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This essential trains for: SAT-I, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.
Non-terminating decimals can be generated by the long division of two integers. In this case, they have the same value as some rational number.
Non-terminating decimals that cannot be generated by dividing integers do not represent rational numbers. They are called irrational numbers. Examples of irrational numbers are:
Note for advanced students: Proofs that some of these numbers are irrational are given in other lessons. Among the irrational numbers above, some are transcendental (π,e). Check the more advanced lessons.
The characteristic difference between decimals that represent rational numbers and decimals that represent irrational numbers is:
This is because, for any integer division, there is only a finite number of remainders possible. Therefore, at some point in the long division algorithm, a remainder is bound to come up a second time - from there on, a whole sequence of digits will repeat indefinitely.
If the decimal does not repeat, it cannot be the result of a division. One easy to understand such irrational decimal is:
Note: This example uses two digits: 0 and 1. They not only repeat, but they do so in a clearly visible pattern. However, this is not what we understand by repeating decimal!
A repeating decimal is one that has a group of decimals of finite length that repeats:
The difference is that, of course, the group of '0's in the irrational decimal keeps increasing in length.
Irrational decimals do not have to look this well organized. The number π for example, is a number for which it is impossible to predict what the next digit can be.
The group of decimals that repeat is called a periodic part (or repetend) and is sometimes denoted with an overline:
Typical questions may ask you to identify rational or irrational numbers.
is a repeating decimal with periodic part 4. The number is rational.
is a repeating decimal with periodic part 41. The number is rational.
is a repeating decimal with fixed part 66 and periodic part 718. The number is rational.
is a repeating decimal with fixed part 000000 and periodic part 5. The number is rational.
is clearly non-repeating. The number is irrational.
looks to be non-repeating. However, we must be aware that, there is no limit set for the length of the periodic part. It must be finite in length, but it can be a very large finite length. So, just by examining a few digits that look like they do not form a repeating pattern, we can generally not decide if the number is rational or not. (Note: in some exam problems the examiner is unclear about this and expects the student to choose 'irrational' from information that is actually inconclusive.)