When writing the following fractions in decimal notation, which have a “terminating” and which have a “non-terminating” decimal?
| , | , | , | , | , | , | , | , | , | , |
| , | , | , | , | , | , | , | , | , | , |
| , | , | , | , | . | |||||
Can you find a pattern in your results from Problem 1? See if you can conjecture a
rule by which we can predict if a fraction will have a terminating or non-terminating
decimal representation.
Argue why a fraction that fits the criteria from Problem 2 for a terminating decimal
must terminate. Argue why a fraction that does not fit the criteria from Problem 2
for a terminating decimal must not terminate.
Take a fraction that does not have a terminating decimal representation (say,
) and use long division to find its decimal representation. What type of
pattern occurred? Why did this happen? Was this a special case or will
this happen for all fractions with a non-terminating decimal representation?
Why?
Using the facts that , , , etc., find a fraction representation for the following
numbers:
- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
We’ve now established (Problems 1 - 5) that all fractions can be written in decimal
notation. Can all decimals be written in fraction notation, i.e. in the form \[ \frac{\text{whole number}}{\text{nonzero whole number}}?\] If
so, argue why. If not, list some that cannot and explain why they are not
fractions.
Can be written as a fraction, i.e., in the form \[ \frac{\text{whole number}}{\text{nonzero whole number}}? \] Argue why or why not.
If is a prime number, can be written as a fraction, i.e., in the form \[ \frac{\text{whole number}}{\text{nonzero whole number}}? \] Argue why or
why not.
2025-08-13 00:45:10