Decimal Representation of Irrational Numbers

Decimal Representation of Irrational Numbers

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Introduction:

Let's recap the decimal representation of rational numbers. A rational number can have two types of decimal representations (expansions):

  • Terminating: It means that the decimal representation or expansion terminates after a certain number of digits.
  • Non-terminating but repeating: It means that although the decimal representation has an infinite number of digits, there is a repetitive pattern to it.

When irrational numbers are written in decimal form, we get non-terminating, non-repeating representations. Let's try to understand it better.


Non-terminating, Non-repeating Decimal Expansion:

We have already seen that any terminating or non-terminating repeating representation corresponds to a rational number. It can be shown that every non-terminating, non-repeating representation is an irrational number (though the proof is advanced). Here are two irrational numbers whose decimal representations have been given up to 10 digits:

\[\sqrt 2 = 1.4142135623 \ldots \]

\[\pi = 3.1415926535 \ldots \]

Non-terminating, Non-repeating Decimal Expansion means that although the decimal representation has an infinite number of digits, there is no pattern to it.

The ellipsis (the three dots at the end of each representation) tells us that the sequence of digits never ends. You can keep evaluating these numbers to more and more decimal digits – there will be no end to it.

What does this really mean? Does this mean that we do not know the exact value of irrational numbers as we do for rational numbers? No, that’s not correct! When we talk about an irrational number, say, \(\sqrt 2 \) or \(\pi \), we do know the exact number (or quantity) we are talking about. For example, we can exactly construct a length of \(\sqrt 2 \) units, given a length of 1 unit. We know that the ratio of the circumference to the diameter in any circle is exactly equal to \(\pi \).

enlightenedThink: If we know these numbers exactly, why do their decimal representations seem less than exact?

⚡Tip: To understand it better, go through the exactness of decimal representations.


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