Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2006-07-20 09:24:55

jd
Member
Registered: 2005-10-02
Posts: 37

2+2 = 3.99 recurring??

I heard somewhere this was true.

Offline

#2 2006-07-20 09:55:29

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: 2+2 = 3.99 recurring??

thats why they say..

n  = 3.99... 

10n = 39.99...   multiply both sides by 10
9n = 36            subtracted n from the left side, and 3.99... from the right.
n = 4.      divide both sides by 9.  QED.


A logarithm is just a misspelled algorithm.

Offline

#3 2006-07-20 11:26:19

numen
Member
Registered: 2006-05-03
Posts: 115

Re: 2+2 = 3.99 recurring??

Is (...) supposed to mean limit? Now that's new.

Besides, I think this would be a valid argument against it:


Edit: 3,999... is not irrational by definition so it has to be rational, so I take the above statement back. tongue

Last edited by numen (2006-07-22 11:09:37)


Bang postponed. Not big enough. Reboot.

Offline

#4 2006-07-20 12:18:26

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,711

Re: 2+2 = 3.99 recurring??

Ahhh ... this one again. Have a read here: Does 0.999... equal 1?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Offline

#5 2006-07-20 17:10:23

jd
Member
Registered: 2005-10-02
Posts: 37

Re: 2+2 = 3.99 recurring??

I heard it on QI something about when you add 2 and 2 using log it turns out to be 3.99 recurring.

Offline

#6 2006-07-20 21:46:18

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: 2+2 = 3.99 recurring??

well 3.99 recurring equals 4 anyways, so it doesnt matter, the only way it matters, is that it could change a result by a very small amount due to calculators, computers whatever not being able to calculate with infinately long fractional parts


The Beginning Of All Things To End.
The End Of All Things To Come.

Offline

#7 2006-07-21 01:16:21

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: 2+2 = 3.99 recurring??

well the difference between 3,9999...and 4 is 0.00............001, but since the 1 is at the end of an endless number of 0s, 3,99999 = 4, right?

Offline

#8 2006-07-21 03:29:11

ben
Member
Registered: 2006-07-12
Posts: 106

Re: 2+2 = 3.99 recurring??

No! Please don't write 0.0.......1, it has no meaning. Rather write 3.9..... or 3.99....... etc. The elipses imply an infinite number of nines. As we really don't have time to write out 9 an infinite number of time by convention we call that number 4.

Or, to be cute. If 3.99.... is not equal to 4, you have two choices. Either tell us what lies between these two numbers, or you assert that the real line is not continuous. I promise you, either option will result in a very serious headache.

Offline

#9 2006-07-21 03:53:20

ben
Member
Registered: 2006-07-12
Posts: 106

Re: 2+2 = 3.99 recurring??

numen wrote:

Is (...) supposed to mean limit? Now that's new.

Besides, I think this would be a valid argument against it:


Certainly not! Here's your homework.

Is 1/3 rational or not?

Is 1/3 + 1/3 +1/3 equal to 1?

Is 1 rational or not?

What is the decimal represenation of 1/3?

Of 1/3 + 1/3 + 1/3?

See where it goes?

Offline

#10 2006-07-21 05:29:26

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: 2+2 = 3.99 recurring??

ben wrote:

Either tell us what lies between these two numbers

for me it sounds like the number that is between those numbers are 10^(-∞) tongue

Offline

#11 2006-07-21 05:33:52

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

Re: 2+2 = 3.99 recurring??

Who was the smart guy who proved you couldn't have something on both ends of an infinite number of things, ben?
I've heard you can't make a set that way, but who proved it?  What mathematician?

Didn't you know infinity isn't really everything?  It's just everything about something.

Last edited by John E. Franklin (2006-07-21 05:35:13)


igloo myrtilles fourmis

Offline

#12 2006-07-21 06:12:45

ben
Member
Registered: 2006-07-12
Posts: 106

Re: 2+2 = 3.99 recurring??

John E. Franklin wrote:

Who was the smart guy who proved you couldn't have something on both ends of an infinite number of things, ben?

Dunno, Cantor maybe? He was interested in different sorts of infininty (!). He also went bonkers......

