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Crazy Math
prove the following:
3.99999... = 4
(by using '...' i mean that the 9's are repeating)
3.99999... = 4
(by using '...' i mean that the 9's are repeating)
Answer
3.9999... = X3.9999... * 10 = 39.9999.... = 10X
10X - 3.9999... = 9X
39.9999... - 3.9999... = 36 = 9X
36 / 9 = 9X / 9 = 4
therefore, X = 4
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Comments
it wil never be true it will always be a tad bit less
Nice try though!
Nice try though!
Well, you can't just leave a comment like that and not point out where the flaw in the logic is (which is actually part of the fun of this puzzle). And yes, i know that it will never actually be exactly equal to four...
it's fun though, isn't it? =)
if X = 3.9999 then 10X - X is not equal to 36!!!!!
dvsatya, you do realize that the equation calls for 3.9999... with the 9's repeating forever, right? not just 3.99999
I think it rocks! Whats it to do with though? Multiplying infinity by 10 or what? Fantastic!
the main thing is the magnitude of infinity. the concept of infinity, what a brain-buster, huh?
Yes the line 10X is where the problem is, because you can't multiply infinity by 10. Its very clever though!
grabeeeee!wheww.....amazing like spider man but not real..
There seem to be a lot of people on this page who haven't taken a math class in a while. There is absolutely nothing wrong with this solution. You CAN multiply something by infinity - it's done all of the time. Plus, the very concept of infinity states that 3.999... is equal to four, because there is no number that you can possibly subtract from 4 to get 3.999...
bluetwo, you shouldn't have backtracked with your comments. There is nothing wrong with your teaser (except for the fact that you don't really understand it).
bluetwo, you shouldn't have backtracked with your comments. There is nothing wrong with your teaser (except for the fact that you don't really understand it).
bobrt, thanks for the comment. I believe there is a problem with doing any sort of arithmetic operation with infinity. no matter what you do, they will never be equal, although the difference is so infinitessimal as to not have any sort f significance. the question then becomes, if you subtract infinity from infinity, is that really zero? hmmmm....
The very concept of infinity means 3.99999..... will NEVER be EQUAL to four
You can't perform operaations on infinity. Infinity + 1 *100 etc.
Assumptions are made all the time when calculations are needed involving infinity, but that does not mean 3.9999 = 4
You can't perform operaations on infinity. Infinity + 1 *100 etc.
Assumptions are made all the time when calculations are needed involving infinity, but that does not mean 3.9999 = 4
Cath, you're absolutely KILLING me! The very concept of infinity means that 3.999... DOES equal four. THAT'S WHAT INFINITY IS! You know the teaser about walking half the distance to a door, and then walking half of that distance, etc.? In real life you would never make it to the door. Mathematically, however, you DO make it to the door AT INFINITY!!!! That's what infinity is - it's the number that takes that final step. It's the number that makes 3.999... EXACTLY equal to four. Do me a favor - take this problem to a calculus teacher. He or she will show you that there is NOTHING mathematically wrong with it.
Oh, and by the way - at no point in this proof is an operation being done on infinity. One step is multiplying 10 times 3.999...(=4), not 10 times infinity.
Oh, and by the way - at no point in this proof is an operation being done on infinity. One step is multiplying 10 times 3.999...(=4), not 10 times infinity.
No sorry I can't agree. Infinity is a point that can never be reached. 3.999... will never reach 4. In calculus do you not let things tend to a number?? limits as something tends to infinity or zero. 3.99... will approach a limit of 4, as the number of decimal places tends to infinity but that is not the same as saying 3.99.. is equal to 4
I'm done trying to explain this. Have a nice day.
Fair enough, this could go on for a while, so we can just stop, but I can't understand why you think what you do. Even a definition of infinity gives something that will never be reached (in this example 4). Nothing personal by the way, I think some of your teasers are very interesting, thought provoking! If you want to post something on the forum about this ...?
aaahhhh, i'm going to have to admit that i actually made a mistake (i can't really believe it myself, as i am a mostly flawless individual) ;) anyway, cathal, i think we made this to be much more complicated than it should be. according to the simple algebraic rules for reducing a repeating decimal to a fraction, this proof should be valid. it took searching the internet and a harsh reprimand from a buddy of mine (BS in some sort of math in college)...
