Spot The First Fallacy!
In which "equation" is the first mistake made?
1 . 0 0 = 0 0
2 . 0 0 = 1 0 0 − 1 0 0 1 0 0 − 1 0 0
3 . 0 0 = 1 0 ( 1 0 − 1 0 ) ( 1 0 2 − 1 0 2 )
4 . 0 0 = 1 0 ( 1 0 − 1 0 ) ( 1 0 + 1 0 ) ( 1 0 − 1 0 )
5 . 0 0 = 1 0 1 0 + 1 0
6 . 0 0 = 2
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9 solutions
Only step 4 to 5 should be wrong since you are cancelling (10-10) , i.e. , 0's.
Step 1 might be mathematically incorrect but logically it says that 2 0/0 ' s have the same value (whatever they may be) which is correct as long as you don't start to compute it (i.e. logically)
...Which is to say that if instead step one would have been 0^0=0/0, in that case it would be both logically and mathematically unsound.
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indeterminate = indeterminate, so 1st step is ok!!!
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In response to Dejan Vuckovic: My solution and answer is also exactly like what you have stated. If 0/0 is indeterminate, then it is also equal to another indetrminate, just as x=x, whatever be the value of x. The first equation containing the mistake in the above equations, according to me is 5th one.
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@Venkatesh Patil – Actually no- just as infinity cannot equal infinity this is the same concept. i.e. 1234567.... (cont.) does not equal 2345678....(cont.). Except we don't know that infinity equals those numbers as there is no value. It is undefined and therefore cannot be equal (got this answer wrong and went for 6 but I see why 1 is correct).
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@MasterPro Sx – I felt just the same as infinity refers to a very large number and therefore isnt necessarily equal, but then 0 seems to be treated just the same way as an infinitesimally small number,so is the expression 0=0 faulty too?
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@Milan Mathew – Infinity isn't a "very large number"; it's greater than any finite large number. You can take an infinite amount away from an infinite set and have nothing, something finite, or something infinite.
And zero isn't "infinitesimally small", either; it's zero. If I do nothing an infinite number of times, I still do nothing. And if I do an infinite number of things zero times, I do nothing.
@Venkatesh Patil – venkatesh is absolutely correct
indeterminate is not a digit its an idea. e.g. tall=tall anger=anger etc.
I agree with this and say number one is correct!
Even if it might sound correct in English to say indeterminate=indeterminate, in math one indeterminate almost never equals to another indeterminate answer. Because all indeterminate answers will have different values. Mostly "0/0" equation means smth like "very small number/different very small number". Example 0.00001/0.0000001or 0.0001/0.001. since all similar equations are approaching 0 in numerator and denominator, its said 0/0 is indeterminate. Indeterminate meaning every 0/0 equation will have different answer depending weather numerator or denominator approaching 0 faster. So, indeterminate != indeterminate.
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@Umida Nurjanova – Jup, but the question 0/0 = 0/0 ? is, not a mathematical question... it is, in fact - a logical question, ...so if you think about meaning of 0/0 , than you think about idea of indeterminate, not about numbers... so for me it is a comparison of the two terms - which are actually the same...
Any way, this problem is not well set, because it causes so many doubts...
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@Dejan Vučković – I agree with Dejan Vuckovic and also decided that the first mistake is 5
@Umida Nurjanova – its funny because actually you just stated that 1/1000 = 10 ... just pointing out that your example .. doesnt exactly support your arguement.
I agree with some of the responses here .. i initially thought 5 was the mistake but i see why 1 has to be the mistake. you cannot make a statement of equality to indefinite values because there is no value at all to 0/0 .. most of the people on this forum are making the mistake of treating 0/0 as a quantity and it, in fact, is not a quantity. similarly to infinity, it is not quantifiable.
The question says: "In which step is the first mistake made?", so, as you noticed, Step 1 is mathematically incorrect (you just can't divide by 0 by no means), which is enough for it to be the answer of the question.
Also, I don't know how "logically" 0/0 = 0/0 and your argument is saying "whatever they may be... as long as you don't start to compute..." both parts of the equation are indeterminate, and one indeterminate isn't the same as other indeterminate.
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0/0 is indeterminant but it is something(meaningless), I mean iota is something, it is imaginary but it is, the mistake is when we write i=2,similarly 0/0=0/0is right, until we divide it or write something else. the answer should be 5.
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@Lalit Som – The answer is five. This site just lost my respect for their mistake.
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@Matt Thibodeau – I see why you think #1 is correct. Logically it may seem correct (EX: x = x), but mathematically it isn't, because when diving 0 by 0, you can never get an output.
