Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations.... Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations. Can you guess what property q must satisfy?
Understand the Problem
The question is asking for examples of rational numbers of the form ( \frac{p}{q} ) where ( p ) and ( q ) are integers with no common factors other than 1, and have terminating decimal representations. The goal is to determine the property that ( q ) must satisfy for these conditions.
Answer
The property that \( q \) must satisfy for \( \frac{p}{q} \) to have a terminating decimal is \( q = 2^m \times 5^n \) where \( m, n \geq 0 \).
Answer for screen readers
The property that ( q ) must satisfy for ( \frac{p}{q} ) to have a terminating decimal representation is that ( q ) can only have the prime factors 2 and 5: $$ q = 2^m \times 5^n $$ where ( m ) and ( n ) are non-negative integers.
Steps to Solve
-
Understanding Terminating Decimals A decimal representation is terminating if it has a finite number of digits after the decimal point. For a rational number $\frac{p}{q}$ to have a terminating decimal, the denominator $q$ in its simplest form must have only the prime factors 2 and/or 5.
-
Examples of Rational Numbers Let's consider a few rational numbers and identify their denominators:
- Example 1: $\frac{1}{2}$ (denominator is 2)
- Example 2: $\frac{3}{10}$ (denominator is 10, factors are $2 \times 5$)
- Example 3: $\frac{7}{25}$ (denominator is 25, factors are $5^2$)
- Example 4: $\frac{1}{8}$ (denominator is 8, which is $2^3$)
-
Analyzing the Denominators Observe that in all the examples above, each denominator only consists of the factors 2 and/or 5. This indicates a possible rule regarding the property of $q$.
-
Identifying the Property of q The property that $q$ must satisfy is that it can only have the prime factors of 2 and 5, expressed mathematically: $$ q = 2^m \times 5^n $$ where $m$ and $n$ are non-negative integers.
The property that ( q ) must satisfy for ( \frac{p}{q} ) to have a terminating decimal representation is that ( q ) can only have the prime factors 2 and 5: $$ q = 2^m \times 5^n $$ where ( m ) and ( n ) are non-negative integers.
More Information
Terminating decimals occur because the base-10 system we use is built upon the factors of 2 and 5. When the denominator ( q ) contains only these factors, it results in a decimal that terminates rather than repeating infinitely.
Tips
- Including other prime factors: Sometimes people incorrectly think that other primes, such as 3 or 7, can also be included in ( q ), but these cause the decimal to be non-terminating.
- Not reducing to simplest form: Always ensure ( p ) and ( q ) have no common factors other than 1 before analyzing the properties of ( q ).