The solution is absolutely right. The implicit point is that "equality" as a concept is misunderstood. Two numbers are considered to be equal if they are mathematically indistinguishable. It is impossible to distinguish between 4 and 3.9999999... as the difference between them is always less than any positive number you could choose. (In otherwords the limit of 3.999999... is 4). Indeed if you ever take a class in "real analysis" you will learn that 3.999999... and 4 are inded the same number.
A better illustration of the question is the following: 1/3 = 0.3333... Next multiply both sides of the equation by 3. This results in 1= 0.99999999.....
good comments... exactly what i was getting at in my last post. ah, algebra, where did you disappear to?
ymiroc, you're my HERO! Why in the heck didn't I think of that? (please don't answer that question)
What is infinity plus infinity??
Blue, You say you can't perform algebra on infinity then you ask what infinity minus infinity is.
By definition the ... goes on forever, so 1/3 = 0.33... is correct (in theory) But that is the point I am trying to make, 1/3 will never reach one third in decimals so really 0.333.... being equal to 1/3 is not true. Its a limit it tends to. In "real analysis" I'm sure 0.3333333333 = 1/3 (no dots) or maybe 20 decimal places or whatever you need for a particular real world application. I study a bit of engineering maths and we round off and assume etc in real terms. But I still don't agree with what you three guys think. I think we are mixing theory with real world assumptions or applications
Blue, You say you can't perform algebra on infinity then you ask what infinity minus infinity is.
By definition the ... goes on forever, so 1/3 = 0.33... is correct (in theory) But that is the point I am trying to make, 1/3 will never reach one third in decimals so really 0.333.... being equal to 1/3 is not true. Its a limit it tends to. In "real analysis" I'm sure 0.3333333333 = 1/3 (no dots) or maybe 20 decimal places or whatever you need for a particular real world application. I study a bit of engineering maths and we round off and assume etc in real terms. But I still don't agree with what you three guys think. I think we are mixing theory with real world assumptions or applications
Cathal, how incredibly huge and fragile is your ego? YOU ARE WRONG!!!!!! The concept that you are failing to grasp is that BY DEFINITION, infinity IS forever. Infinity is a word developed by man to describe a situation that NEVER ends. Change your wording a little...instead of saying that 0.333... will NEVER be equal to 1/3, say that 0.333... will take FOREVER to equal 1/3. Guess what - infinity IS FOREVER!!! So by definition if you say that the 3's go on forever, then you are saying that 0.333... equals EXACTLY 1/3. You're right - in reality a human being could not possibly write down enough 3's to make this equation true. But the "..." is saying that an infinite number of 3's are being used. It's saying that "forever" has been reached. Not approached. REACHED. Now where's my ulcer medication...?
...oh, I would like to take this opportunity to apologize for my first comment (referring to people not taking a math class in a while). I guess that with all of the idiotic comments made on this web site, I get a little swept up in the moment sometimes. I would also like to state for the record that I have nothing against Cathal - we're actually real good buddies. We grew up together in south central Reykiavik, we spend every other weekend playing canasta with our proctologists, and I'm even dating his half sister Lulubelle. Hey, remember that time we spray painted "Stalin was a woman" on the Kremlin? Ah...good times...good times.
hahaha, all this reminds me of my college (or university, for some of you) days when my roommates would argue over the purpose of math in engineering. my one roommate (math geek extraordinaire) said that math was the foundation of engineering; we countered that math was merely a tool. he didn't like that and we spent many hours arguing with each other... fun times... =)
OK I'm not going to read through everything again! Quick message, I'm sorry I think you are all right! I was talking to someone, who said in relation to limits :"until we define an infinite decimal in such a way, it
has no meaning" I think this is the bit that I was having problems with. He also said "in THEORY we can show that it APPROACHES 1, and that is what we mean by its value."
I think I get it now, rather than saying 3 is equal to 4 (which is obviously wrong) what is actually being said is talking about the limit of 3.9999... not literally 3.99
has no meaning" I think this is the bit that I was having problems with. He also said "in THEORY we can show that it APPROACHES 1, and that is what we mean by its value."