@Matt Thibodeau – Even if #1 was correct (it isn't, because you can never divide 0/0 and get an output), #5 still wouldn't be the first to be incorrect, #4 would be...
@Lalit Som – It really depends on how you look at the things and how they defined. For example an alien or a baby (interestingly both examples work out well) could look at the terms: 0 0 not as a ratio, but exactly what they are: Abstract symbol denoting any something. Thus various definitions could be viewed as "legit". Raging from the symbol has 2 symmetries to "upper"half" and the "lower half" look the same. However this leaves the plane of arithmetic to which this problem belongs to and flies on to the bottomless pit of logic, philosophy and other things.
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@Meer Kat – This is how I looked at it, picking (3) as my answer.
@Meer Kat – That's entirely why I chose 3.
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@Luke McCallum – there is nothing wrong with three ..
you can go from 3 -> 2 .. using distributive property of Parenthesis in the denominator and computing the squares in the numerator.. likewise 2->3 simply substituting the 100 for 10^2 is a valid and simple, going out on a limb to call it an, operation... Factoring out a 10 from (100-100) to get 10(10-10) is also valid and trivial.
i have no idea what your seeing to label 3 as a mistake.
@Meer Kat – However, the problem does ask what the first fallacy is.
@Lalit Som – What about step 3 you can modify it to say (10-10)/10
0 / 0 = 0. Because 0 X 0 is still going to equal 0 whether it's indeterminate or not. A is still going to equal A.
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@Joseph Largé – When dividing 0 by 0, you can never get an output. The answer to 0/0 is not 0, because it is undefined. For example, if there are 0 of the variable X, and you split it evenly among 0, you wouldn't get an output... (It wouldn't work).
But Andronikus id right, you can't divide anything by zero.
if i consider your argument that means step 2 is correct that implies that 0/0=(100-100)/(100-100)=1 but it is not correct and 0/0 is undefined that it has no defined value hence they are not equal thus step 1 is incorrect
I also did the same mistake.
Ver good explanation. I was also thinking that step 4&5 should be wrong
Step 4 to 5 is wrong as 10-10/(10-10) is a undefined case so he can't write 1
x/x=1, if you take it as a fraction, and disregard the fact that you can't divide by zero (which is just a big mathematical cop out), it works. Equations five and six are telling us that 1=2 which is clearly false. The rest would work out to 1=1.
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Why would you disregard the fact that you can't divide by zero? Math and science are all about facts.
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@Whitney Clark – Because you can divide by zero... the answer is always one. Say you have a pile with X objects in it. Now with that original pile of X, divide it into more piles with Y pieces each. Once that stops having any noticeable effect on the value of X (usually due to reaching zero, but in this case because of no change), count how many piles you have left, that is Z.
If you divide by 0, you are not making any new piles, but you are still left with your original, single pile of X pieces.
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@Christopher Felch – That's not what division is. X divided by Y is defined as the (unique) number that fill in the blank with Y x _ = X. If no number works (as in 1234/0) or more than one number works (as in 0/0) then there is no answer.
Right? That's what I learned in elementary school. If not, please explain.
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@Whitney Clark – It is division in name only, same as dividing by one. If you don't end up with more portions than you started with, how can it truly be division? No, it is just a mathematical procedure that most closely resembles literal 'division', so it is called that for ease of communication. Otherwise, can you explain why my definition is not division?
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@Christopher Felch – Well, in algebra, I was taught that a/b was a b − 1 , or a times the inverse of b . But zero has no inverse, so you can't divide by zero.
And you don't have to have more portions in the end. Five divided by ten is one half, and five divided by one half is ten.
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@Whitney Clark – The concepts of multiplication and division may be reciprocal, this is true, but that does not mean there is a practical means of application for every real world situation.
Five apples, divided into ten portions, is half an apple each.
You do in fact end up with more portions.
But you cannot divide things into half portions.
You can only redefine portion size.
$portion=($portion/2)
Division, is sharing.
Practical division requires plural, positive integers.
You cannot have equal parts if you do not have a whole number of places to divide it between.
Otherwise it is just proportionally equivalent.
Negative numbers cannot be divided.
There is no such thing as -2 apples.
However, the principle of division can be applied to debt.
Try not to confuse mathematical division with actual division.
The problem I have here is that the first step of a proof is to start with a given piece of information. If the problem (which I note here is not given) states: Given 0/0 = 0/0... This first step is not incorrect as it is a given. Whether the problem itself would have issue is another thing, but I honestly think this question is on really shaky grounds.
But step 4 is also wrong.
