Point nine recurring equals one

[Shannon]

      x = 0.9999...
    10x = 9.9999...
10x - x = 9.9999... - 0.9999...
     9x = 9.0000...
     9x = 9
      x = 1

I've seen some divided opinions on this...

 

[Touring Mars]

Doesn't it all come down to the line 10x - x = 9.9999... - 0.9999 .... if you limit the number of decimal places in each term, i.e. say to 5 decimal places, x = 0.99999 , then 10x = 9.99990 (i.e. NOT 9.99999), which means that 9x doesn't equal exactly 9.00000.... by not limiting x to any number of decimal places, i.e. recurring 9's to infinity, x will tend closer to 1 asymptotically...

right, lunch time.... you hurt my brain

 

[niky]

Yeah. Especially if you're asking for change from the nice lady and the candy bar only costs 99 cents.

 

[Famine]

1/3 = 0.33333333...
3 x 0.33333333... = 0.99999999...

Therefore 3/3 = 0.99999999...

But 3 x 1/3 = 3/3 = 1

Therefore 1 = 0.99999999...


And no messing around with algebra is required.

 

[Danoff]

1/3 = 0.33333333...
3 x 0.33333333... = 0.99999999...

Therefore 3/3 = 0.99999999...

But 3 x 1/3 = 3/3 = 1

Therefore 1 = 0.99999999...


And no messing around with algebra is required.

The problem with this is that 1/3=0.3333 infinitely repeating - which isn't properly represented in decimal form. When you include remainders 1/3 = 0.3 remainder .1

3 times 0.3 + .1 = 1

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9.0000...
9x = 9
x = 1

Again, the inability to represent an infinitely repeating series (which isn't a real number afterall because it constantly changes as you go one more decimal out). Do the experiment with x=.999

10x=9.99
10x-x=8.991
9x=8.991
x=8.991/9=.999

 

[Kent]

So this is a social or political issue how?

Also, 9 is the magic number...

In western systems we tend to think of things interms of 1-10 but in acutality the number system is more like 0-9... and so, 9 has that special place in all of our hearts.

btw, thread moved.

 

[Famine]

The problem with this is that 1/3=0.3333 infinitely repeating - which isn't properly represented in decimal form. When you include remainders 1/3 = 0.3 remainder .1

3 times 0.3 + .1 = 1



Again, the inability to represent an infinitely repeating series (which isn't a real number afterall because it constantly changes as you go one more decimal out). Do the experiment with x=.999

10x=9.99
10x-x=8.991
9x=8.991
x=8.991/9=.999

Nevertheless, it is universally accepted by mathematicians that it is true.

1 divided equally into 3 parts gives 3 parts of equal size, each of which is 0.3 recurring of the original parts. Add the three 0.3 recurrings together and you get 0.9 recurring, because no single part has that extra microscopic fragment required to tick over to the full 1. We don't like this, because we like rounding.

It, like radioactive half lives, is a modern day Zeno's Paradox (pick one - they're all essentially the same. Something can never get somewhere, because it has to travel half the distance first, then half that distance, then half that distance, and so on).

 

[amp88]

Well, since the 9 is recurring, you can't really faithfully represent it using decimals...

0.99999999999999999999999999999999999999999999999999999999

is not the same as 1. Not technically anyway...

Just in the same way that

3.33333333333333333333333333333333333333333333333333333333

isn't the same as 10/3

 

[Magic069]

I imagine this is exactly what members of an engineering frat do all day.

I think I like being dumb, but creative. It seems more... fun.

 

[Danoff]

Nevertheless, it is universally accepted by mathematicians that it is true.

It's an accepted error, but it remains an error. As soon as you perform a mathematical function on 0.99999... (repeating) you've introduced an error in the system. Pointing out the inconsistencies that error causes makes sense only from the point of view of reminding us that the error exists due to convenience.

It, like radioactive half lives, is a modern day Zeno's Paradox (pick one - they're all essentially the same. Something can never get somewhere, because it has to travel half the distance first, then half that distance, then half that distance, and so on).

The paradox is faulty. I tried (and failed) to explain this to my philosphy instructor in college. When you travel half the distance, you take half the time... and so on and so forth until you're travelling an infinitesimal distance in an infintesimal time. The paradox is ruined by considering time because as distance goes to zero so does time.

 

[skip0110]

Not reading what has been said on this before, I can state the facts:

1) Decimal representation is non-unique, so having 0.999.... = 1 is acceptable. If you want a unique representation, use fractions in lowest terms (which is a clean definition).

2) The best way (no matematical holes) to determine the value of 0.999... is to express it as an infinite geometric series, i.e. 9/(10^i) for i=1 to infinity and sum.

 

[Famine]

It's an accepted error, but it remains an error. As soon as you perform a mathematical function on 0.99999... (repeating) you've introduced an error in the system. Pointing out the inconsistencies that error causes makes sense only from the point of view of reminding us that the error exists due to convenience.

It's actually extremely inconvenient - and a direct consequence of the decimalisation of fractions

The paradox is faulty. I tried (and failed) to explain this to my philosphy instructor in college. When you travel half the distance, you take half the time... and so on and so forth until you're travelling an infinitesimal distance in an infintesimal time. The paradox is ruined by considering time because as distance goes to zero so does time.

Yes - Zeno's paradoxes are flawed, but amusing.

Consider, however, radioactive half-lives. If you have eight atoms of a radioactive particle, with a half life of a minute, how long until there are no atoms of that particle left?

