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(Some Guy)   6/2(1+2) = 9 no no, 6/2(1+2) = 1 no no, 6/2(1+2) = 2   (mathmagical.co.uk) divider line 821
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30203 clicks; posted to Main » on 30 Mar 2012 at 9:56 AM (2 years ago)   |  Favorite    |   share:  Share on Twitter share via Email Share on Facebook   more»



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2012-03-30 06:51:46 PM

Kahabut: FizixJunkee: When I see

6/2(1+2) =

I interpret it as

6
-------------- =
2 (1 + 2)

which would be equal to 1.

That's idiotic. Lets go step by step for a minute.
6/2(1+2)=
which can be written
6/2*(1+2)=
which can be written
3*(1+2)=
which can be written
3*3=

Order of operation is irrelevant if you can read the farking equation. Didn't anyone learn simplified math? Operators make for sections, sections can be solved independent of each other outside of parenthesis.

You looked at 6/2(1+2)= and decided that it was actually 6/(2*(1+2))= that makes no sense, and it's just simply wrong. The syntax for this stuff was worked out before you were born, and it's been pretty straight forward since then.

It is kinda disturbing that you weren't taught this in school, at least not correctly.


Or

6/2(1+2) =

can be written as

6/(2(1+2)) =

which becomes

6/(2(3))=

which is

6/6 = 1


To me, the arithmetic, as presented, is ambiguous. I'd never plug something like that into my calculator (or Mathematica, for that matter). Heck, I can't remember the last time I even used a calculator.
 
2012-03-30 07:07:48 PM

FizixJunkee: To me, the arithmetic, as presented, is ambiguous.


It's not ambiguous, though. The convention is perfectly specified: multiply and divide associate left-to-right. It's just that the rules sometimes come into conflict with people's intuition, and this is one of those cases. You are seeing an explicit division but an implied multiplication, and your intuition is to apply to implicit operation first.

It makes some sense: things that are written closer together ought to be considered a tighter group. Unfortunately (perhaps), that's not what the convention is. The convention is left-to-right, whether the operatior is implicit or explicit. All serious mathematical texts treat it this way, and all serious programming languages I know of that use infic operators except one do it that way, Mathematica included.
 
2012-03-30 07:08:26 PM

Grables'Daughter: Thus, the reason that girls are bad at math.


Huh. So that's why I think Kleenex comes in b-cup and c-cup sized boxes.
 
2012-03-30 07:13:29 PM

FizixJunkee: Or

6/2(1+2) =

can be written as

6/(2(1+2)) =

which becomes


No, it really can't.

6
-----
2(1+2)

can be written as

6/(2(1+2)

But it can't be written as
6/2(1+2)

Where there is no operator between the parentheses, multiplication is always assumed, so

6/2*(1+2)

It's assumed that unless there's an additional set of parentheses, the operation has been escaped.

While division, fractions, and multiplication are all logically the same, they are NOT presenting the same information syntactically. In other words, a dividend is NOT a numerator, and a divisor is NOT a denominator.
 
2012-03-30 07:13:50 PM
The derp is strong here. No wonder the current generation has failed to engineer ANYTHING.
 
2012-03-30 07:16:38 PM
OK, I now understand the how people get 1 or 9, and I even understand why 9 is correct (though I disagree with people who think the syntax is obvious).

Can someone explain what rules would result in 6/2(1+2) =2? That one still stumps me.
 
2012-03-30 07:21:27 PM

FizixJunkee: Or

6/2(1+2) =

can be written as

6/(2(1+2)) =


No, it can't. You're moving the second piece from the numerator into the denominator by that.

6/2(1+2) could also be expressed as (6/2)(1+2) because it doesn't change order of operations, but it absolutely is NOT equivalent to 6/(2(1+2)).

That's the same reason that 1/3*3/4 is equal to 1/4, and not 4/9. Unless parentheses are explicitly used, you perform operations from left to right.
 
2012-03-30 07:24:21 PM
I posted this to a few of my friends and the first answer I got back was "27". SMH.
 
2012-03-30 07:25:31 PM

Aunt Crabby: (though I disagree with people who think the syntax is obvious).


Just remember this:

Dr. Mojo PhD: a dividend is NOT a numerator, and a divisor is NOT a denominator.


