will every fraction terminate or repeat????

Question

explain w/ example

i dont get this

this is a precal question

Answer 1

The answer is yes. Every fraction when converted to decimal form will either terminate or repeat. For example

1/2=.5

That means it terminated

1/3 = .333333333333..... (3's forever)

That means it repeated.

Every fraction will do one or the other

Answer 2

Rational numbers are numbers that can be expressed as a fraction with integers. For example, 3/11 is a rational number (a fraction using the integers 3 and 11).

All rational numbers, when written as decimal expansions, will either terminate or repeat. If it repeats, the cycle will have a finite number.


For example, all fractions with 7 in the denominator have the cycle 142857 repeating in the decimal expansion:

1/7 = 0.14285714285714...
2/7 = 0.28571428571428...
3/7 = 0.42857142857142...
4/7 = 0.57142857142857...
etc.
The "cycle" has 6 digits.

Numbers that cannot be expressed as a fraction are not rational (the word rational comes from "ratio" which means proportion or fraction). For example, it is impossible to write the square root of 2 as a precise fraction.
It is also impossible to find a finite cycle in the decimal expansion of the square root of 2 (it goes on forever and there is no "repeat")

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Going the other way:

It turns out that any number whose decimal expansion terminates must be rational:

0.1234567890123 can always be written as
1,234,567,890,123 / 10,000,000,000,000

This is a rational number (fraction made up of 2 integers).

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It is more difficult to show that any decimal expansion that repeats in a finite cycle must be rational.

I'll show it for a repeating cycle of 2:
We have a decimal expansion that looks like
0.abababababababababababab... forever.

Is there a rational number r to represent this expansion?

r = 0.abababababababababababab...
multiply both sides by 100 (10^2, where 2 is the length of the cycle)
100 r = ab.abababababababababab...

Subtract the first from the second:

100 r - r = (100-1) r = 99 r

ab.abababababab... - 0.abababababab... = ab
(the infinite decimal expansion disappears)
ab is an integer.

We are left with:
99 r = ab
r = ab/99

ab is an integer (by the way we found it)
99 is an integer.

Therefore, ab/99 is a rational number (fulfills the definition).
Since r = ab/99, then r must also be a rational number.

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Example with 0.857142857142...

r = 0.857142857142...
multiply by a million (10^6)
1,000,000 r = 857142.857142857142...
subtract the first from the second:

1,000,000 r - r = 0.857142857142... - 857142.857142857142...
999,999 r = 857142
r = 857,142 / 999,999

If you divide the numerator and the denominator by 142857, you'll get 6/7 (as expected).

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pi is NOT 22/7. 22/7 is only an approximation that you can use in everyday life, instead of pi.

22/7 = 3.142857142857142857...
pi = 3.14159265358979323846264338327...

Millions of decimals have been determined for pi and still no repeat cycle has been found (it is difficult to "prove" that an infinitely long number will have no repeat).

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If your fraction includes a number that is not rational (like 1 divided by pi), the decimal expansion will not terminate and it will not repeat.

With real numbers, this is true for any irrational number.

Answer 3

Hey there!

Every fraction WILL either terminate or repeat. Fractions are commonly represented in the form a/b.

All rational numbers, are in the form a/b. Since sqrt(2) is an irrational number, it is not in the form a/b.

Here's an example.

1/2=0.5, terminates.
1/3=0.333..., repeats.

If you're saying that a fraction neither terminates, nor repeats, then you are saying that the fraction is not rational i.e. not in the form a/b, which is impossible.

Any rational number, in the form a/b, will either terminate or repeat.

The word terminate, means to end. So, literally, when you say that 0.5 terminates, it "ends" right after 5.

Hope it helps!

Answer 4

Every fraction is a rational number, so it must eventually terminate or have a repeating sequence. I cannot prove this for you without advance calculus, but you can 'feel' this through examples of the following sort:

x = 0.33333... then 10x = 3.333... so 10x -x =3.333... - 0.3333... = 3

Hence 9x = 3 and x =3/9 = 1/3

If the decimal expansion is finite, then the number it represents must be rational (a fraction). For instance, 0.354 means 2/10 + 5/100 + 4/1000 which is a fraction.

Hope this helps

Answer 5

Yes. Given any such decimal fraction, there is a simple algorithm for obtaining the numerator and denominator of the corresponding rational fraction. Consider 0.1428571428..., for example. Write a rational fraction with the repeating sequence in the numerator, an equivalent number of 9's in the denominator, and simply: 142857/999999 = 1/7.

Answer 6

A fraction divided out will terminate or endlessly repeat if and only if both numerator and denominator are rational numbers.
1/√2 will neither repeat nor terminate, nor will numbers such as x/π or π/x. (Of course if the fraction is composed of irrationals which can all be canceled, it then becomes a rational number.)

Answer 7

every rational number will repeat or terminate!
fractions are rational numbers!

Things like pi, phi and e (irrational numbers) never repeat or terminate!

Answer 8

Some do, some don't and most never end, or repeat. Evgen 1/2 has endless zeros.

Answer 9

1/4 is .25
1/3 is .3333333333333 (and continues as far out as you wish to compute.

So NO, every fraction will not terminate or repeat. Some will terminate and some will repeat.

so now let's play with 22/7 that's 3.14385714........
it doesn't terminate or repeat. It's pi

Answer 10

@kayakdudeus, pi is represented as APPROXIMATELY 22/7 it is not exactly 22/7, which is why it is irrational and not rational. 22/7 is rational. pi is irrational but very close to 22/7. Not exactly.

@Leen, same thing here, pi is irrational because it does not repeat.