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Question indigenous lil, a student:

Why room repeating decimals taken into consideration rational numbers?

We have actually two responses because that you

Hi Lil,

The price is yes, but prior to I illustrate why ns am going come quibble v the method you request the question. A repeating decimal is not considered to it is in a reasonable number the is a rational number. Us have different ways of representing numbers, for instance the variety of fingers on my left hand have the right to be represented by the English native five, or the French indigenous cinq or the price 5 or the Roman character V or the fraction 10/2 or numerous other ways. An in similar way the portion 1/3 have the right to be represented by the decimal number 0.3333... These room two various ways that representing the exact same number.

A rational number is a number that deserve to be stood for a/b where a and b are integers and also b is no equal come 0.

You are watching: Is 0.3 repeating a rational number

A reasonable number can likewise be represented in decimal form and the result decimal is a repeating decimal. (I see the decimal 0.25 together repeating since it have the right to be created 0.25000...) likewise any decimal number that is repeating deserve to be created in the kind a/b through b no equal to zero so that is a rational number. Permit me illustrate with one example.

Consider the repeating decimal n = 2.135135135... The repeating component (135) is 3 digits lengthy so i am walk to main point n by 103 to obtain 103 n = 2135.135.135... Now I subtract


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Thus n = 2133/999 and since 9 split both the numerator and denominator this can be created n = 237/11.

Penny

Hi there,

Repeating decimal are thought about rational numbers due to the fact that they can be represented as a proportion of 2 integers.

To represent any pattern the repeating decimals, divide the section of the pattern to be repetitive by 9"s, in the adhering to way:

0.2222222222... = 2/9

0.252525252525... = 25/99

0.1234567123456712345671234567... = 1234567/9999999

The number of 9"s in the denominator should be the same as the number of digits in the repeated block. These rational numbers may of course be reducible, if the optimal is divisible by 9, or both the top and bottom space divisible by another number. But this is a starting point which will always get friend what you want.

Why go this work? Well, us can enter a bit much more detail and also write the end our repeating decimal, speak 0.252525252525..., as an infinite collection of decreasing fractions, like so

0.252525252525... = 2/10 + 5/100 + 2/1000 + 5/10000 + 2/100000 + 5/1000000 + ...

Now allow this series be equal to x, that is

x = 2/10 + 5/100 + 2/1000 + 5/10000 + 2/100000 + 5/1000000 + ...

now multiply both sides by 100

100x = 20 + 5 + 2/10 + 5/100 + 2/1000 + 5/10000 + ...

Now subtract the 1st equation native the second like so:

100x = 20 + 5 + 2/10 + 5/100 + 2/1000 + 5/10000 + 2/100000 + 5/1000000 + ... -x = - 2/10 - 5/100 - 2/1000 - 5/10000 - 2/100000 - 5/1000000 - ... -------------------------------------------------------------------------------------------------------------------- 99x = 25

now rearrange because that x and also get

x = 25/99

which is what we were feather for! so 25/99 really does equal 0.252525252525...

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I expect this helps!

Gabe


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