I've heard you can't make a set that way

You mean you can't have an infinite set? Of course you can

Kurre wrote:

for me it sounds like the number that is between those numbers are 10^(-∞)

I realize this was tongue in cheek, but it does show a common difficulty. Infinity can not be treated like "just some hugely big number". It is an abstract concept.

Like, infinity is what there is when you run out of counting numbers. Huh? You simply cannot manipulate it as though it were an element of R. So 10^(-∞) has no meaning. Sorry to appear tough!

(Actually, there is a real line, called, I think, the projective real line, or something like that, which does precisely that. Try googling, as I know FA about it)

Offline

#13 2006-07-21 06:36:03

numen
Member
Registered: 2006-05-03
Posts: 115

Re: 2+2 = 3.99 recurring??

ben,

1/3+1/3+1/3=1.

But note that 0,333... is not 1/3, it's an approximation.
It's just slightly less than 1/3. It approaches 1/3 like a limit, but never becomes it. 4 is in Q, 3,999... isn't.

Keep in mind that an infinitely large amount of zero's adds up to zero, while infinitely small is larger than 0. The difference is infinitely small, thus not equal to zero, so they're strictly not the same.


Bang postponed. Not big enough. Reboot.

Offline

#14 2006-07-21 06:57:36

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: 2+2 = 3.99 recurring??

If you are talking about infinity without talking about limits, then you aren't talking about the real numbers.  The only way to even have the concept of infinity with a real number is to use limits.

It can be proven that any number with repeating decimals is rational, in the reals.  I'll have to find this proof later on and post it here.  But 3.999... certainly has repeating decimals, so it must be rational.  So there has to exist and a and b such that a/b = 3.9999...

Further more, there aren't any real numbers between 3.999... and 4.  But between every two distinct real numbers lies another real number, specifically their average.  So it must be the case that 3.999... and 4 are not distinct.

When you deal with things such as infinitesimals, then 3.999... does not equal 4.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

#15 2006-07-21 07:04:31

ben
Member
Registered: 2006-07-12
Posts: 106

Re: 2+2 = 3.99 recurring??

numen wrote:

But note that 0,333... is not 1/3, it's an approximation.

No, it's the best our notation can render. Everyone except you understands that. The onus is on you to show which is an approximation. 1/3 or 0.33.... Mathematically, that is.

It's just slightly less than 1/3.

Watch your logic here. If you want to argue that 1/3 in not equal to 0.33.. you are not allowed to use that as an assumption.

It approaches 1/3 like a limit, but never becomes it.

It has absolutely nothing, nothing to do with limits.

Anyway, I'm running away from this thread. The like get me cross, and I have no wish to fall out with anyone here.

Offline

#16 2006-07-21 07:29:17

numen
Member
Registered: 2006-05-03
Posts: 115

Re: 2+2 = 3.99 recurring??

If that proof is true, then 3,999... is in Q.

If 1,999... = 2, then we could multiply both by 10000...
thus, 1999... = 2000... right?

Actually, I don't think I'll continue from here, I'm really confused now... tongue


Bang postponed. Not big enough. Reboot.

Offline

#17 2006-07-21 09:00:36

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: 2+2 = 3.99 recurring??

numen wrote:

If that proof is true, then 3,999... is in Q.

If 1,999... = 2, then we could multiply both by 10000...
thus, 1999... = 2000... right?

Actually, I don't think I'll continue from here, I'm really confused now... tongue

1.999.... = 2
1999.999... = 2000

remember the infinate number of decimals


The Beginning Of All Things To End.
The End Of All Things To Come.

Offline

#18 2006-07-22 17:26:29

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: 2+2 = 3.99 recurring??

Infinite digits is tricky. The only way to deny 3.999...=4 is to say infinite digits and infinite division is illegal, meaningless or unpratical.

Humans can set up a real axis, but nature wouldn't bother to do the trick.


X'(y-Xβ)=0

Offline

#19 2006-07-22 19:45:20

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: 2+2 = 3.99 recurring??

