.9 repeating = 1? Is our numerical system flawed?

I'd missed that one!

0.999... is definately not a fraction of infinity.

Originally posted by The Vagabond
I'm not really into math, but a friend brought something up to me today that really seemed very strange. (For the duration of this post, .999 will mean .9 repeating unless otherwise specified- just for the sake of ease)
.999=x
10x=9.999
10x - x = 9x
9x=9
1x=1.
.999 = 1.

Nope. 9(0.999) = 8.991

So 1x still equals 0.999 and 0.999 ≠ 1

Sorry to break your heart.

[edit on 31-10-2007 by NRen2k5]

All this just proves the inadequacy of the digital system versus anolog. Digital requires some 'rounding' to express things with repeating digits like Pi or 1/3 whereas analog maths has no problem with it.

3 x 1/3 = 1 for example
but 3 x 0.333333 = 0.999999

(2^0.5)^2 = 2
but your calculator may tell you it's 1.999999999

So the precision is enhanced by using more significant digits to minimise the error.

It's just a concept after all.

[edit on 31/10/2007 by Pilgrum]

Originally posted by Pilgrum
All this just proves the inadequacy of the digital system versus anolog. Digital requires some 'rounding' to express things with repeating digits like Pi or 1/3 whereas analog maths has no problem with it.

3 x 1/3 = 1 for example
but 3 x 0.333333 = 0.999999

(2^0.5)^2 = 2
but your calculator may tell you it's 1.999999999

So the precision is enhanced by using more significant digits to minimise the error.

It's just a concept after all.

You hit the nail right on the head there.

In computers, you want to have the work done at a significantly higher level of precision than the answers you want. This is also good practice when doing math in pencil and paper. If you want an accurate answer to two decimal places, then do your work with three or four and round the final answer.

Though I’d think good software will recognize a fraction like 1/3 and keep it as the operation 1÷3 for as long as it can before getting an actual decimal number out of it.

As to the calculator giving you 1.999… nah, it won’t. When it reaches the last digit it can display, it rounds rather than truncating. Which pushes the 1.9999999999999 to 2.0.

Though on a computer, when working with code, you might get such behaviour. Though the only time I ever really fooled around with maths in a programming language was a long time ago in QBasic.

[edit on 31-10-2007 by NRen2k5]

Originally posted by NRen2k5

Originally posted by The Vagabond
I'm not really into math, but a friend brought something up to me today that really seemed very strange. (For the duration of this post, .999 will mean .9 repeating unless otherwise specified- just for the sake of ease)
.999=x
10x=9.999
10x - x = 9x
9x=9
1x=1.
.999 = 1.

Nope. 9(0.999) = 8.991

So 1x still equals 0.999 and 0.999 ≠ 1

Sorry to break your heart.

[edit on 31-10-2007 by NRen2k5]

By 0.999 he means o.9 recuring, his math is correct, o.9 recurring does equal 1, 4.9 recurring equals 5 and so on.

This is not a flaw in maths its a flaw in how you are looking at the numbers, just because it's written differently doesn't mean they cannot be the same.

10/3 = 3.3 recurring.
3.3 recurring*3 = 9.9 recurring
(10/3)*3 = 10

x = 0.9 reccuring
10x = 9.9 recurring
9x = 10x - x AKA 9x = 9.9 recurring - 0.9 reccuring AKA 9x = 9
9x/9 = x AKA 9/9 = x
x = 1

If you're any good at maths you'll know that if x can = 2 things then those 2 things are both the same.

[edit on 4-11-2007 by b309302]

reply to post by SANTARII


No. 3.3 repeating times 3 does not equal 9.99 repeating and 9.99 repeating does not equal ten. Only when you truncate the 3.3 repeating or round the 0.9 repeating does this happen.

It isn’t a special property of numbers we’re seeing. It’s a limitation of how we record them.

If .999(infnity)=1.000 I guess we are saying that it is infinitely close to 1

then what is one third, .333(infinity)? It has to work with other numbers too right?

using the same logic:

.999(infinity)=1 then .333(infinity) =?


[edit on 4-11-2007 by b309302]

Hmm, you know what, I think you’re onto something.

Assuming a 0.999 that goes on to infinity, then the difference between it and 1 would have to be 0. Zero is sort of the opposite to infinity.

Still, I say this is a matter of how we record numbers, not the actual quantities they represent. An imaginary substance does not care if its measurement s 0.999(infinity). It’s a finite quantity.

Dude, you forgot to round.

This one is much easier:
ur = ID*10*T

As was posted earlier, .999 repeating exactly equals 1. It's not close to 1, but exactly 1. This is why-

To keep this from being too messy .333 and .999 are repeating

1/3= 0.333

Which means

0.333 x 3 = 1/3 x 3

Since 1/3 x 3 = 1, then .333 x 3 also equals 1.

