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Magazine Chapter 2: Problem 40
Terminator II. Show that if the denominator of a fraction has only factors of 2 and 5 , then the decimal expansion for that number must terminate in a tail of zeros.
Short Answer
Expert verified
If a fraction has 2 and 5 as its only denominator factors, its decimal form ends in zeros due to its denominator becoming a power of 10.
Step by step solution
01
Express the Fraction
Consider a fraction \( \frac{a}{b} \), where the numerator \( a \) is an integer and the denominator \( b \) consists only of factors of 2 and 5. This means \( b = 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.
02
Decimal Expansion
To express \( \frac{a}{b} \) as a decimal, divide \( a \) by \( b \). The decimal representation will move into trailing zeros if the fraction can be represented as a finite decimal.
03
Condition for Terminating Decimal
A fraction \( \frac{a}{b} \) has a decimal expansion that terminates if the denominator \( b \), after simplifying, is of the form \( 10^k \), where \( k \) is a non-negative integer. This is because dividing by \( 10^k \) shifts the decimal point \( k \) places to the left, resulting in a finite decimal.
04
Convert to Base 10
Since \( b = 2^m \times 5^n \), it can be expressed as \( 10^k \) if \( m = n \). If \( m eq n \), adjust by multiplying the numerator and denominator by either \( 5^{n-m} \) or \( 2^{m-n} \) to equalize them, resulting in \( 10^k \).
05
Demonstrate Termination
After equalizing the powers of 2 and 5 in \( b \), \( b \) becomes \( 10^k \). Dividing a number by \( 10^k \) results in a decimal number that terminates, indicating that the original fraction has a decimal expansion that ends in trailing zeros.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factors of 2 and 5
When we talk about the factors of a number, we're essentially referring to the building blocks that, when multiplied together, create that number. For the purpose of this discussion, we're focusing on the factors 2 and 5. When the denominator of a fraction is composed solely of these factors, it means it can be expressed in the form \( b = 2^m \times 5^n \). Here, \( b \) represents the denominator, while \( m \) and \( n \) are non-negative integers that signify the power of 2 and 5 respectively.
- Significance: These specific factors are key to understanding why the fraction's decimal expansion terminates.
- Why 2 and 5? Multiplying 2 and 5 together gives 10, the base of our numbering system, which is crucial for forming finite decimals.
Finite Decimal
A finite decimal is a decimal that has a limited number of digits after the decimal point. In simpler terms, it means the decimal ends after a few places without continuing indefinitely. This is a contrast to repeating decimals, where the numbers after the decimal point continue in a repeating pattern.
- Key Insight: If a fraction's denominator has factors only of 2 and 5, the decimal must be finite.
- Example: \( \frac{1}{8} = 0.125 \) is finite, as it ends here.
Decimal Expansion
Decimal expansion is the process of representing a fraction as a decimal. It's essentially how a fraction is expressed in numerical form beyond the basic fraction notation. When you divide the numerator by the denominator, you'll see the decimal representation.
- Terminating Decimal: Means the decimal expansion is complete and ends with zeros.
- Connection: If the denominator consists of only the factors 2 and 5, the decimal expansion will be simple and finite.
Denominator
The denominator is the bottom part of a fraction, representing the total number of parts the whole is divided into. In our discussion, we are particularly interested when the denominator consists only of the factors 2 and 5. Such a denominator suggests that the fraction's decimal expansion will terminate.
- Formulation: A denominator can be written as \( 2^m \times 5^n \).
- Conversion: It can become \( 10^k \) by equating \( m \) and \( n \), ensuring it only has 10 (as a factor).
Numerator
The numerator is the top part of a fraction, indicating how many parts of the denominator are being considered. In the process of expressing a fraction as a terminating decimal, attention is often focused more on the denominator, but the numerator plays a crucial role in this calculation.
- Simplifying: It's often necessary to adjust the numerator when transforming the denominator into \( 10^k \).
- Balancing: Multiplying the numerator by the necessary power of 2 or 5 can help simplify calculations.