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the FIRST mistake
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I support patni
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@Divjot Singh Manchanda – What do you mean? This isn't a democracy.
No. Step 4 also comes out to 0/0
No it isn't (distributive property) (10 + 10)(10 - 10) Multiply first 10 with the 3rd and 4th 10. Do the same with the 2nd 10 and then the bottom 10. You have (100-100)(100-100) over (100-100). It all cancels out.
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Um, but no. (10+10)*(10-10) = 1st times 3rd +1st time 4th + 2nd times 3rd + 2nd times 4th. Or 10 times 10+10 times -10+10 times 10+10 times -10 or 100-100+100-100 which equals 100-100, or 0. You did 1st times 3rd + 1st times 2nd and then multiplied that to 2nd times 3rd + 2nd times 4th, which is incorrect.
Yes, we know that it is not right to say 0/0 equals 0/0, but in this case, this is just an assumption, a thing that you may prove or disprove using correct mathematical procedures. Assumptions are just assumptions, they are not part of the proof, so we could say that the first mistake was done on the 5th "equation" where (10-10) was divided by (10-10). Please enlighten me if I am wrong. I am willing to learn more.
I initially thought that step 1 must be incorrect since 0/0 is indeterminate. However, I then thought that that could not be the answer for 3 reasons
First reason, it's an assumption, not part of the proof. It's as though you were proving by contradiction. You start off by putting something down and find out if it's correct or not by logical conclusion. If you say this is a mistake for this reason, you are saying all proofs by contradiction are mistaken
Second reason, whether something is indeterminate or not, if it's written down twice with an equals sign between, you are saying that something is equal to itself. If one side is indeterminate, to is the other.
Third reason, it's too obvious!.
The first true mistake is step 4 when you divide through by (10-10)
Saying the first step is wrong feels like you are ignoring the reflexive property... indeterminate are messy
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no, the First step is wrong and I can show you why. A division (and incidentally any operation) can be thought as a function which takes 2 values and outputs 1 value, i.e., we can think of a/b as f(a,b) = a/b. In our case, our function is defined ∀ ( a , b ) s u c h t h a t b = 0 ) .I.e., f cannot be considered a function when b=0. Here, by saying 0/0=0/0, you are assuming that the function, at this critical point (0,0) has still the properties of a function and this is not the case. (a function fulfills 3 properties, one of which says that the output has to give only one value, i.e. for each (a,b) where f is defined, f attributes to it exactly one value)
Let us give a concrete example, what is the definition of a/b in the first place? it says that a/b = c iff a = bc. It is easy to see that for b = 0 , this function f is well defined. If b = 0, we have (recall, for given a & b) : a = 0*c.
We have 2 cases : Either a = 0 , in which case, no such c exists, and the function cannot assign any number to the pair (a,0). Or a = 0 , and there, we have the opposite, since c can be any finite number, we find that f(0,0) can virtually be anything, i.e. 0/0 can be any number at all, and does not need to be a fixed, unknown number. So I could take 0/0=1 and 0/0=0, and we see that 1 = 0 and thus 0/0 = 0/0 .
There's no problem putting a=a. There is a mistake between 4 and 5 since that's the process dividing with 0.
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5 is clearly the first, you can divide and multiply by zero. 5. you have 20/10 doesn't equal 0.
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Gotta disagree with the statement "you can divide by zero". This is one of the first rules of math and that is you can not have zero in the denominator. The statement is "undefined" - that is you can not have parts of nothing (which is what you are saying when zero appears in the denominator - even if you want to say "zero parts of zero"). To go on to say "undefined = undefined" is incorrect at the first "undefined", but also does not pass the laws of logic. Granted that mathematically #5 is the first incorrect statement adhering to the rules of defined math... it does not make #1 correct. Rather #1 is incorrect before you even get to the equal sign.
0/0 =0/0 Read it as indeterminate = indeterminate, or meaningless = meaningless. True
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Question then is, if the indeterminate (something which cannot be determined) is equal on both the sides of the equation ??....Also, Curious to know the reasons as to why we cannot determine this value ??
Well, not to get out of the area of mathematics, but this is like in a relational/SQL database: does null = null? And the answer is no, it doesn't.
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One, why not? Two, comparisons to math, if they're not math, are not math, plain and simple.