I think I get it now, rather than saying 3 is equal to 4 (which is obviously wrong) what is actually being said is talking about the limit of 3.9999... not literally 3.99
Sorry by the way to anyone who got annoyed over this. I don't know why you think I have an "ego" problem. I just didn't agree, and to be honest I still don't really agree with your arguments. I still don't think you can say 10*3.99... but its easy to say 10* the limit of 3.99.. which was what I misunderstood in the question. In my opinion, 3.99... will never equal 4, but in maths if you write 3.99... = 4 thats OK because you mean a limit.
Another way is this:
1/9 =.11........
9/9 =.99........
9/9 = 1
I hope you can understand it.
1/9 =.11........
9/9 =.99........
9/9 = 1
I hope you can understand it.
Fun teaser. I haven't taken a math class in over ten years, but I remembered the fraction thing, so I understood the point of the teaser. From the comments posted, however, I think perhaps my mathematical ignorance is indeed bliss.
You know you could always just round 3.99 to 4 and there you get 3.99 to equal 4 and there wasn't a whole lot of thought put inot that either.
Ah, dont worry everyone, its understandable but puzzling unless you accept it. But interestlingly enough, my calculator decided that 3* (3)^2 = 81.00001. Go figure.
Simply put, when you multiply it by 10, instead of still having 3.9999... you get 3.9999...990. When you divide both sides again, you will get the same answer that you started with.
Hey kids! Don't try to learn Mathematics from the comments here - you'll end up seriously disturbed. Bobbrt is correct up to a point except that he has indicated that you can do mathematical operations on infinity, thereby suggesting that infinity is a number. Infinity is a concept. It is not a number.
For any real number x times 2 = 2x. If there was such a thing as a number infinity then doubling it would still give infinity and therefore by the rules of Algebra, infinity is NOT a real number.
The real elephant in the room here is the definition of a repeating decimal which is that the digit keeps on repeating TO INFINITY. That's the concept of infinity, not the number infinity. Here is the commonly accepted "proof" that 0.99999999.. (repeater) is exactly equal to 1.
Let x = 0.999999999....(repeater) [1]
Then 10x = 9.99999999...(repeater) [2]
{You will note here that you cannot shift the decimal point one place to the left and claim that you have one less than an infinite number of nines! It's still infinite!!}
Subtracting [1] from [2]
9x = 9
x = 1
There are other ways such as considering it as the sum of an infinite series.
0.9999.....= 9/10 + 9/100 + 9/1000 +...
Using the sum to infinite of a geometric series
S = a/(1 -r) = 0.9 / 0.9 = 1
Of course if you still think of infinity as a "really big number" then you will probably think think that the sum to infinite of a geometric series is also "approximately" right. NO! It is exactly right otherwise Physics would work! {See the comments on other teasers about Zeno's paradoxes - search Zeno on Braingle}
For any real number x times 2 = 2x. If there was such a thing as a number infinity then doubling it would still give infinity and therefore by the rules of Algebra, infinity is NOT a real number.
The real elephant in the room here is the definition of a repeating decimal which is that the digit keeps on repeating TO INFINITY. That's the concept of infinity, not the number infinity. Here is the commonly accepted "proof" that 0.99999999.. (repeater) is exactly equal to 1.
Let x = 0.999999999....(repeater) [1]
Then 10x = 9.99999999...(repeater) [2]
{You will note here that you cannot shift the decimal point one place to the left and claim that you have one less than an infinite number of nines! It's still infinite!!}
Subtracting [1] from [2]
9x = 9
x = 1
There are other ways such as considering it as the sum of an infinite series.
0.9999.....= 9/10 + 9/100 + 9/1000 +...
Using the sum to infinite of a geometric series
S = a/(1 -r) = 0.9 / 0.9 = 1
Of course if you still think of infinity as a "really big number" then you will probably think think that the sum to infinite of a geometric series is also "approximately" right. NO! It is exactly right otherwise Physics would work! {See the comments on other teasers about Zeno's paradoxes - search Zeno on Braingle}
Sorry - shift the decimal point one place to the RIGHT
Sorry again - otherwise Physics would NOT work.. was what I meant to type - Whew
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