Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 22
Show that the decimal expansion of a rational number must, after some point, become periodic or stop. [Hint: Think about the remainders in the process of long division.]
Short Answer
Expert verified
A rational number's decimal expansion either terminates or becomes periodic due to limited remainders in long division.
Step by step solution
01
Understanding Rational Numbers
A rational number is any number that can be expressed as the fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). We use long division to convert this fraction into a decimal.
02
Setup Long Division of a Fraction
Perform the long division of \( a \div b \). While dividing, you get a series of remainders at each step along with the quotient digits.
03
Count Possible Remainders
Since the divisor (\( b \)) can be any integer greater than zero, the possible remainders from division are integers from 0 to \( b-1 \). Thus, there are at most \( b \) possible remainders.
04
Examine Position of Remainders
As the division continues, each remainder determines the next digit in the quotient and the subsequent remainder. By the Pigeonhole Principle, if the division process continues for more than \( b \) divisions, some remainder must repeat.
05
Identify Periodicity or Termination
If a remainder repeats, the digits of the quotient (after the repeated remainder) must repeat as well, indicating a periodic decimal. If a remainder becomes zero, the decimal terminates, as the division results in a quotient with no remainder.
06
Conclusion
There are only two outcomes for a rational number's decimal representation: it either terminates when the remainder becomes zero or becomes periodic when a remainder repeats.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Periodic Decimal
A periodic decimal is a type of decimal expansion where a sequence of digits repeats indefinitely. This repetition happens after a certain point in the decimal sequence, creating a repeating pattern. For example, in the decimal representation 0.333... for the fraction \( \frac{1}{3} \), the '3' repeats indefinitely.
Why do periodic decimals occur? When you perform long division to convert a fraction into a decimal, you work with a limited set of remainders. Because there can only be a certain number of remainders, eventually, one of them must repeat. When it does, the corresponding digits in the quotient will also start to repeat, creating a periodic decimal.
Why do periodic decimals occur? When you perform long division to convert a fraction into a decimal, you work with a limited set of remainders. Because there can only be a certain number of remainders, eventually, one of them must repeat. When it does, the corresponding digits in the quotient will also start to repeat, creating a periodic decimal.
- Periodic decimals are common in the representation of rational numbers.
- The repeated pattern can be one or more digits long.
- To identify the repeating sequence, observe the digits in the decimal that appear more than once in the same order.
Long Division
Long division is a process used to divide numbers and determine their decimal representation. For rational numbers, this involves dividing the numerator of the fraction by its denominator.
To perform long division on a fraction \( \frac{a}{b} \):
To perform long division on a fraction \( \frac{a}{b} \):
- Start by writing the dividend inside the long division symbol and the divisor outside.
- Determine how many times the divisor can fit into the first digit or group of digits from the dividend.
- Subtract the result from this part of the dividend and bring down the next digit.
- Repeat the process until you reach a remainder of zero or a repeating sequence of remainders.
Pigeonhole Principle
The Pigeonhole Principle is a simple but powerful concept used in many mathematical proofs. It states that if you have more pigeons than pigeonholes and you want to place each pigeon in a hole, at least one hole must contain more than one pigeon.
In the context of rational numbers and their decimal expansions, consider the remainders in long division as the pigeons and the possible remainder values (0 to \( b-1 \)) as the pigeonholes.
In the context of rational numbers and their decimal expansions, consider the remainders in long division as the pigeons and the possible remainder values (0 to \( b-1 \)) as the pigeonholes.
- There are only \( b \) possible remainders since the divisor is \( b \).
- If the division continues for more than \( b \) steps, some remainder must repeat.
- Once a remainder repeats, the resulting decimal becomes periodic since the same sequence of digits will continue.
Decimal Expansion
Decimal expansion is the representation of a number in its decimal form. For rational numbers, decimal expansion is obtained by dividing the numerator by the denominator using long division.
The result can be twofold:
The result can be twofold:
- A **terminating decimal**, where the division process ends with a remainder of zero.
- A **non-terminating, repeating decimal**, where the decimals continue endlessly in a repeating pattern.
