Rational Number Mystery

harpazo

harpazo

Full Member
Joined
Jan 31, 2013
Messages
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Why is the decimal number 8.32104104104104... a rational number? The digits 104 repeat. I thought this to be an irrational, repeating decimal.
 
\(\displaystyle
8.32\overline{104} = \\
8.32 + 0.00\overline{104} = \\
\dfrac{832}{100} + \dfrac{1}{100}\cdot \dfrac{104}{999} = \dfrac{207818}{24975}
\)

repeating -> rational
non-repeating -> irrational
 
Last edited:
There's no such thing as an irrational, repeating number. One of the properties of an irrational number is that they have an infinite number of digits after the decimal point and these digits never stabilize into any repeating pattern. This property holds true, even if you write the number in a different base (e.g. binary is base 2), so long as it's an integer base.

Rational numbers, on the other hand, come in three basic "flavors." First, a rational number might have no digits after the decimal point. Typically we call these "integers," or if the number in question is non-negative, various sources may also call them "whole numbers," "natural numbers," or "counting numbers." However, the integers are, by definition, a subset of the rational numbers - that is, every integer is also a rational number (but not vice versa).

Second, a rational number might have finitely many digits after the decimal point. Any such number can clearly be written as a fraction where the numerator and denominator are both integers, simply by multiplying by the appropriate power of 10, in order to shift the decimal point over the required number of places. And, by definition, any number that can be written as a fraction where both the numerator and denominator are integers is a rational number (you may also hear it said that such a fraction is the ratio of two integers, hence the term rational).

Third, a rational number might have infinitely many digits after the decimal point, provided these digits eventually stabilize into some repeating pattern. These numbers can also be written as the ratio of two integers, but the process of finding this representation not always as clear as it is for terminating decimals. Some examples of repeating rational numbers include:

\(\displaystyle 0.3333 \dots = 0.\overline{3} = \frac{1}{3} \)
\(\displaystyle 1.446428571428571 \dots = 1.446\overline{428571} = \frac{81}{56} \)
\(\displaystyle 8.32104104 \dots = 8.32\overline{104} = \frac{207818}{24975} \)
 
Romsek said:
\(\displaystyle
8.32\overline{104} = \\
8.32 + 0.00\overline{104} = \\
\dfrac{832}{100} + \dfrac{1}{100}\cdot \dfrac{104}{999} = \dfrac{207818}{24975}
\)

repeating -> rational
non-repeating -> irrational
Click to expand...

I like when you said:

repeating -> rational
non-repeating -> irrational
 
ksdhart2 said:
There's no such thing as an irrational, repeating number. One of the properties of an irrational number is that they have an infinite number of digits after the decimal point and these digits never stabilize into any repeating pattern. This property holds true, even if you write the number in a different base (e.g. binary is base 2), so long as it's an integer base.

Rational numbers, on the other hand, come in three basic "flavors." First, a rational number might have no digits after the decimal point. Typically we call these "integers," or if the number in question is non-negative, various sources may also call them "whole numbers," "natural numbers," or "counting numbers." However, the integers are, by definition, a subset of the rational numbers - that is, every integer is also a rational number (but not vice versa).

Second, a rational number might have finitely many digits after the decimal point. Any such number can clearly be written as a fraction where the numerator and denominator are both integers, simply by multiplying by the appropriate power of 10, in order to shift the decimal point over the required number of places. And, by definition, any number that can be written as a fraction where both the numerator and denominator are integers is a rational number (you may also hear it said that such a fraction is the ratio of two integers, hence the term rational).

Third, a rational number might have infinitely many digits after the decimal point, provided these digits eventually stabilize into some repeating pattern. These numbers can also be written as the ratio of two integers, but the process of finding this representation not always as clear as it is for terminating decimals. Some examples of repeating rational numbers include:

\(\displaystyle 0.3333 \dots = 0.\overline{3} = \frac{1}{3} \)
\(\displaystyle 1.446428571428571 \dots = 1.446\overline{428571} = \frac{81}{56} \)
\(\displaystyle 8.32104104 \dots = 8.32\overline{104} = \frac{207818}{24975} \)
Click to expand...

Thank you so much for this information. I will store this data in my math files.
 
harpazo said:
Thank you so much for this information. I will store this data in my math files.
Click to expand...
Why was this a mystery, harpazo? We went over this last spring. And you posted last month that you know repeating decimal fractions are rational numbers! I think your math files are basically clutter.

:rolleyes:
 
Otis said:
Why was this a mystery, harpazo? We went over this last spring. And you posted last month that you know repeating decimal fractions are rational numbers! I think your math files are basically clutter.

:rolleyes:
Click to expand...

Last spring is far behind.

1. I forgot what I had for breakfast this morning much less the properties of zero learned last year.

2. I have struggled with learning disability for most of my life. People like me should not have a B.A. degree but through God's help, I made it through CUNY.

3. I post questions here to review what has been forgotten not to start a debate. This is quite obvious to others.
 
harpazo said:
… I forgot what I had for breakfast this morning much less the properties of zero learned last year …
Click to expand...
I didn't post about what you learned last year. I said "last spring" and "last month". You're trying to change the subject, and your claim about breakfast this morning is a fib.


… I post questions here to review what has been forgotten …
Click to expand...
You could answer a lot of these review questions yourself, by using the index in your textbook or simply googling the question. (Do you ever look at your math files?)

Based on your past behavior here and at other math sites over the past few years, I'm not always convinced that you're very serious about review. It often seems like you want to spend your free time recreationally dabbling in whatever math topic suits your fancy in the moment. After awhile, your repeated threads, thread bumps and unwillingness to make reasonable efforts before posting become tedious.


… People like me should not have a B.A. degree …
Click to expand...
Careful now, harpazo. You ought to let people "like you" speak for themselves!

\(\;\)
 
harpazo said:
Why is the decimal number 8.32104104104104... a rational number? The digits 104 repeat. I thought this to be an irrational, repeating decimal.
Click to expand...
All of you need to get it together.
\(\displaystyle 8.32\overline{104}=\frac{832}{10^2}+\sum\limits_{k = 1}^\infty {\frac{{104}}{{{{10}^{3k + 2}}}}} \)
See HERE
That shows the ratio of two integers. That is the very definition of a rational number.
 
pka said:
All of you need to get it together.
\(\displaystyle 8.32\overline{104}=\frac{832}{10^2}+\sum\limits_{k = 1}^\infty {\frac{{104}}{{{{10}^{3k + 2}}}}} \)
See HERE
That shows the ratio of two integers. That is the very definition of a rational number.
Click to expand...

Very cool. Thank you. Back to math.
 
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