Is 0.354355435554 rational or irrational or integer?

Oct 5, 2016

You have to look carefully at the decimal expression

Explanation:

Integer numbers do not have a decimal part, that is, they do not have decimals after the dot $' . '$. Examples are $- 2 , - 53 , 0 , 4 , 75$ etc.

Rational numbers are the ones that can be written as a quotient of two integers $\frac{p}{q}$. Integers are in particular rational, because they can be written as $\frac{p}{1}$, as in $\frac{4}{1} , \frac{- 3}{1}$, etc.

However, in terms of the decimal expression such as the one given in the problem, rational numbers can be expressed either with a finite number of decimals (such as $2.35 , 79.5465989$), or periodic, such as $\frac{1}{3} = 0.33333 \ldots .$.

Irrational numbers cannot be written in the way above. Examples are $\pi , \sqrt{2} , 1.12131415162728192021 \ldots .$.

From all this you can say that the number given is not a integer and it is a rational number as it has a finite decimal expression

Oct 5, 2016

If $0.354355435554$ ends after the last digit $4$, it is a rational number but if $0.354355435554 \ldots \ldots \ldots \ldots \ldots \ldots . .$ repeats the pattern endlessly, it is an irrational number.

Explanation:

If the number $0.354355435554$ is limiting after $12$ places of decimals, it is a rational number as

$0.354355435554 = \frac{354355435554}{1000000000000}$.

However, apparently questioner is rather looking at

$0.354355435554 \ldots \ldots \ldots \ldots \ldots \ldots . .$, which is clearly irrational as

grouping them as under reveals the pattern as follows:

$0. \underline{354} \textcolor{red}{3554} \underline{35554} \ldots \ldots \ldots \ldots \ldots \ldots . .$

Here we first have one $5$ between $3$ and $4$,

then we have two $5 ' s$ between $3$ and $4$,

and then we have three $5 ' s$ between $3$ and $4$.

Hence the number of $5 ' s$ between $3$ and $4$ is continuously increasing

and there is no group of numbers repeating endlessly

Hence $0.354355435554 \ldots \ldots \ldots \ldots \ldots \ldots . .$ is an irrational number.