A dividend (the left side of a division operation) is not read the same as a numerator (the top half of a fraction), and the divisor (the right hand of a division operation) is not read the same as a denominator (the bottom half of a fraction). Both operations will result in the same number, but the equation should never, ever be read the same.

Aunt Crabby: Can someone explain what rules would result in 6/2(1+2) =2? That one still stumps me.


Uh, nope. Can't help you with that one. Wolfram Alpha just tells me (6/2)(1+2)=2 is false, and 6/(2(1+2))=2 is also false. No matter how I consider it, I cannot possibly fathom how the hell they arrived at that, unless the iPhone kid has sausage fingers and poked the wrong operator, or just forgot one -- 6/(1+2) = 2, for example.
 
2012-03-30 07:30:15 PM

Aunt Crabby: OK, I now understand the how people get 1 or 9, and I even understand why 9 is correct (though I disagree with people who think the syntax is obvious).

Can someone explain what rules would result in 6/2(1+2) =2? That one still stumps me.


Comment at the bottom of TFA explains that the iPhone straight up ignores whatever number you put in front of the parenthesis, so its just doing 6/(1+2). I guess its an unsupported syntax, but Apple chose to hide the error message.
 
2012-03-30 07:38:32 PM
Maths.

;)
 
2012-03-30 07:39:35 PM

Gaumond: Just to reinforce the obvious, TI-89 says:

[img.photobucket.com image 600x800]

Anything else is just WRONG.


I was taught to always get rid of the parentheses first for PEDMAS. If you still have parentheses, you need to take care of that operation first, both inside and outside the parentheses. That calculator in the pic from the quote above is adding a * to change the equation from 6/2(1+2) = to 6/2*(1+2) = to get around the fact that implied multiplication is supposed to take precedence over normal multiplication and division. 6/2(1+2) = and 6/2*(1+2) = are not the same equation. The instructor in TFA put that second equation in his Acorn Risc PC. Of COURSE he got 9 from that second equation. I would have been interested to see the result of the first equation from the Acorn Risc PC.

Meet Us at the Stick: optikeye: a poorly presented equation is open to interpretations.

As it's written in the headline it's not poorly presented at all

6/2(1+2)

Order of Operation says perform the operation in parentheses first

6/2(3)

Now since the only operations left are division and multiplication, perform those from left to right


I bolded where I don't agree. For 6/2(3) you still have that set of parentheses around the 3. You can't move on to the rest of the equation until all of the parentheses have been resolved both inside and outside. 6/2(3) becomes 6/6 which equals 1.


I think it depends on how you were taught. It's more of a grammar question than a math question.
 
2012-03-30 07:41:20 PM

Aunt Crabby: OK, I now understand the how people get 1 or 9, and I even understand why 9 is correct (though I disagree with people who think the syntax is obvious).

Can someone explain what rules would result in 6/2(1+2) =2? That one still stumps me.


Fuzzy logic. Those iPhones have really advanced stuff going on in them.
 
2012-03-30 07:43:14 PM

Dr. Mojo PhD: Wolfram Alpha just tells me (6/2)(1+2)=2 is false, and 6/(2(1+2))=2 is also false.


*Disclaimer: I thought WolframAlpha considered varying radices without needing to specify. It returns 1+1=10 as false, so it clearly does not.
 
2012-03-30 07:43:22 PM

Aunt Crabby: OK, I now understand the how people get 1 or 9, and I even understand why 9 is correct (though I disagree with people who think the syntax is obvious).

Can someone explain what rules would result in 6/2(1+2) =2? That one still stumps me.


.
.
6/2(1=gibberish, so the computer is only picking up the +2.
 
2012-03-30 07:45:48 PM

Mind the Gap: I bolded where I don't agree. For 6/2(3) you still have that set of parentheses around the 3. You can't move on to the rest of the equation until all of the parentheses have been resolved both inside and outside. 6/2(3) becomes 6/6 which equals 1.


But since any parenthesis without an operator to its left is considered to have an implied multiplication operator, 6/2(3) becomes 6/2*(3) becomes 6/2*3. Interpreting that as 6/6=1 involves solving right to left.
 
