and is therefore rational. So we  ask the following question:

Are all non-terminating, repeating decimals rational numbers?

The answer is yes. We give several examples below, but the proof is left as an exercise.

Example 1: Show that 0.666 \cdots is a rational number.

Let x = 0.666 \cdots
10 x = 6.666 \cdots
10x - x = 6.666 \cdots - 0.666 \cdots
9x = 6
x = 2/3

Example 2: Show that 0.3737 \cdots is a rational number.

Let x = 0.3737 \cdots
100x = 37.3737 \cdots
100x - x = 37.3737 \cdots - 0.3737 \cdots
99x = 37 \cdots
x= 37/99

Example 3: Show that 4.275275 \cdots is a rational number

Let x = 4.275275 \cdots.
1000x = 4275.275
999x = 4271
x = 4721/999

From the discussion above, we have shown that (2) irrational numbers are non-repeating decimals. Therefore, the decimal representations of irrational numbers satisfy conditions 1 and 2; that is, irrational numbers are decimals that do not terminate and do not repeat. Knowing these properties, it is now possible for us to construct irrational numbers in decimal form at will.  Of course, it is impossible to write all the digits of non-terminating decimals, so we always write the symbol \cdots.  For example 0.101001000100001 \cdots is an irrational number since it does not terminate and does not repeat. Also, 0.12123123412345 \cdots is an irrational number. A decimal number with random digits, which do not repeat (infinitely) and do not terminate, like 0.712845023 \cdots is also an irrational number.

The popular numbers \pi, e, and \phi are also irrational numbers.

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