Originally posted by Esoterica
As was posted earlier, .999 repeating exactly equals 1. It's not close to 1, but exactly 1. This is why-

To keep this from being too messy .333 and .999 are repeating

1/3= 0.333

Which means

0.333 x 3 = 1/3 x 3

Since 1/3 x 3 = 1, then .333 x 3 also equals 1.

No. The .333s don’t add up to .999, they add up to 1.

Originally posted by NRen2k5
reply to post by SANTARII

 

No. 3.3 repeating times 3 does not equal 9.99 repeating and 9.99 repeating does not equal ten. Only when you truncate the 3.3 repeating or round the 0.9 repeating does this happen.

It isn’t a special property of numbers we’re seeing. It’s a limitation of how we record them.

9.9 recurring does equal 10, the only reason why it is not written as this is because the decimal system is not calculating it fully, 9.9 reccuring is 10 but written in a rather stupid way.

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No flaw, just a little misunderstanding

There is no flaw in our system of arithmetic. The flaw is in the statement of the problem.

Look:

.999=x
10x=9.999
10x - x = 9x

is all very well, but

9x=9 and
1x=1

are wrong.

To see where it goes wrong, write x = 0.999...9, where the dots in the middle refer to a variable number of decimal places.

If x = 0.999...9

to any number of decimal places you please, then

1x is (obviously) also = 0.999...9

again, to any number of decimal places you please.

Even if the line stretches out the crack of doom like Banquo's heirs in the witches' mirror, this number can never be equal to 1.

Likewise,

9x = 9(0.999...9)

= 8.999...1

which is definitely not the same number as 9, no matter how many decimal places you care to stretch it out to.

Another, perhaps simpler, way of exposing the flaw in this little arithmetical conundrum is to point out that in your post you treat x as a constant of finite value 0.999, when actually it is a variable whose range lies between 0.900...1 and 0.999...9.

Also,

the obvious conclusion in my mind is that you can't substract an infinite number.

A number carried to an infinite number of decimal places is not an infinite number. It may not be accurately expressible in decimal notation (as Pi and your example, 1/3, are not) but these are still numbers with finite values.

Originally posted by SANTARII

9.9 recurring does equal 10, the only reason why it is not written as this is because the decimal system is not calculating it fully, 9.9 reccuring is 10 but written in a rather stupid way.

Thats exactly my thoughts on the matter. I don't understand what the big deal is here. The numbers as we know them are simply symbols to represent something. 9.9... is simply another way of writing 10.

.3... is simply another way of writing a fraction.

A simple representation / symbol / alias / character / mark / sign of the fraction 1/3

1/3 , One Third , .3... , .33333333... all mean the same thing.

Its not as if we calculate only using decimal system numerals. We calculate using symbols and fractions that are there for this very reason , as symbols. Even if a number is impossible to write down using decimals , one of three equal parts is expressed easily by any of the symbols above.

[edit on 5-12-2007 by Heckman]

operations based on mathematical or boolean logic using real numbers are not applicable to all situations.

there are other forms of mathematics and logic, involving different types of number systems, real and "imaginary" numbers, vectors, handedness, and multiple dimensions, including "fractional" or fractal dimensions.

the complete understanding of these systems has not been worked out by humanity, but their applications are virtually unlimited.

i understand your confusion when looking at mathematics, and if there's really anything i can tell you about it, it wouldn't be that there are definite answers to these dilemmas today, but there is always pleasure in asking the questions.

scratching your head when looking at this, you are taking a step towards becoming a "mathematician".

it's a rewarding and fascinating study, even if the repercussions of the knowledge gained cannot be felt immediately. these subjects are useful and pertinent to all of history and all of mankind.

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Am I going insane? can't anyone else see that the original post was nothing but a case of poor math!? if you think
10x=9.999 and 10x - x= 9x you are wroong you can not add something from one side and take away from another. that changes the original equation so it's not true anymore

This one confused me at first, but here is how I thought about it.

This is the OP's idea... (I use r to signify recurring)

1/3=0.3r
0.3r x 3=0.9r

What I do to think about this is not write the recurring numbers, but use a variable

x = 1/3

so in that case x signifies our 0.3r number

we want to get one side of that to be 1 (after all the idea here seems to be to prove what 1 equals), so what we do is rearrange that by multiplying both sides by 3 (standard maths), to get rid of the 3 on the right side to leave 1

3x = 1

we already defined that x = 0.3r (1/3), so we can just swap x in that for 0.3r to show

3 x 0.3r = 1

another way to look at this problem (better in my opinion)

the OP wants to find the result for 0.3r x 3 = ?, well we know 0.3r is the same as 1/3, so

3 x (1/3) = 1

heh, sorry if that gets even more confusing, but if this just helps one person understand this, i'll be happy!