12 7 9 16 71 50 91 Look at some decimal expressions of fractions such as 3 25' ' 15' 20' 14' 50 by carrying out long divisions or using your calculator (Careful, calculators can give mis- leading results!), and see which numbers have a terminating decimal expression, while paying attention to the prime factorisation of the denominator. Hint: Take a terminating decimal expression and see what happens when you convert it to a fraction in lowest terms. (Don't include this part in your homework submission.) Now come up with a conjecture for which fractions have a terminating decimal expres- sion. Your conjecture should be of the following form: Let a, beN with b>1 and with gcd(a, b) = 1. Then a has a terminating decimal expression iff .... b Now prove your conjecture. Note that this is an “if and only if” statement, requiring proofs for both implications.
12 7 9 16 71 50 91 Look at some decimal expressions of fractions such as 3 25' ' 15' 20' 14' 50 by carrying out long divisions or using your calculator (Careful, calculators can give mis- leading results!), and see which numbers have a terminating decimal expression, while paying attention to the prime factorisation of the denominator. Hint: Take a terminating decimal expression and see what happens when you convert it to a fraction in lowest terms. (Don't include this part in your homework submission.) Now come up with a conjecture for which fractions have a terminating decimal expres- sion. Your conjecture should be of the following form: Let a, beN with b>1 and with gcd(a, b) = 1. Then a has a terminating decimal expression iff .... b Now prove your conjecture. Note that this is an “if and only if” statement, requiring proofs for both implications.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Look at some decimal expressions of fractions such as 2,3 25 8' 15' 20' 14' 50
1
7 9 16 71 50 91 by
carrying out long divisions or using your calculator (Careful, calculators can give mis-
leading results!), and see which numbers have a terminating decimal expression, while
paying attention to the prime factorisation of the denominator.
Hint: Take a terminating decimal expression and see what happens when you convert
it to a fraction in lowest terms. (Don't include this part in your homework submission.)
Now come up with a conjecture for which fractions have a terminating decimal expres-
sion. Your conjecture should be of the following form:
Let a,
bɛ N with b > 1 and with gcd(a, b) = 1. Then
a
has a terminating decimal expression iff ....
b
Now prove your conjecture. Note that this is an “if and only if” statement, requiring
proofs for both implications.
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