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Math: .9 repeating equals 1

F

Fearless Leader

macrumors 68020
Original poster
Mar 21, 2006
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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.
 
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.
 
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.
 
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 :)
 
Jasonbot said:
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.
Click to expand...

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
 
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
Click to expand...

Aaah! The memories :D Thanks!
luckily in grade 11 my casio FX 82-ES does fractions for me these days.
 
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
 
bemylover said:
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 :)
Click to expand...

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.
 
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