$0.333333$ - a recurring or non-terminating decimal?

Question

I have read like,

1.All terminating and recurring decimals are RATIONAL NUMBERS.

2.All non-terminating and non recurring decimals are IRRATIONAL NUMBERS.

if the statements are right, then here comes my doubt.

$1/3=0.333333$ Here $3$ is recurring , so from statement 1) $0.3333$ or $1/3$ is a rational number.

And also $0.3333$ is non-terminating as the decimal is not ending or the remainder for 1/3 is not zero. So from 2) $0.333$ is an irrational and it is non terminating.

So please clarify what is $0.3333$ - a recurring or non terminating?

Update: I got the answer from Mohokhbh -

0.3333 is both recurring and non terminating - it's a rational number .

My observation

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Rational number = Terminating or recurring (anyone should suffice)

Irrational number = Non-terminating + Non-recurring(both should satisfy)

THANKS ALL, we can stop messaging this thread

Answer 1

It is recurring non-terminating number.

Answer 2

Your second point says that ".All non-terminating and non recurring decimals are IRRATIONAL NUMBERS." And $0.3333\dots$ is recurring, so it doesn't fit the second point. A recurring and non-terminating decimal is rational. So

• recurring (terminal or not) means rational,
• non-recurring (and non terminating) means irrational.

Answer 3

The first point you have stated should actually be all terminating or recurring decimals are rational numbers. Hence $0.33333...$ is actually a rational number.

Definition of Rationality: A number that can be represented in the form $\frac{p}{q}$ where $p$ and $q$ are integers ($q$ not equal to zero) is a rational number.

$0.3333...=\frac{1}{3}$; hence it is a rational number.

Answer 4

The second point menas "All numbers that ar both non-terminating and non-recurring ..." not "All non-terminating numbers as well as all non-recurring numbers ..."

Answer 5

It is a recurring number, you have some digits that repeat themselves with a fixed period (in this case is just one digit, 3, and the period is then one).

The problem that you have is that you're not considering the second part of the definition of irrational numbers that you give, to be a non recurring decimal, so both definitions are mutually exclusive.

For instance, $\pi$ is an irrational number because, no matter how many decimals you take, there are no periodicity in those digits.

Answer 6

The problem you face stems from an imprecision of the language that makes the situation quite confusing when described as you do.

The first sentence "All terminating and recurring decimals are RATIONAL NUMBERS." is supposed to say that all terminating decimals are rational and in addition all recurring decimals are rational. Put differently and as mentioned in another answer if a number is terminating or recurring then it is rational.

By contrast the second sentence "All non-terminating and non recurring decimals are IRRATIONAL NUMBERS." is supposed to say that a number that is (both) non-terminating and non-recurring is irrational.

I think the first sentence is just not phrased well it ought to read at least "All terminating and all recurring decimals are RATIONAL NUMBERS."

Answer 7

1) A terminating decimal representation means a number can be represented by a finite string of digits in base $10$ notation, e.g. $0.5$, $0.25$, $0.8$, $2.4$

2) A non-terminating decimal representation means that your number will have an infinite number of digits to the right of the decimal point. There are two sorts of non-terminating decimal numbers.

2a) The first sort are called recurring non-terminating decimals. The decimal representations of these numbers consist of an infinite number of periodic repeats of a fixed string of digits to the right of the decimal point. Note that the repeating string can be composed of any number of digits - in the case of $\frac 13 = 0.333... = 0.\overline{3}$, the periodic string is just one digit long. But you can also have a number like $\frac 17 = 0.142857142857... = 0.\overline{142857}$. And in fact, to the immediate right of the decimal point, you can start with a finite string of non-repeating digits before the number goes into its periodic repeats, e.g. $3.1230980709807... = 3.123\overline{09807}$.

2b) The second sort are called non-recurring non-terminating decimals. The number cannot be represented as a repeat of any finite fixed string of digits. Put another way, the decimal representation is aperiodic (i.e. lacking a period). You will see numbers like $\sqrt 2 = 1.4142135623730950488016887242097...$ and $\pi = 3.14159265358979...$ in this category and these numbers appear to have "random" digits (no obvious pattern) to the right of the decimal point. However, and this is a fairly subtle point - the only requirement for this sort of decimal is aperiodicity, not randomness. An example of a clearly non-random non-recurring non-terminating decimal representation is this number: $0.123456789101112131415161718192021...$, which is a famous number known as Champernowne's constant and which is formed by concatenating the digits of the natural numbers in sequence.

And finally, you should note that Types 1 and 2a) are always rational whereas Type 2b) is always irrational.