Math Is Fun Forum

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Rational Number

A rational number is defined as the quotient of two integers. When written as a decimal, the decimal will either repeat or terminate. By looking at the denominator of the rational number, there is a way to tell in advance whether it's decimal representation will repeat or terminate. What about the denominator of a rational number indicates that its decimal representation will repeat or terminate?

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,830

Re: Rational Number

Links: Terminating decimal and Recurring decimal.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Rational Number

Links: Terminating decimal and Recurring decimal.

Thanks for the links. I will check it out.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Rational Number

Look at the following fractions:

1/4 = 25/100
1/5 = 2/10
1/25 = 4/100

If you choose a fraction which has only 2s and 5s as prime factors you can always re-write it with a denominator which is a power of ten.

eg.  40 = 2 x 2 x 2 x 5.  1/40 = 1/(2x2x2x5) = (5x5)/(2x5x2x5x2x5) = 25/1000

Fractions which have a denominator which is a power of ten will always result in a terminating decimal.

1/4 = 0.25
1/5 = 0.2
1/25 = 0.04
1/40 = 0.025

No other prime factor can be manipulated in this way to give a terminating decimal so all other denominators result in recurring decimals.

eg.

1/3 = 0.3333333.....
1/7 = 0.142857142857......
1/11 = 0.090909.....

If you have a recurring decimal it is always possible to turn it into a fraction.

eg.

0.16161616......

let a/b = 0.16161616.....

100a/b = 16.16161616.....

Subtract
99a/b = 16

a/b = 16/99

eg.

0.7666666....

= 7/10 + 0.0666666

so use the above technique to get a/b for the recurring part and then sum the two fractions to get another rational.

So if all terminating and recurring fractions are rationals everything else must be irrational. So the irrationals are non-terminating, non-recurring decimals.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
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sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Rational Number

Look at the following fractions:

1/4 = 25/100
1/5 = 2/10
1/25 = 4/100

If you choose a fraction which has only 2s and 5s as prime factors you can always re-write it with a denominator which is a power of ten.

eg.  40 = 2 x 2 x 2 x 5.  1/40 = 1/(2x2x2x5) = (5x5)/(2x5x2x5x2x5) = 25/1000

Fractions which have a denominator which is a power of ten will always result in a terminating decimal.

1/4 = 0.25
1/5 = 0.2
1/25 = 0.04
1/40 = 0.025

No other prime factor can be manipulated in this way to give a terminating decimal so all other denominators result in recurring decimals.

eg.

1/3 = 0.3333333.....
1/7 = 0.142857142857......
1/11 = 0.090909.....

If you have a recurring decimal it is always possible to turn it into a fraction.

eg.

0.16161616......

let a/b = 0.16161616.....

100a/b = 16.16161616.....

Subtract
99a/b = 16

a/b = 16/99

eg.

0.7666666....

= 7/10 + 0.0666666

so use the above technique to get a/b for the recurring part and then sum the two fractions to get another rational.

So if all terminating and recurring fractions are rationals everything else must be irrational. So the irrationals are non-terminating, non-recurring decimals.

Bob

I totally get it. Thanks.

I do have one question.

You said subtract but there is no subtraction symbol in your work. I assume you made a typo. Yes? Also, can you answer my thread "Future Time". No one has replied. It's a rather short word problem but a bit tricky for me.

Subtract
99a/b = 16

a/b = 16/99

eg.

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Rational Number

If you have a recurring decimal it is always possible to turn it into a fraction.

eg.

0.16161616......

let a/b = 0.16161616.....

100a/b = 16.16161616.....

Subtract
99a/b = 16

a/b = 16/99

I do have one question.

You said subtract but there is no subtraction symbol in your work.

its hard to show without formatting

subtracting

    a/b  =  0.16161616...
100(a/b) = 16.16161616...

100(a/b) = 16.16161616...
 -1(a/b) = -0.16161616...
-------------------------
 99(a/b) = 16
 
    a/b  = 16/99

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Rational Number

If you have a recurring decimal it is always possible to turn it into a fraction.

eg.

0.16161616......

let a/b = 0.16161616.....

100a/b = 16.16161616.....

Subtract
99a/b = 16

a/b = 16/99

I do have one question.

You said subtract but there is no subtraction symbol in your work.

its hard to show without formatting

subtracting

    a/b  =  0.16161616...
100(a/b) = 16.16161616...

100(a/b) = 16.16161616...
 -1(a/b) = -0.16161616...
-------------------------
 99(a/b) = 16
 
    a/b  = 16/99

Interesting problem for which Sullivan does not give a sample.