(5 Points) Express 5.57575757576... As A Rational Number, In The Form Where P And Q Are Positive Integers
Understanding how to express repeating decimals as rational numbers is a fundamental concept in mathematics, particularly in the study of rational and irrational numbers. This process not only helps in simplifying complex decimal expressions but also deepens our comprehension of the relationship between decimals and fractions. In this article, we will explore the step-by-step method to convert the repeating decimal 5.57575757576... into a rational number in the form where P and Q are positive integers, with P/Q representing the fraction.
Understanding Repeating Decimals
Before diving into the conversion process, it is essential to understand what repeating decimals are and how they relate to rational numbers.
What Are Repeating Decimals?
Repeating decimals are decimal numbers in which a certain sequence of digits repeats infinitely. For example, in the decimal 0.333..., the digit 3 repeats endlessly. Similarly, in 5.575757..., the sequence 75 repeats infinitely.
Rational Numbers and Repeating Decimals
It is a well-established mathematical fact that every repeating decimal can be expressed as a rational number — that is, as a ratio of two integers. Conversely, every rational number has a decimal expansion that either terminates or repeats periodically.
Step-by-Step Conversion of 5.57575757576... to a Fraction
Given the decimal 5.57575757576..., our goal is to express it as a fraction P/Q in the lowest terms, where P and Q are positive integers.
Step 1: Recognize the Decimal Structure
The decimal is:
5.57575757576...
Note that the repeating part is '75', and the non-repeating part is '5'. The digits after the decimal point are:
- Non-repeating part: 5
- Repeating part: 75
Since the decimal begins with 5, the integer part is 5, and the fractional part is the repeating decimal.
Step 2: Separate the Non-Repeating and Repeating Parts
Express the decimal as the sum of two parts:
- The non-repeating part: 5.5
- The repeating part: 0.07575757576...
Alternatively, to make calculations clearer, we can consider the entire decimal as:
X = 5.57575757576...
Now, focus on isolating the repeating decimal.
Step 3: Define the Variable and Create Equations
Let:
X = 5.57575757576...
Since the repeating sequence is '75', and it repeats after the decimal, we can manipulate the decimal to eliminate the repeating part.
To do this, multiply X by a power of 10 that moves the decimal point just after the repeating part.
The repeating sequence '75' is two digits, so:
- Multiply X by 100 to shift two digits to the left:
\[ 100X = 557.57575757576... \]
- Multiply X by 1 to keep the original:
\[ X = 5.57575757576... \]
Now, subtract the original from the multiplied:
\[ 100X - X = 557.57575757576... - 5.57575757576... \]
Calculations:
\[ 99X = 552 \]
Because the decimal parts cancel out due to the repeating nature:
\[ 99X = 552 \]
Step 4: Solve for X
Divide both sides by 99:
\[ X = \frac{552}{99} \]
Now, simplify the fraction:
- Find the greatest common divisor (GCD) of 552 and 99.
Prime factorization:
- 552 = 2^3 3 23
- 99 = 3^2 11
Common factors: 3
Divide numerator and denominator by 3:
\[ X = \frac{552 ÷ 3}{99 ÷ 3} = \frac{184}{33} \]
Therefore,
\[ X = \frac{184}{33} \]
Final Expression in Lowest Terms
Expressed as a fraction, the decimal 5.57575757576... equals:
\[
\boxed{\frac{184}{33}}
\]
This fraction is in simplest form since the numerator and denominator share no common divisors other than 1.
Conclusion: Rational Number Representation
The decimal 5.57575757576... can be precisely expressed as the rational number \(\frac{184}{33}\). Here, both 184 and 33 are positive integers, satisfying the requirement for P and Q in the form \(\frac{P}{Q}\). This conversion exemplifies the process of translating a repeating decimal into a fraction, reinforcing the foundational concept that all repeating decimals are rational numbers.
Additional Tips for Converting Repeating Decimals to Fractions
- Always identify the repeating part and the non-repeating part clearly.
- Use multiplication by powers of 10 to shift the decimal point beyond the repeating sequence.
- Subtract to eliminate the repeating component and solve for the original decimal.
- Simplify the resulting fraction to its lowest terms by dividing numerator and denominator by their GCD.
Summary of the Conversion Process
| Step | Description | Mathematical Expression |
|-------|--------------|-------------------------|
| 1 | Identify the decimal structure | 5.575757... |
| 2 | Multiply to shift repeating part | 100X = 557.57575... |
| 3 | Subtract equations | 99X = 552 |
| 4 | Solve for X | \(X = \frac{552}{99} = \frac{184}{33}\) |
| 5 | Simplify fraction | \(\frac{184}{33}\) |
By following this systematic approach, converting any repeating decimal into a rational number becomes straightforward and reliable.
In conclusion, expressing 5.57575757576... as a rational number yields \(\frac{184}{33}\), with P and Q being positive integers. This method underscores the importance of understanding decimal structures and algebraic manipulation in number theory and mathematics at large.