2012-03-30 07:47:49 PM

Skyrmion: God forbid they ever read a real historian's version of the first Thanksgiving.


The problem with "real historians" is that they get major stuff just plain *WRONG* because they aren't familiar with the technology of the age they write about. I don't know how many times I've seen real howlers in books written by purely intellectual historians. Either that, or they are writing with a social agenda in mind (I'm looking at you, zombie Howard Zinn and Michael Bellesiles), and thus they only give part of the story, if they aren't committing outright fraud.

Now, you want to read a *REAL* historian? Go read "Sons Of A Trackless Forest" by Mark A. Baker. Not only does he do the intellectual side, checking original sources and such, but he duplicates everything they had, by hand, and goes out and lives like they did in order to see things in the correct frame of reference.
 
2012-03-30 07:50:52 PM

boomm: Aunt Crabby: OK, I now understand the how people get 1 or 9, and I even understand why 9 is correct (though I disagree with people who think the syntax is obvious).

Can someone explain what rules would result in 6/2(1+2) =2? That one still stumps me.

Fuzzy logic. Those iPhones have really advanced stuff going on in them.


They're so advanced, they know to make the same mistakes as their typical user.

/"SIRI" stands for "Stupid Idiots, Rubes, Imbeciles".
 
2012-03-30 07:51:01 PM

Dr. Mojo PhD: Aunt Crabby: (though I disagree with people who think the syntax is obvious).

Just remember this:

Dr. Mojo PhD: a dividend is NOT a numerator, and a divisor is NOT a denominator.

A dividend (the left side of a division operation) is not read the same as a numerator (the top half of a fraction), and the divisor (the right hand of a division operation) is not read the same as a denominator (the bottom half of a fraction). Both operations will result in the same number, but the equation should never, ever be read the same.


Oh, I'll remember it now forever. Mainly, I'll remember because I'll be wondering about my grade school math teacher's credentials (something I never questioned at the time). Apparently, it's never come up enough for me to have faced the issue before. If I am ever faced with such an equation (which I doubt) I'm going to trust my calculator and be damned.

I do like understanding it though. Thanks. I am more interesting in the reasoning than the actual math part. I'm a bit odd like that.

/I was strangely gleeful when I heard about lies my history teacher told me, but I assumed that my math teacher was playing it straight
//You can't trust anybody (except for strangers on Fark)
 
2012-03-30 07:52:01 PM

Dr. Mojo PhD: Aunt Crabby: (though I disagree with people who think the syntax is obvious).

Just remember this:


A kiss is just a kiss...
 
2012-03-30 07:54:32 PM
My LG won't even let me type it in as written... it requires you put in the multiplication operator before it will let you put in parenthesis. And it gets the right answer.
 
2012-03-30 07:56:30 PM

dittybopper: Dr. Mojo PhD: Aunt Crabby: (though I disagree with people who think the syntax is obvious).

Just remember this:

A kiss is just a kiss...


Hah, was just talking to my wife last night about Casablanca.

Math to Louis Armstrong to Dooley Wilson as Sam to Casablanca, with a little synchronicity thrown in, just to get a reference to the Police. Which brings us back full circle to our regular news cycle focus of George Zimmerman.

Well shiat.
 
2012-03-30 08:08:02 PM
Hi everyone!

Remember, 0.99999.... = 1!
 
2012-03-30 08:11:55 PM
And remember to switch when Monty Hall gives you the chance!
 
2012-03-30 08:14:28 PM

Dr. Mojo PhD: Mind the Gap: I bolded where I don't agree. For 6/2(3) you still have that set of parentheses around the 3. You can't move on to the rest of the equation until all of the parentheses have been resolved both inside and outside. 6/2(3) becomes 6/6 which equals 1.

But since any parenthesis without an operator to its left is considered to have an implied multiplication operator, 6/2(3) becomes 6/2*(3) becomes 6/2*3. Interpreting that as 6/6=1 involves solving right to left.


That's where you get into the "math grammar" question of whether implied multiplication gets grouped with the P portion of PEDMAS or the M. Interestingly, Texas Instruments sometimes give implied a higher priority and sometimes it doesn't, depending on the model.

Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators? (new window)

From that link:

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.

This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication is given the same priority.