Non of them is 100% incorrect!!! because indeterminate doesn't mean that it equals nothing (that's zero) ,actually 0/0 equals everything ,but we don't have a specified value for it. it can equal 1,100,1000,0,29,47,.00047. and one way to prove it is the idea of the division as whole ,for instance 12/4=3 as we teach small kids and as same as I've learnt when I was a kid we can say this as: What is the number that we multiple it by 4 and gives us 12 the answer is 3 ,but if it come to 0/0 and we asked our self what is the number that we multiple by 0 and would gives 0 as a result the answer is any number!!! because any number we multiple by 0 equals 0 therefore the whole set of R can be answers to 0/0 (notice that infinite and -infinite don't belong to R) in other words if 0/0=x then 0=0*x and any number in the R set can be an answer to x and actually that's the reason why we call it indeterminate ,but the problem is that we must take only one value and we don't know which one is right one that choose because we can't say that 0/0=1=0.0001=247953 because these values aren't equivalent , but 0/0 can equal each one of then alone. So it is kind of right to say that the first is incorrect because not every indeterminate equales the other but it doesn't mean that it won't.mathematically if x=y then it is not nessecery that x/0=y/0 but doesn't mean it don't.
Shouldn't that be "undefined"? That's what I was taught; that 0/0 has no meaning. In calculus, the indeterminate form is different.
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Hi! A nonzero value over zero is undefined. Zero over zero is indeterminate. It's not covered very well below Calculus level in school.
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What's the difference? A quotient can't equal two different things at once, can it?
A very poor question.
Step 4 was the first mistake. In the form of algerbra lines 1-3 were saying " undefined = undefined " that is correct when it was shown in the math written. The mistake was when it said " undefined = (-400\0) " it is still undefined. However the algebra at its most simplistic form before it was undefined was not the same as the previous 3 steps and because your problem is given in an algebraic statement, the answer must be in an algebraic statement.
The question itself is flawed. Does "first mistake" refer to the ordered numbering equation (ie: 1 is first, 6 is last) or does it ask something more specific like "when evaluating the equations, which of the 6, non-ordered equations first comes upon a 'mistake'."
Thus, since they all begin with LHS "0/0 =", this is indeterminate, thus a well-defined, yet indeterminate value. Then we begin with RHS where the precedence of operations comes into play. Equation 6 is the first to evaluate, since there is no operation order to consider; it is determined. Meanwhile, the assertion that "2" is equal to an indeterminate is a "mistake" and occurs "first" during evaluation.
To extend the justification, it is uncommon (rare?, poorly formatted? incorrect?, insufficiently stated to evaluate?) that a single choice (as opposed to multiple choice, where more than one answer may by correct) requires an ordered evaluation of the choices.
"Zero to the Zero Power" =0/0 =1 (google "Zero to the Zero Power" ). That makes #1 and #2 ok. #3 have 10(10-10) = 10 * 0, multiplication with zero, which is the first mistake.
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How is 10 (10-10) incorrect? It is the same as (100-100) all you doing is pulling a 10 out of the equation, making it 10 (10-10). I.e. if you distribute the 10 back in you get 10 times 10 - 10 times 10 which is 100-100.
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0/0 is by definition made legal in math. Multiplication by zero is the added "dangerous" part in this equation. 10 * (10 - 10) = 10 * 0, multiplication with zero, which is the first mistake.
The parenthesis in #3, (10^2-10^2), is wrong because 10^2 - 10^2 is (10 * 10) - (10 * 10), not (10+10) * (10-10), which makes #4 wrong too.
These 2 errors together make #3 the wrong equation .
Multiplication by zero is added and removed apparently according to well known rules in math and that is the confusing part of this equation.
almost my thoughts but you stumble at the first hurdle (0 to the zero power would be 0x0 not 0/0).However I agree that the mistake is in 3, saying 1 is false is saying the problem doesn't exist. PEMDAS folks....the Parenthetical in the denominator would be solved first, turning it into 0
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Zero to the zero power would be 0^0, not 0x0 OR 0/0.
Two fractions with the same values in the numerator and he denominator set equal to one another is the same?
Not a fan of a question with multiple right answers where only one counts...
it means the question is wrong ...........if you have give the question to prove based on wrong assumptions.....in Step 5 0/0=10(1+1)/10=2 ...........
I think it's better to say that using #1 as the answer is a bit pedantic: indeterminate = indeterminate isn't an invalid statement, logically, unless we're going to posit that there is different values of indeterminacy. While it's technically correct, it's not where the first calculation error was made.
actually 0/0=0/0 isn't wrong. It's right. Because 0/0=1 then 0/0=0/0 would be 1=1 so the actual first mistake was on the 5th equation because 0/0 or 1 does not equal 10+10/10 or 20/10 or 2/1 or 2. So the 5th equation is the first one with a mistake since 1 doesn't equal 2.
in which context of zero is the equation using? the original use of zero was as a placeholder to denote "nothing"... if that is the case then simply put: nothing divided by nothing should equal nothing and the first equation is right... or are we using zero as "undefined"? an example would be like helping a friend move and carrying an item into a room and asking the question "where should i put this?" as there is an undetermined number of locations in that space the object can be placed. in either case there is not enough information given to solve the problem as there is no defined context of the use of zero.
