Rational Number Mystery

[harpazo]

Why is the decimal number 8.32104104104104... a rational number? The digits 104 repeat. I thought this to be an irrational, repeating decimal.

 

[Romsek]

\(\displaystyle
8.32\overline{104} = \\
8.32 + 0.00\overline{104} = \\
\dfrac{832}{100} + \dfrac{1}{100}\cdot \dfrac{104}{999} = \dfrac{207818}{24975}
\)

repeating -> rational
non-repeating -> irrational

 

[willyengland]

It's a fraction: 831272/99900, and fractions are rational numbers.

 

[ksdhart2]

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:

$\displaystyle 0.3333 \dots = 0.\overline{3} = \frac{1}{3}$
$\displaystyle 1.446428571428571 \dots = 1.446\overline{428571} = \frac{81}{56}$
$\displaystyle 8.32104104 \dots = 8.32\overline{104} = \frac{207818}{24975}$

 

[harpazo]

\(\displaystyle
8.32\overline{104} = \\
8.32 + 0.00\overline{104} = \\
\dfrac{832}{100} + \dfrac{1}{100}\cdot \dfrac{104}{999} = \dfrac{207818}{24975}
\)

repeating -> rational
non-repeating -> irrational

I like when you said:

repeating -> rational
non-repeating -> irrational

 

[harpazo]

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:

$\displaystyle 0.3333 \dots = 0.\overline{3} = \frac{1}{3}$
$\displaystyle 1.446428571428571 \dots = 1.446\overline{428571} = \frac{81}{56}$
$\displaystyle 8.32104104 \dots = 8.32\overline{104} = \frac{207818}{24975}$

Thank you so much for this information. I will store this data in my math files.

 

[Otis]

Thank you so much for this information. I will store this data in my math files.

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.

 

[harpazo]

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.

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.

 

[Otis]

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

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 …

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 …

Careful now, harpazo. You ought to let people "like you" speak for themselves!

$\;$

 

[Deleted member 4993]

repeating -> rational
non-repeating -> irrational

Exactly opposite of HUMAN behavior!!!

 

[harpazo]

Exactly opposite of HUMAN behavior!!!

I know what to do in terms of this site. All my threads are classified as a threat or some ongoing, personal nonsense. I got it. Trust me, I got it now.

 

[pka]

Why is the decimal number 8.32104104104104... a rational number? The digits 104 repeat. I thought this to be an irrational, repeating decimal.

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.

 

[harpazo]

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.

Very cool. Thank you. Back to math.

 

[Steven G]

Exactly opposite of HUMAN behavior!!!

Are you trying to rational?

 

[Deleted member 4993]

Are you trying to rational?

Did you mean: "Are you trying to be rational?"

In that case - No! I have already been rationalized through conjugate "functions"!!