Please see the graphing calculator guidebooks for additional information.
 
2012-03-30 08:17:08 PM

evaned: Hi everyone!

Remember, 0.99999.... = 1!


Oh you're a mean bastard. Here's the proof for this if anybody's curious and unaware of it already:

x = 0.99999...
10x = 9.99999...
10x - x = 9.99999... - 0.99999...
9x = 9
x = 1

(where all .99999... are assumed to trail into perpetuity as a repeating decimal)
 
2012-03-30 08:19:05 PM

Mind the Gap: That's where you get into the "math grammar" question of whether implied multiplication gets grouped with the P portion of PEDMAS or the M. Interestingly, Texas Instruments sometimes give implied a higher priority and sometimes it doesn't, depending on the model.


Implied is not a higher priority. Prioritizing implied multiplication assumes that the parentheses move. They don't. It's simply wrong. I know it's taught, and it's wrong.
 
2012-03-30 08:24:01 PM

profplump: Xcott: Spelling is not simply a communication skill; it is a rudimentary cognitive skill that a college student should be able to exhibit without help.

It is? What part of cognition requires that I can spell?


Presumably you meant to ask, "what part of cognition is required for spelling?" I suppose you need the ability to acquire, remember and recall short sequences, the ability to map phonics to letters, and the ability to recognize basic patterns in your native language, for example that a lot of words ending in "tion" are spelled "tion."

If you are unable to learn how to spell words, then you may be deficient in any of a number of basic cognitive abilities.

And which languages are the "right" ones that make me cognitively sufficient -- because spelling certainly isn't the same across languages, and most people don't know how to spell in most languages.

I'm surprised I even need to answer this, but obviously you need to exhibit the ability to spell words in your native language.

I'd argue instead that spelling is of no value except for the role it plays in interpersonal communication. Spelling is important for transcribing my thoughts to paper in a way that other people can understand, but I don't see what cognitive function spelling provides at all.

I think you completely misunderstand what other people are trying to tell you: regardless of the utility of spelling or adding, if you can't do it then you need remedial tutoring in very basic cognitive skills.

Furthermore, people who barely practice these skills to the point where they need a computer to help them are generally going to be lousy at any job that requires verbal or quantitative reasoning, both due to their inability to work when away from their crutch, and due to being mentally sedentary.
 
2012-03-30 08:29:36 PM

Dr. Mojo PhD: Oh you're a mean bastard. Here's the proof for this if anybody's curious and unaware of it already:


Vi Hart gives an awesome, beyond-the-normal exposition. (Just watch, like all of her channel, seriously. She even has a video about this very thread. I feel it's not quite on-point, but it's better than most of what you see. I linked it in an earlier post where I also demonstrated some awesome arithmetic skills to arrive at the answer 3 for this. (I think I was going to say 3*3=9 and forgot a step.)) Her video is not as rigorous as some of the explanations of 0.9999...=1 thing, but it goes beyond them in ways I don't think I've seen nearby it before, hinting at things like hyperreal numbers.
 
2012-03-30 08:35:37 PM

profplump: Xcott: Calculators, like cars, are great for the general population; but if your major and career requires you to be good at mathematics, then you must have skills that you don't get by having a machine do your work for you.

I'm saying you can do mathematics efficiently without necessarily being a to do arithmetic in your head, because tools exist to do that arithmetic quickly and reliably outside your head.


Earlier, I gave an example of mentally working out the temperature of a 1200W electric oven while driving home. How could I do that if I had to use a calculator or computer?

Clearly those tools are not that efficient for simple arithmetic, once you factor in the time penalty you pay having to go find a crutch every time you need to know what 17*3 is.
 
2012-03-30 08:36:36 PM
Oh, by the way, here's another question. I will withhold my motivation for why I'm asking this until later.

What is the value of 2(1+2)^3?

Does your answer change if I actually write it nicely?
i.imgur.com
 
2012-03-30 08:37:21 PM

evaned: Vi Hart gives an awesome, beyond-the-normal exposition. (Just watch, like all of her channel, seriously.


I got into Vi Hart (not literally) a while back , maybe a week or two before a few Vi Hart links started showing up on Fark (there's that tingling sense of synchronicity again). They're always great, amusing, and highly entertaining stuff.