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Zero is a number. When you multiply anything by zero, the answer is zero; you are not multiplying by nothing and getting the same thing. 7x0 = 0, not 7.
I disagree you.
I don't care what math lingo you want to use, if the right and left side of an equation are the exact same, then they are equal. The problem comes in when you start manipulating. That's like if I asked if a/a=a/a and you would actually respond, "not if a equals 0." give me a break. Also, writing the equation isn't part of the proof.
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does this apply to infinity = infinity? then if we use transposition we can assume that infinity - infinity = 0? which is i know that is false.. please help.. thanks
I'm not a mathematical genius but any number divided by its self is 1. correct?
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Any number other than zero , actually. You can't divide by zero.
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Ya you can. If the number your dividing by zero is zero then it's like saying 1/1
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@Antonio Caro – Zero is different. Zero divided by anything else is zero, so shouldn't the answer also be zero? But does zero equal one?
but 0/0 =1,
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Zero is not like any other number You can't divide zero by zero as the result would be indeterminate And you can't divid any number by zero as it's also indeterminate but you can divid zero by any number and the answer will be zero.
i think 0/0 is not equal to 1.. indeterminate?
Both Steps 1 and 4 to 5 are wrong, but as per the question we have to find first mistake , So Step 1 is wrong since no two indeterminate 0/0 can be equal and cannot be written as equations
Wrong, 1,2,3,4 are essentially the same thing. It leads to 0/0. 5 is clearly wrong because you have 20/10, which doesn't equal 0.
YOU CAN'T DIVIDE BY ZERO!!
Yes you can, do it on any calculator.
Have you taken calculus?
0 can b divides by 0 like 1 by 1
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No you can't. No matter what number you put.
You need to be careful with the number zero. It's like the most dangerous number.
Here's a contradiction that 0 0 = 1
We have 0 = 0 ⟹ 0 × 0 = 0 × 1
Dividing both sides by 0 , we have:
0 = 1
Well 0 does not equal 1. Therefore, 0 0 is indeterminate.
To everybody who says that Step 4 is the answer: Ask yourself, if you cannot accept (10-10)/(10-10) because it is indeterminate, then Step 1 is basically the same thing. If you accept that in step 4 (10-10)/(10-10) is wrong, then I don't see how you can justify that 0/0=0/0 is "logically" correct. So the correct answer should be Step 1, as 0/0 is indeterminate in the first place.
0 divided by 0 is indeterminate aka UNDEFINED. This means that it cannot be one definite thing. E.g: 0/0=1 because x/x= 1 BUT! 0=0 0 And since 0/0=1 Then 0 (0/0)=1 But that means 0*1=1 Which is false. Why? Because 0/0 is ANYTHING you want it to be. Therefore, step 1 IS wrong because the 2 values are undefined to begin with. It's like saying that since a 6 foot person is tall and a 7 foot person is tall, then 6 foot= 7 foot. It's just flat out wrong,
0/0 it is undefined form,so u can't say that 0/0=0/0
You can't divide anything by 0.
0 0 is undefined (not defined by any definition in math)
Also, this question should explain more details, (i.e. define "first misktade made")
The problem is that everyone is taking 0 to be a number when it's technically an idea. It's the idea of nothing just like infinity is the idea of everything. So to read 0/0 is "How much nothing is in nothing?" The question doesn't make sense. One could answer everything or nothing and not technically be incorrect. The 1st is the mistake because 0/0 doesn't make sense to say.
The proof that step 1 is the error is to do algebraic substitution on it, it if you say A=0 then it becomes A/A=A/A, which would normally always equal 1.... except when A=0, which proves that normal algebra isn't equipped to handle 0/0=0/0, so anything that follows is also going to be flawed.
Cross multiply... And then no dividing. Hmm. We can find the limit of something indeterminate, so as long as I don't attempt the division, you are wrong.
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It's not a limit, it's division. There are no "limit" signs here; this isn't calculus.
0 0 is indeterminate. So it's a mistake if you put it in an equation such as 0 0 = 0 0 . And because the first step is wrong, the whole calculation must be wrong.
Actualy the whole step is wrong, because 0 0 always stated as an equation. But, just say that the 1 s t step is the primary mistake.