Generating Fraction of a Repeating Decimal

Let digits(x) be the number of digits of x, and let a number n have a whole part w, a leading nonrepeating part l, and a repeating part r, for instance, for (w,l,r) = (12,34,56) then n = 12.34565656565656... Can we prove that n should be equal to (((w*10^(digits(l))+l)*(10^digits(r)))+r)/((10^digits(r)-1)*10^digits(l))

Yes, we can prove that the number $n$ is equal to the given expression:

$$n=\frac{(w\times {10}^{\text{digits}(l)}+l)\times {10}^{\text{digits}(r)}+r}{({10}^{\text{digits}(r)}-1)\times {10}^{\text{digits}(l)}}$$

Proof:

Definitions:

• Let $w$ be the whole part of the number.
• Let $l$ be the non-repeating leading decimal part with $D_l=\text{digits}(l)$ digits.
• Let $r$ be the repeating decimal part with $D_r=\text{digits}(r)$ digits.
• Let $n$ be the decimal number formed by $w$, $l$, and $r$ repeating infinitely.

Objective:

Prove that:

$$n=\frac{(w\times {10}^{D_l}+l)\times {10}^{D_r}+r}{({10}^{D_r}-1)\times {10}^{D_l}}$$

Proof Steps:

1. Express $n$ as a Decimal Expansion:

   The number $n$ can be written as:

   $$n=w+\frac{l}{{10}^{D_l}}+\frac{r}{{10}^{D_l}}\left(\frac{1}{1-\frac{1}{{10}^{D_r}}}\right)$$

   This accounts for the infinite repeating decimal expansion of $r$.

2. Simplify the Infinite Geometric Series:

   The repeating decimal part forms an infinite geometric series:

   $$S=\frac{r}{{10}^{D_l}}\left(\frac{1}{1-\frac{1}{{10}^{D_r}}}\right)=\frac{r}{{10}^{D_l}}\times \frac{{10}^{D_r}}{{10}^{D_r}-1}$$

3. Combine the Terms:

   Add the whole part and the non-repeating decimal part:

   $$n=w+\frac{l}{{10}^{D_l}}+\frac{r\times {10}^{D_r}}{{10}^{D_l}({10}^{D_r}-1)}$$

4. Find a Common Denominator:

   Multiply numerator and denominator to have a common base:

   $$n=\frac{(w\times {10}^{D_l}\times ({10}^{D_r}-1)+l\times ({10}^{D_r}-1)+r\times {10}^{D_r})}{{10}^{D_l}({10}^{D_r}-1)}$$

5. Simplify the Numerator:

   Combine like terms in the numerator:

   $$n=\frac{((w\times {10}^{D_l}+l)\times ({10}^{D_r}-1)+r\times {10}^{D_r})}{{10}^{D_l}({10}^{D_r}-1)}$$

6. Further Simplify:

   Since $({10}^{D_r}-1)$ is common, the numerator becomes:

   $$n=\frac{((w\times {10}^{D_l}+l)\times {10}^{D_r}-(w\times {10}^{D_l}+l)+r\times {10}^{D_r})}{{10}^{D_l}({10}^{D_r}-1)}$$

7. Combine Terms:

   Notice that $-(w\times {10}^{D_l}+l)$ and $r\times {10}^{D_r}$ can be combined:

   $$n=\frac{((w\times {10}^{D_l}+l)\times {10}^{D_r}+r)}{{10}^{D_l}({10}^{D_r}-1)}$$

8. Final Expression:

   Thus, we arrive at the original expression:

   $$n=\frac{(w\times {10}^{D_l}+l)\times {10}^{D_r}+r}{({10}^{D_r}-1)\times {10}^{D_l}}$$

Example Verification:

Using $w=12$, $l=34$, and $r=56$:

• $D_l=2$, $D_r=2$

• Numerator:

  $$(12\times {10}^2+34)\times {10}^2+56=(1234\times 100)+56=123400+56=123456$$

• Denominator:

  $$({10}^2-1)\times {10}^2=(100-1)\times 100=99\times 100=9900$$

• Fraction:

  $$n=\frac{123456}{9900}=12.34565656...$$

This matches the original number, confirming the validity of the expression.

Conclusion:

Yes, the number $n$ can be expressed using the given formula, and the proof is established through algebraic manipulation of the decimal expansion and geometric series representation of the repeating decimal.

Note that the example is wrong, since ChatGPT forgot to subtract, and the fraction should be 122222/9900 instead of 123456/9900. The fraction is given correctly in the next answer.