And in the interest of many a disclaimer:

Dr. Mojo PhD: Here's the a proof...


While it is the proof in the same sense that when I say "are you taking the car" to my wife I am referring to our car, it in no way means to imply that our car is the only car, nor the only one she can take. Moving right along, I picked that one only because it was the easiest to do in basic ASCII, and because it's fairly straight-forward. Others may like other ones, but I always liked the simple and the elegant.
 
2012-03-30 08:37:34 PM

evaned: Oh, by the way, here's another question. I will withhold my motivation for why I'm asking this until later.

What is the value of 2(1+2)^3?

Does your answer change if I actually write it nicely?
[i.imgur.com image 68x20]


54.
No.
 
2012-03-30 08:44:09 PM

evaned: Oh, by the way, here's another question. I will withhold my motivation for why I'm asking this until later.

What is the value of 2(1+2)^3?

Does your answer change if I actually write it nicely?
[i.imgur.com image 68x20]


54, and no.

2(1+2)^3
2(3)^3
2*3^3
2*27
54

Regardless of if you use the caret to indicate superscript or just write it as superscript, I get 54.
 
2012-03-30 08:44:25 PM

Dr. Mojo PhD: While it is the proof in the same sense that when I say "are you taking the car" to my wife I am referring to our car, it in no way means to imply that our car is the only car


Oh, sorry; I wasn't at all trying to say "there's more than one way to skin a cat". The link was directed at anyone who was interested; replying to you was just a bit of a springboard.

/I'm leaving for a couple hours. After I get back, hopefully I'll have an answer to my 2(1+2)^3 question from a couple people from both camps.
 
2012-03-30 08:49:20 PM

Xcott: Furthermore, people who barely practice these skills to the point where they need a computer to help them are generally going to be lousy at any job that requires verbal or quantitative reasoning, both due to their inability to work when away from their crutch, and due to being mentally sedentary.


You are making an assumption about people who use technological aids and the many types of cognitive skills a person may develop. I'm dyslexic and dysgraphic, but I compensate with other abilities. Some of these skills involve higher reasoning. Some involve running Spellcheck and having a proofreader for important documents. I suppose hanging out on Fark makes me mentally sedentary, but I use my technology/proofreader "crutch" for the same reason a crippled person uses a wheel chair. For what it's worth, I did try to practice spelling and have seen experts about my issues as a child. I still can't spell consistently, though if you ask me the correct spelling of most words I will probably know the letters and even say them out loud in the correct order.

Despite the dour predictions of your hypothesis about people who need computers to spell and do math, I'm fairly high functioning given my situation and I have been successful in my career (despite time spent on Fark). I can even be logical when properly motivated. Though people should be taught basics, using technological or personal aids is not a sign of weakness. It is a sign of accepting reality and doing the best you can with what you have.

/Most offices have computers, so mental crutches are handy for all types of people.
//Verbal reasoning doesn't always equate to written work either.
 
2012-03-30 08:49:56 PM

evaned: Oh, sorry; I wasn't at all trying to say "there's more than one way to skin a cat". The link was directed at anyone who was interested; replying to you was just a bit of a springboard.


No worries. I'm trying to say there's more than one way to skin a cat. A proof is a proof is a proof, and if there's twenty ways to prove things that are all valid, well, they're all proofs. I like blue, you like red, they're both colours, tomayto and tomahto are the same fruit, even if some people call it a vegetable, and that vegetable is the fruit, and all fruits are vegetables anyway, cellular-wise, if not culinary-wise, that was better than sex, I need a cigarette.
 
2012-03-30 09:01:56 PM

Dr. Mojo PhD: evaned: Oh, by the way, here's another question. I will withhold my motivation for why I'm asking this until later.

What is the value of 2(1+2)^3?

Does your answer change if I actually write it nicely?
[i.imgur.com image 68x20]

54, and no.

2(1+2)^3
2(3)^3
2*3^3
2*27
54

Regardless of if you use the caret to indicate superscript or just write it as superscript, I get 54.


I think both the 6/2(1+2)=9 and 6/2(1+2)=1 crowds will agree on the answer to 2(1+2)^3 is 54, since exponents come before multiplication, whether the multiplication is implied or not. The only way to get a different answer than 54 is to group and multiply the 2(3) first rather than using the exponent on (3)^3 and that would clearly be out of operational order.
 