Again, it's extremely inconvenient, but all you can say is "probably 4-5 minutes". We're attempting to divide the indivisible. But I digress.


Three lots of 0.3 recurring do not equal 1, because no part of them contains the extra microscopic fraction required to push the result up to 1 from 0.9 recurring. However, since each item of 0.3 recurring is 1/3, three lots of it DO equal 1. Thus 0.9 recurring = 1. The error is to round up the 0.9 recurring to 1.


Let us now consider Base3 mathematics, and run the sum again.

1/10 = 0.1
10 x 0.1 = 1
Therefore 1 = 1

This sum is identical to the one I first posted, but in Base3 instead of Base10.

As you can see, the problem isn't any error, but the attempt to express in decimals numbers which cannot be expressed in decimals - it's a flaw inherent to the decimal number system.

 

[Blackbird.]

Famine if you're so smart. Type all the numbers in Pi. You'd be dead before you were done.

 

[Famine]

The universe would have ended before I was done - Pi is infinite.

However, it's only infinite from a decimal point of view. In BasePi it's 10.

 

[Blackbird.]

Yeah I know it's infinite. The longest i've ever seen Pi to be written/printed is 2,000,000 numbers.

Even the universe wouldn't fit all the paper needed to print Pi. It's quite mind boggling.

 

[Kylehnat]

I imagine this is exactly what members of an engineering frat do all day.

I think I like being dumb, but creative. It seems more... fun.

Wrong. Engineers don't give a rat's ass how many recurring decimal places 1/3 has. When push comes to shove, we usually can't use more than two decimal places anyway (if that).

A mathematician and an engineer were put in a room with a buffet table on the other end. They were told that they could walk towards the table, but only by walking half the distance to the table, then half of the remaining distance, and then half of that remaining distance, and so on... The mathematician starved to death, reasoning that he could never reach the table. The engineer did this procedure several times, said "close enough", then reached out his arm and ate.

These finnicky number games don't keep engineers up at night. I say .3 is 1/3, .9 is 1, and a million doesn't matter if the answer is a billion.

 

[Famine]

While the scientist said "Screw that." and dived head first into the roast suckling pig.

 

[Blackbird.]

I suck at algebra so right now this is all going right over my head.

 

[amp88]

Famine if you're so smart. Type all the numbers in Pi. You'd be dead before you were done.

22/7 (I believe)...

Da-da!

 

[Event]

22/7 (I believe)...

Da-da!

3.1415926535859 is as far as I know. 22/7=3.142857 (bold is repeating)

 

[Event]

Yeah I know it's infinite. The longest i've ever seen Pi to be written/printed is 2,000,000 numbers.

Even the universe wouldn't fit all the paper needed to print Pi. It's quite mind boggling.

IThere's a program you can download that calculated Pi to like 128 million digits.

I suck at algebra so right now this is all going right over my head.

It's actually more like calculus than algebra.

 

[Omnis]

Yeah...so many digits.


But, that's because nobody can ever have enough Pi. It's just SO ****ING GOOD.

 

[Kylehnat]

(Professor Frink) "Pi is exactly three!!!"

 

[Danoff]

As you can see, the problem isn't any error, but the attempt to express in decimals numbers which cannot be expressed in decimals - it's a flaw inherent to the decimal number system.

Agreed. By trying to represent the number in decimal form we introduce an error - and so logic (algebra) breaks down and does not give the correct result.

 

[skip0110]

Agreed. By trying to represent the number in decimal form we introduce an error - and so logic (algebra) breaks down and does not give the correct result.

I don't see any error--nor any place where algebra/logic breaks down.

.9999.... = 1 in every sense. They are alternate representations for the same concept (conceptual "one").

Unique representations are nice, but not needed.

Show me a computation where I replace 1 with 0.99999.... and the result changes.

 

[amp88]

Show me a computation where I replace 1 with 0.99999.... and the result changes.

...

10 / 1 = 10
10 / 0.99999 = 10.00010000100001000010000100001
10 / 0.9999999999 = 10.00000000100000000010000000001

...

 

[DeLoreanBrown]

3 x 0.33333333... = 0.99999999...

three times point three recurring equals one & not a recurring decimal

Therefore 3/3 = 0.99999999...

does'nt really follow

point nine recurring is not proved for unity

but in most quantative respects it would have the value of a totality ( almost all of one gallon is usually a gallon )

 

[skip0110]

...

10 / 1 = 10
10 / 0.99999 = 10.00010000100001000010000100001
10 / 0.9999999999 = 10.00000000100000000010000000001

...

But we are talking about .999... recurring to infinity, not any sort of terminating decimal. Without being rigorus, you can see in that computation that 10/0.999.... = 10 exactly, when you account for an infinite muber of digits.

0.9999 is not 1, it's 9999/10000.
0.9999.... is 1.

 

[gt_masta]

1/3 = 0.33333333...
3 x 0.33333333... = 0.99999999...

Therefore 3/3 = 0.99999999...

But 3 x 1/3 = 3/3 = 1

Therefore 1 = 0.99999999...


And no messing around with algebra is required.

1/3 approximatively = 0.33333
3/3 approximatively = 0.99999
1 approximatively = 0.99999

 

[skip0110]

point nine recurring is not proved for unity

but in most quantative respects it would have the value of a totality ( almost all of one gallon is usually a gallon )

It is proven to be unity. I will do it right here.

Geometric series sum is taught in every college algebra/calculus textbook, btw.

0.999.... (repeating) is identical to 1