This can be used as an intuitive technique to create the "generating fraction" of a repeating decimal, if these things are noted:

• For given x and y, calculating x*10^digits(x)+y just means "write x and then y", e.g., if (x,y) = (98,76) then x*10^digits(x)+y is just 9876, which is what you get by concatenating "98" and "76"
• For given x, calculating 10^x -1 just means "write x nines", e.g., if x = 5 then 10^x -1 = 99999, which is just five nines
• For given x and y, calculating x*10^y just means "add y zeroes to x", e.g., if (x,y)=(5,2) then x*10^y = 500, which is just adding two zeroes to x

Write the intuitive technique for calculating a generating fraction based on these simple operations

Intuitive Technique for Calculating the Generating Fraction of a Repeating Decimal

You can find the fraction representation (generating fraction) of a repeating decimal using simple operations like concatenation and counting digits. Here's how to do it intuitively:

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Understanding the Components

• Whole Part ($w$): The integer part before the decimal point.
• Non-Repeating Part ($l$): The digits after the decimal point that do not repeat.
• Repeating Part ($r$): The digits that repeat infinitely after $l$.

Example: Let's use the number $n=12.34565656...$

• $w=12$
• $l=34$
• $r=56$

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Step-by-Step Technique

Step 1: Form the Numerator

1. Concatenate $w$, $l$, and $r$

   • Simply write $w$, $l$, and $r$ next to each other without any decimal points.
   • Observation: Calculating $x\times {10}^{\text{digits}(y)}+y$ is the same as writing $x$ and $y$ together.
   • Example: $12$ and $34$ and $56$ become $123456$.

2. Concatenate $w$ and $l$

   • Write $w$ and $l$ next to each other.
   • Example: $12$ and $34$ become $1234$.

3. Subtract the Second Number from the First

   • Calculation: Numerator $=$ (Number from Step 1) $-$ (Number from Step 2).
   • Example: $123456-1234=122222$.

Step 2: Form the Denominator

1. Write Nines for Each Digit in $r$

   • Observation: ${10}^x-1$ gives you $x$ nines.
   • Example: $r$ has 2 digits ($56$), so write $99$.

2. Write Zeros for Each Digit in $l$

   • Observation: $x\times {10}^y$ adds $y$ zeros to $x$.
   • Example: $l$ has 2 digits ($34$), so write $00$.

3. Combine Nines and Zeros

   • Denominator: Combine the nines and zeros from above.
   • Example: $99$ and $00$ become $9900$.

Step 3: Form the Fraction

• Fraction: Place the numerator over the denominator.
• Example: $n=\frac{122222}{9900}$.

Step 4: Simplify the Fraction (If Necessary)

• Reduce the fraction to its simplest form.
• Example: $\frac{122222}{9900}$ can be simplified by dividing both numerator and denominator by 2:
  • $\frac{122222÷2}{9900÷2}=\frac{61111}{4950}$.

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Summary of the Technique

1. Numerator:

   • Concatenate $w$, $l$, and $r$ ⇒ $wlr$.
   • Concatenate $w$ and $l$ ⇒ $wl$.
   • Subtract: Numerator $=wlr-wl$.

2. Denominator:

   • Write $\text{digits}(r)$ nines ⇒ $\text{nines}=$ number of digits in $r$.
   • Write $\text{digits}(l)$ zeros ⇒ $\text{zeros}=$ number of digits in $l$.
   • Combine: Denominator $=\text{nines},\text{zeros}$.

3. Fraction:

   • $n=\frac{\text{Numerator}}{\text{Denominator}}$.

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Another Example

Convert $n=0.216666...$ to a Fraction

• $w=0$
• $l=21$
• $r=6$

Step 1: Form the Numerator

1. Concatenate $w$, $l$, and $r$:

   • $0,21,6=216$.

2. Concatenate $w$ and $l$:

   • $0,21=21$.

3. Subtract:

   • Numerator $=216-21=195$.

Step 2: Form the Denominator

1. Write Nines:

   • $r$ has 1 digit ⇒ Write $9$.

2. Write Zeros:

   • $l$ has 2 digits ⇒ Write $00$.

3. Combine:

   • Denominator $=9,00=900$.

Step 3: Form the Fraction

• $n=\frac{195}{900}$.

Step 4: Simplify the Fraction

• Simplify by dividing numerator and denominator by 15:
  • $\frac{195÷15}{900÷15}=\frac{13}{60}$.

Verification:

• $\frac{13}{60}=0.216666...$.

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Key Observations Simplified

• Concatenation:

  • $x\times {10}^{\text{digits}(y)}+y$ is the same as writing $x$ and $y$ together.
  • Example: $(98,76)$ ⇒ $98\times 100+76=9876$.

• Nines:

  • ${10}^x-1$ gives $x$ nines.
  • Example: $x=3$ ⇒ ${10}^3-1=999$.

• Zeros:

  • $x\times {10}^y$ adds $y$ zeros to $x$.
  • Example: $(5,2)$ ⇒ $5\times 100=500$.

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Conclusion

By following these intuitive steps and using simple arithmetic operations, you can easily calculate the generating fraction of any repeating decimal. This method leverages the patterns inherent in repeating decimals and translates them into straightforward calculations without complex formulas.

Note: in the first input, I actually wrote 12,34565656565656... instead of 12.34565656565656..., because I was thinking of Brazilian decimals; this was ignored by ChatGPT, and I have “corrected” it in the first prompt to make the post easier for foreigners to read.