General / Off-Topic Does 0.999999999... =1?

[AluminumAlloy]

This question has been lurking in the depths of my head since my childhood.

The 9's can go on infinitely, out of the monitor to my neighbor's house across the street to our neighbor Andromeda galaxy and so on, and yet, does this make 1.0? It's ridiculously confusing.

1 - .9 = .1
1 - .99 = 0.1
1 - .999 = .001
1 - .9999999 = ????
1 - .9999999 = 0000000...1????

Someone please share their thoughts on this, thanks!

 

[Asp Explorer]

Yes and no.

0.999 repeating and 1 are in all aspects identical. There is never a teeny bit missing from 0.999 to add to make it 1. It's also why 1 is not really an integer.

Mathematicians will surely point out just how wrong I am

It's easily (but wrongly) proven by 1/3=0.333 repeating. Balance demands equivalency, so 0.333x3=0.999=1

As you can probably tell, I completely failed mathematics on every level. I don't care - I'm a hardware guy, and +- 0.5 volts is good enough for me

 

[Cody]

<cachinnates>

 

[Roybe]

Nope...1=1 any other number...does not.

If you want to think about it in Elite terms....with the distances involved....a very small error can mean a very large expanse of space missed.

This type of error becomes really problematic if it occurs over even short periods of time...thus the need for peer to peer networking...as the time it takes information to travel from a server to a player and back could translate into large chunks of kilometers that would need to be addressed...which would mean MORE jumping ships.

 

[Zieman]

Yes and no.

0.999 repeating and 1 are in all aspects identical. There is never a teeny bit missing from 0.999 to add to make it 1. It's also why 1 is not really an integer.

Mathematicians will surely point out just how wrong I am

It's easily (but wrongly) proven by 1/3=0.333 repeating. Balance demands equivalency, so 0.333x3=0.999=1

As you can probably tell, I completely failed mathematics on every level. I don't care - I'm a hardware guy, and +- 0.5 volts is good enough for me

The decimal one is always an approximation. Fractions are exact.
0.333x3 = 0.999 < 1 but 1/3x3 = 3/3 = 1

That said, in ~all practical (and most/many theoretic) applications, an approxmiation with 3 decimals is good enough, so 0.333x3 = 0.999 ~ 1

 

[Andy Kappa]

Not sure if this answers your question, but 0.9999999... (i.e. 0.9 recurring) is a mathematically acceptable way of writing 1.

1.000000000... would do as well.

 

[bubbleawsome]

Some people say 0.999...=1 because
0.999...=x
9.999...=10x
9=9x
1=x

But integers are very imprecise like that.

 

[Patrick_68000]

Sunday is the day of rest ----- http://math.wikia.com/wiki/Proof:The_Decimal_0.999..._is_Equivalent_to_1

 

[Futuristic Kung Fu]

I'm 99.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
percent sure that it isn't

 

[CyFur]

..Cough.. not quite exact or finite..

22 / 7

 

[Parmo]

This question has been lurking in the depths of my head since my childhood.

The 9's can go on infinitely, out of the monitor to my neighbor's house across the street to our neighbor Andromeda galaxy and so on, and yet, does this make 1.0? It's ridiculously confusing.

1 - .9 = .1
1 - .99 = 0.1
1 - .999 = .001
1 - .9999999 = ????
1 - .9999999 = 0000000...1????

Someone please share their thoughts on this, thanks!

Not in hex.

 

[Surfinjo]

I recall way back when I was still at school, reading a story about a debate in Ancient Greece where it was discussed.

It was proposed that a totoise and a hare were in a race. The tortoise ran at half the speed of the hare but was given a head start of exactly half the distance given to the hare.

 

[Frank]

Pentium FDIV bug

I recall way back when I was still at school, reading a story about a debate in Ancient Greece where it was discussed.

It was proposed that a totoise and a hare were in a race. The tortoise ran at half the speed of the hare but was given a head start of exactly half the distance given to the hare.

