Math: .9 repeating equals 1

Fearless Leader

macrumors 68020
Original poster
Mar 21, 2006
2,360
0
Hoosiertown
I got bored after a math test and was just messing around and realized something that makes sense and doesn't make sense at the same time.

1/3 = .3 repeating
2/3 = .6 repeating
following the trend
3/3 = .9 repeating
but
3/3 = 1/1 which is 1.

Where Did I go wrong? or did I?

also: psst Q Add a math section. It'll be great. You know you want one.
 

ChrisBrightwell

macrumors 68020
Apr 5, 2004
2,294
0
Huntsville, AL
Basically, the "lightbulb" moment for me was when I realized that an infinitessimal value, on the real number system, is equal to zero.

That is:

0.000...[infinitely many zeroes]...0001 is, on the real number line, equal to zero.

So ...
1/3 = 0.333...

1/3 * 3
= 0.333... * 3
= 0.999...
= 1 - 0.000...001
= 1 - 0
= 1

I feel like I grossly over-simplified it, but I hope that makes sense.
 

Rodimus Prime

macrumors G4
Oct 9, 2006
10,132
4
ah I used to think about stuff like this. Then I got into college and engineering work. That when I learned anything past 3-4 decimal is pretty worthless any how. That includes when I am using scientific notation. I will only go 3-4 points past the decimal in my work and my answer will be 2-3 past.
Reason I know it is worthless is it effects the answer so little that it does not matter. My though is I am with in one hundredth of the correct answer so it is close enough.
 

bemylover

macrumors regular
Jun 20, 2005
124
0
yep, 0.999999... is exactly 1. If it was not exactly 1 then there would be some other number that would be greater than 0.99999... and less then 1 and as there is no such number these two are the same.

Nice observation :)
 

swiftaw

macrumors 603
Jan 31, 2005
6,309
20
Omaha, NE, USA
There was this funny simplification of fractions thing I did in maths a few years ago that showed how .999... is equal to 1. I can't remember it though.
Probably something like this

x = 0.99999....

so 10x = 9.99999........

so 10x - x = 9.99999..... - 0.99999.... = 9

so 9x = 9

so x = 9/9 = 1


Actually this method works to convert any reoccurring decimal into a fraction, for example
x = 0.838383.......
100x = 83.838383.....
99x = 83
x = 83/99
 

Jasonbot

macrumors 68020
Aug 15, 2006
2,468
0
The Rainbow Nation RSA
swiftaw said:
Probably something like this

x = 0.99999....

so 10x = 9.99999........

so 10x - x = 9.99999..... - 0.99999.... = 9

so 9x = 9

so x = 9/9 = 1


Actually this method works to convert any reoccurring decimal into a fraction, for example
x = 0.838383.......
100x = 83.838383.....
99x = 83
x = 83/99
Aaah! The memories :D Thanks!
luckily in grade 11 my casio FX 82-ES does fractions for me these days.
 

gertruded

macrumors 6502
Jul 5, 2007
300
983
Northwestern Illinois
In the days when we used slide rules to do all our calculations, we were limited to 3 significant places. In that practical context .999 definitely is not equal to 1.00. I still laugh when I see calculations done computers displayed to 5 or more significant figures for ordinary engineering calculations.

The slide rule was an extraordinary device. Mine was a Post log log duplex versilog, made of bamboo. I still have it and it still works without electricity.

Gertrude
 

Fearless Leader

macrumors 68020
Original poster
Mar 21, 2006
2,360
0
Hoosiertown
yep, 0.999999... is exactly 1. If it was not exactly 1 then there would be some other number that would be greater than 0.99999... and less then 1 and as there is no such number these two are the same.

Nice observation :)
ohh I really like that one. I went over it with my Math teacher today. He went from not believing me, to helping me to get him to agree, then agreeing.