3.99... is a number, like the others, and it has some interestion properties, which allow to "calculate" it:
I'm writing 3.(9) instead of 3.999... , because it doesn't use this "...".
For example:
1.101010(234)=1.101010234234234...,

Call 3.(9) C:
C=3.(9)

Now C has an infinite number of digits, but it also have a property, given with the definition:
C 10^n has the same fractional part as C, so we may exclude it, by substracting:
For example:
10C-C=39.(9)-3.(9)=36.(0)=36
C=36/9=4.
That kind of method is used for evaluating different kinds of numbers:
For example






Last edited by krassi_holmz (2006-07-22 19:48:29)


IPBLE:  Increasing Performance By Lowering Expectations.

Offline

#20 2006-07-24 02:36:42

Heldensheld
Member
Registered: 2006-07-01
Posts: 25

Re: 2+2 = 3.99 recurring??

My first contribution smile .

One number can simply not equal another!

All the numbers are different!

Is that not right?

Offline

#21 2006-07-24 04:45:01

numen
Member
Registered: 2006-05-03
Posts: 115

Re: 2+2 = 3.99 recurring??

Yes, but these are the same number smile (yes, I've realized that now, sorry!)
They're just presented differently, 4 or 7-3 or 3,999... but if a number like 3 would equal another number 4 you'd have big problems!


Bang postponed. Not big enough. Reboot.

Offline

#22 2006-07-24 07:24:18

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: 2+2 = 3.99 recurring??

3 = 4 in the integers mod 1.  In other words, they both have the same remainder when divided by 1, so they are equal.

In the same sense, 2 = 5 in the integers mod 3, because when divided by 3 they both have a remainder of 2.

But overall, what you should understand that whether or not numbers are equal depends on why number system you are talking about.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

#23 2006-07-24 07:48:50

ben
Member
Registered: 2006-07-12
Posts: 106

Re: 2+2 = 3.99 recurring??

numen wrote:

Yes, but these are the same number smile (yes, I've realized that now, sorry!)
They're just presented differently, 4 or 7-3 or 3,999... but if a number like 3 would equal another number 4 you'd have big problems!

Hey, don't be sorry. This is still a deep discussion in the philosophy of math! In what sense is 2 + 3 the same as 4 + 1? They share only one element, the = sign. In philosophy, it is by no means a trivial question whether 0.99... = 1 or not.

In math, we just define it thus, and move on to the really good stuff

Offline

#24 2006-07-24 14:40:35

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: 2+2 = 3.99 recurring??

The only explaination for the phenomenon that comes to my mind immediately is this:

For one of the steps, you multiply n by 10, and assume you have a new digit from nowhere in saying 10n = 39.999... In reality, when we multiply a number by 10, we lose a decimal place, like so:

10 × 3.99 = 39.9.

So with this thinking, in our problem here we would have 10n = 39.999...9, where this 9 at the end is one decimal place closer to the ones place than the original "final" 9. (of course this is a foolish thing to say, how can there be a "final" 9 in an infinite sequence of 9's? But this is all for the sake of argument, and in the end it seems to be a valid explaination, so follow me still) Now this is a different number than the 39.999... erroneously declared earlier. If we subtracted 3.999... from this number, we'd get 35.999...91. (once again, I realize that it's foolish to assume this can be done with an infinite decimal, but I feel that it shows the real workings of this problem) Divide this by 9 and you get n = 3.999..., and everyone is happy and the world of mathematics is safe and sound once again. Now, the procedure I described is something we aren't sure works for an infinite amount of digits (or maybe it does and I don't know that is does for sure), so let's see how this works for a finite amount of digits. Let m = 3.99. Then 10m = 39.9, and subtracting m, 9m = 35.91. Divide both sides by 9 and you get m = 3.99, as expected. No matter how many decimal places of 9's we have after the 3, we can still work this out and m doesn't magically round up, unless you're using a calculator that rounds up after so many digits. Assuming that this trend continues into the realm of infinite 9's after the decimal is a lot safer than assuming 3.999... = 4. I think this is the correct way to look at the problem.

Offline

#25 2006-07-24 16:13:08

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: 2+2 = 3.99 recurring??

Assuming that this trend continues into the realm of infinite

There are very few, if any, mathimatical properties that continue when you go from finite to infinite.  This is not one of them.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

Board footer

Powered by FluxBB