repeating integers

Janis

Janis

New member
Joined
Oct 27, 2012
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I know what repeating decimals are.
They can be written as a fraction.
I know how to calculate the fraction.
But what are repeating integers?

My homework problem says to calculate the fraction represented by the following repeating integer.
...857142857143
 
Janis said:
I know what repeating decimals are.
They can be written as a fraction.
I know how to calculate the fraction.
Click to expand...

Yes, good for you -- I would only add that repeating decimals are Rational numbers (a subset of Real numbers).


But what are repeating integers?

My homework problem says to calculate the fraction represented by the following repeating integer.

...857142857143
Click to expand...

That is not a repeating integer! Your materials misspoke. Replace the word integer with the word decimal.

The only way that I know of to express an integer as repeating is this:

1.999999…

That is the repeating integer 2, for example.

There must be an infinite number of 9s following the decimal point, to the right of an integer, in order to say "repeating integer".

What is the significance of the three periods in front of the number? Are they ellipsis dots? (The exercise does not show ellipsis dots at the right end? Is there a line drawn over the repeating part? Did YOU round it off? If your post shows the exact exercise, then you have substandard materials.)

Edit: Oh, I get it now. The ellipsis dots indicate that any integer may be placed to the left of the decimal point. Too bad, they left off the decimal point. I'll fix these things for them:

… .857142857142857142…

So, my guess is to answer the exercise by first converting the Rational number 0.857142857142… from decimal to fraction form ?/?. Then write the answer as:

Z + ?/?, where Z is an Integer
 
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Janis said:
Oh, thank you Bob Brown.
The answer is 1/7
Click to expand...

Well, then I have totally misunderstood the exercise.

The number 1/7 does not appear in my work!

Nor does the decimal form of the Rational number 1/7 look like what you posted!

It would be nice, if I were there to look at your materials. :cool:
 
mmm4444bot said:
Well, then I have totally misunderstood the exercise.

The number 1/7 does not appear in my work!

Nor does the decimal form of the Rational number 1/7 look like what you posted!

It would be nice, if I were there to look at your materials. :cool:
Click to expand...


Hi Bot,
Janis took my hint and swept it under infinity.
He multiplied by 7 and got 1.
Therefore, the "repeating integer" is 1/7.

Not sure if that is what the book intended, but it is what the problem said without a re-write.
Kind of interesting!
 
I have no idea what schools are doing these days. That part of a decimal number is called Integer?

lol

My answer would have been Z+6/7, where Z is an Integer

And that is a repeating decimal (Rational number) in mixed form.
 
Janis, disregard this section

\(\displaystyle \begin{array}{c}
\text{Set}\text{ }A = \text{...}+x^2+x+1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}\text{...} \\
A x=A \\
A (x-1)=0 \\
A=0\text{ }\text{when}\text{ }x\neq 1 \\
\text{So}\text{ }\text{...}+x^2+x+1=-\left(\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}\text{...}\right) \\
\text{and}\text{ }\left(\text{...}+x^2+x+1\right)k=-\left(\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}\text{...}\right)k \\
\end{array}\)

Let x=1000000 and k=142857
We recognise RHS = -1/7
So (... 142857) = -1/7
to get the answer we need a repeating integer that sums to zero with -1/7.

\(\displaystyle \begin{array}{c}
\text{...}142857142857 \\
\text{...}857142857143 \\
\text{--}\text{--}\text{--}\text{--}\text{--}\text{--} \\
\text{...} 000000000000 \\
\end{array}\)

Again the result indicates that ...857142857143 = 1/7
 
What is the definition of a repeating Integer?

I have seen that name only in computer science.
 
Looks like they are teaching the p-adic (in this case 1000000-adic) number field.
In ELEMENTARY school!!!!!!!!!!!!!!!
 
The nagging sense that it's about time to hang up my hat continues, lol
 
Kinky (Janis close your eyes)

As you cram more digits into an integer, it just gets bigger.
UNTIL you reach an infinite number of digits.
Then it is a positive or negative FINITE rational number.

Of course, The last few infinite crams must be the same finite integer stuck to the left hand side, over and over.
In this problem, Janis' book started with 3 and kept appending 285714 on the left until they finally got 1/7
 
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