Math help...Expressing repeating decimals into fractions

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pray4mojo

pray4mojo

Diamond Member
Mar 8, 2003
3,647
0
0
How do you write
1.060606...
1.4272727...
0.4999999...
as fractions?

I also came across expressing 0.9999.... as a fraction ;)
 
A

agnitrate

Diamond Member
Jul 2, 2001
3,761
1
0
Originally posted by: pray4mojo
How do you write
1.060606...
1.4272727...
0.4999999...
as fractions?

I also came across expressing 0.9999.... as a fraction ;)
Click to expand...

Think geometric series ;)

-silver
 
F

fatbaby

Banned
May 7, 2001
6,427
1
0
It comes from a well-known result in series. If you have the series

1 1 1 1
x = - + -- + -- + ... + --
n a 2 3 n
a a a

as n increases, the sum gets closer to 1/(a-1). So as you
keep going through the digits of .111111..., which is

1 1 1 1
--- + --- + --- + --- + ...
10 2 3 4
10 10 10

your sum gets closer and closer to 1/(10-1) = 1/9. Since .5555 is 5
times that, the sum is 5 times (1/9) or 5/9.
 
A

agnitrate

Diamond Member
Jul 2, 2001
3,761
1
0
Originally posted by: pray4mojo
that didn't make any more sense than my book did.
Click to expand...

lol, I know how to do this and I have no idea what fatbaby just said.

Ok, here you go.

Take .060606 repeating. Split it up into .06 + .0006 + .000006 + etc, etc These terms all have something in common (hint: divide a term by the previous one) and you will see there is a common ratio.

You can express this sum as a series (a geometric one actually). Isn't it convenient that a geometric series converges to a / (1-r) where a = the first term and r = the number to the nth power.

;)

-silver
 
R

RavnShield

Member
Jul 18, 2002
142
0
0
Example: 2.838383838

2.838383838=2 + 83/100 + 83/100^2 + 83/100^3 + 83/100^4 +83/100^n

Sum (from 1 to infinity) of 83/100^n

This is a geometric series so a=83/100 and r=1/100

sum=a/(1-r)
=2+(83/100)/(1-(1/100))=281/99
 
M

Muzzan

Member
Apr 15, 2003
169
0
0
A lowtech version ;) :

x = 1.060606...

100x = 106.060606...
10000x = 10606.060606...

10000x - 100x = 10606.060606... - 106.060606... (notice how the decimal parts cancel each other out, which leaves you with an integer)
9900x = 10500
x = 10500 / 9900 = 35 / 33

------

x = 1.4272727...
10x = 14.272727... (the repeating part is 27, the 4 is not part of that).
1000x = 1427.272727...
100000x = 142727.272727...

100000x - 1000x = 142727.272727... - 1427.272727...
99000x = 141300
x = 141300 / 99000 = 157 / 110

Is the idea clear to you?
 
Z

Zugzwang152

Lifer
Oct 30, 2001
12,134
1
0
Originally posted by: Muzzan
x = 1.060606...

100x = 106.060606...
10000x = 10606.060606...

10000x - 100x = 10606.060606... - 106.060606... (notice how the decimal parts cancel each other out, which leaves you with an integer)
9900x = 10500
x = 10500 / 9900 = 35 / 33

------

x = 1.4272727...
10x = 14.272727... (the repeating part is 27, the 4 is not part of that).
1000x = 1427.272727...
100000x = 142727.272727...

100000x - 1000x = 142727.272727... - 1427.272727...
99000x = 141300
x = 141300 / 99000 = 157 / 110

Is the idea clear to you?
Click to expand...


your first ATOT post! and it was constructive...it's all down hill from here :)

 
S

Syringer

Lifer
Aug 2, 2001
19,333
2
71
Basically, if you don't have to show any work or anything, it's just the number divided by 10^n - 1 where n is the length of each repeating string.

For example, 1.060606...

would be 1 + 6/99

.354354354...

would be 354/999
 
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