2012-03-30 09:04:29 PM
If you came up with something other than 1 you fail at reading and math. Your professors should be proud.
 
2012-03-30 09:10:42 PM
Grandmother of nine shoots hole in one.

/TSG roundup, please
 
2012-03-30 09:18:48 PM
Wow. Hipsters incorrectly explaining third grade math.....
 
2012-03-30 09:20:01 PM

dittybopper: Either that, or they are writing with a social agenda in mind (I'm looking at you, zombie Howard Zinn and Michael Bellesiles), and thus they only give part of the story, if they aren't committing outright fraud.

Now, you want to read a *REAL* historian? Go read "Sons Of A Trackless Forest" by Mark A. Baker. Not only does he do the intellectual side, checking original sources and such, but he duplicates everything they had, by hand, and goes out and lives like they did in order to see things in the correct frame of reference.


I don't think it's fair to put Zinn and Bellesiles in the same category, though I do agree with you on the problems of political bias in general. I haven't heard of Mark Baker, but he sounds interesting. I don't know if you're interested in military history, but I always liked Hans Delbruk's writing. He had a very analytical approach and would have made a good natural scientist.
 
2012-03-30 09:20:53 PM

evaned: Oh, by the way, here's another question. I will withhold my motivation for why I'm asking this until later.

What is the value of 2(1+2)^3?

Does your answer change if I actually write it nicely?
[i.imgur.com image 68x20]


Even if I use the incorrect "multiply first then divide" 6 step version of PEMDAS that I had been using, you would have 2 * 27 because it still starts out parenthesis then exponents.The only way you could get a different answer is if you multiply before you do the exponent. I don't think anyone was taught that. I suppose it's about syntax though. Maybe it would matter to people who argue an implied parenthesis to multiplication in front of the parenthetical expression.

/People tell me math and science are more straightforward than law and politics
//Is nothing ever simple?
 
2012-03-30 09:23:01 PM

Aunt Crabby: //Is nothing ever simple?


The answer, of course, is yes.
 
2012-03-30 09:39:14 PM
i1169.photobucket.com

HP 48 GX throws poorly written equations back in your face.
 
2012-03-30 09:54:11 PM
BEDMAS

B
rackets
Exponents
Division
Multiplication
Addition
Subtraction
 
2012-03-30 10:05:35 PM
Handycalc on my Android phone is sufficiently well-designed that it won't even let me enter that ambiguous piece of crap equation. It forced me to pick whether the (1+2) went in the denominator, or out to the side.
 
2012-03-30 10:21:10 PM
The calculator is correct. The syntax is poor. You never write a fraction to be multiplied by some other function that way. To get 9, you would write it like 6/2*(2+1). The * forces 6/2 to be seen as an entity which is then multiplied by (2+1) to get 9. 6/2(2+1) is interpreted as 6/(2(2+1)). Which would satisfy your mnemonic. Any time you see a parentheses or brackets, you solve what is inside and multiply it by any number directly in contact with it, FIRST.
a(b+c)=(a(b+c))=(ab+ac) 6/2(2+1) is actually 6/(2(2)+2(1))=6/(4+2)=6/6=1.
 
2012-03-30 10:25:29 PM
Not every calculator and/or programing language uses the same notation as they do in standard algebra. Consult the manual or be explicate about what you want.

BurnShrike: Think of how stupid the average person is, and realize half of them are stupider than that.

-George Carlin


It is ironic that this quote includes a basic math error.
 
2012-03-30 10:25:58 PM

Dr. Mojo PhD: evaned: Hi everyone!

Remember, 0.99999.... = 1!

Oh you're a mean bastard. Here's the proof for this if anybody's curious and unaware of it already:

x = 0.99999...
10x = 9.99999...
10x - x = 9.99999... - 0.99999...
9x = 9
x = 1

(where all .99999... are assumed to trail into perpetuity as a repeating decimal)


My simple proof for this.

1/3 = 0.333...
3 * 0.333... = 0.999...
3 * 1/3 = 3/3
3/3 = 1
0.999... = 1
 
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