I know the answer to this one.. The Hare is sure he's got lots of time so he takes a nap against a tree, only to be overtaken by the doggedly determined tortoise who crosses the finish line first. It's not a story about whether an infinitesimally small difference can be considered zero. It's about how "slow and sure" can win the race.

 

[EdzUp]

This all boils down to CPU math if for example you add 0.00000001 to 0.00000001 over a period of cycles it will glitch and you will always get extras where they shouldn't. This problem been known about for decades and its always been put down to floating point inaccuracies of using binary math.

To cut it short computer's aren't always good at math unless you use two integers for the main and one for the fraction components of a number.

https://en.m.wikipedia.org/wiki/Floating_point#Accuracy_problems

 

[Surfinjo]

I know the answer to this one.. The Hare is sure he's got lots of time so he takes a nap against a tree, only to be overtaken by the doggedly determined tortoise who crosses the finish line first. It's not a story about whether an infinitesimally small difference can be considered zero. It's about how "slow and sure" can win the race.

Nope. You're thinking of Aesop's fable of the tortoise and the hare.

This however, while a nod to Aesop, was not the same thing at all.

The hare runs twice as fast as the tortoise. But must run twice the distance.

So, after the hare has run 50% of his distance, the tortoise will have run 50% of his, exactly half the distance of the hare.

This is repeated over and over, but while neither will ever actually finish the race, the hare will never quite catch up with the tortoise.

 

[GJ51]

.999... means that 1 is the limit approached but never reached. Depending on how it's used it may in fact have a finite limit i.e as you get down to Planck quantum measurements.

For distance, here is the smallest theoretical possible measurement. There is also Planck time, the smallest theoretical time interval.

1.61622837 × 10^-35 meters


The Planck time is the time it would take a photon travelling at the speed of light to across a distance equal to the Planck length. This is the �quantum of time�, the smallest measurement of time that has any meaning, and is equal to 10^-43 seconds.

For all practical purposes, this is where infinity ends...

 

[Futuristic Kung Fu]

Darth Vader: "INFINITY ENDS HERE FOR YOU, LUKE."

 

[LCplCombat]

I recall way back when I was still at school, reading a story about a debate in Ancient Greece where it was discussed.

It was proposed that a totoise and a hare were in a race. The tortoise ran at half the speed of the hare but was given a head start of exactly half the distance given to the hare.

The Ephebians had a similar quandry. One of them worked out that it was impossible to kill a tortoise with an arrow shot at it from a bow. The reasoning was because the arrow had to travel half the distance from the firer to the tortoise, when it reached that distance, it would then have to travel half of the remaining distance again. This half way distance to travel would repeat an infinate number of times, resulting in the arrow NEVER reaching the tortoise. They almost caused the extincion of the local tortoise population during experiments

 

[Surfinjo]

The Ephebians had a similar quandry. One of them worked out that it was impossible to kill a tortoise with an arrow shot at it from a bow. The reasoning was because the arrow had to travel half the distance from the firer to the tortoise, when it reached that distance, it would then have to travel half of the remaining distance again. This half way distance to travel would repeat an infinate number of times, resulting in the arrow NEVER reaching the tortoise. They almost caused the extincion of the local tortoise population during experiments

Not heard that one, nor the term Ephebians used in that context nor that they were a separate culture. Sadly, as you know, all of our knowledge of ancient Greece is second hand.

Even still, the more interesting aspect of your tale is the revelation that the ancient Greeks, or even the ancient Romans, practised a brand of humour normally associated with post 1980s UK and USA.

We live to learn!

 

[Andy Kappa]

Q. What do you get if you put 20 software developers in a room and shut the door?

A. A discussion like this one.


Mathematics is capable of embracing concepts that may or may not have any correspondence in the real world. From a Mathematician's perspective 0.9 recurring and 1.0 recurring are both mathematically equivalent in value to 1.

0.99999999... + 0.99999999... = 1.99999999...


*Sits back and waits for counterexamples...*