Is 0.354355435554 rational or irrational or integer?

2 Answers
Oct 5, 2016

Answer:

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 #p/q#. Integers are in particular rational, because they can be written as #p/1#, as in #4/1, (-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 #1/3=0.33333 ....#.

Irrational numbers cannot be written in the way above. Examples are #pi, sqrt(2), 1.12131415162728192021 ....#.

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

Answer:

If #0.354355435554# ends after the last digit #4#, it is a rational number but if #0.354355435554....................# 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=354355435554/1000000000000#.

However, apparently questioner is rather looking at

#0.354355435554....................#, which is clearly irrational as

grouping them as under reveals the pattern as follows:

#0.ul(354)color(red)(3554)ul(35554)....................#

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....................# is an irrational number.