An article in the Onion from 1907 reported that a record breaking number of American children are staying in school beyond third grade. They are learning advanced skills such as multiplication, which we are told, is a powerful form of adding, resulting in numbers so large that three or even sometimes four figures are required to write them.
My 7th grade algebra teacher would only whisper of dividing by zero because it would “upset the calculator gods”. He was one of my favorite teachers ever.
Do not touch the operational end of The Device. Do not submerge The Device in liquid, even partially. Most importantly, under no circumstances should you divide The Device by zero.
An accountant, an engineer, and a mathematician are asked how much is 1 + 1: Mathematician: "1 + 1 is 2 and I can prove it" Engineer: "Well, 1 + 1 is anything between 1.8 & 2.1" Accountant: "It depends. How much do you want 1 + 1 to equal?"
in high school I was doing a problem on the blackboard in an algebra class and I was finishing it fast so I was writing both sides of the equation (my way of doing it) and the teacher saw it and yelled "algebraic sacrilege!!!!" and that scared me lol and everyone else in the classroom. I swear that we almost hear thunders falling on us. Sufficient to say that I never had the opportunity to explain to him that I was still writing my answer, so I just completed the answer as a "correction".
@@circuit10 I meant that when you start writing it, you begin with the right side, you finish writing that side and then move on to the right side. I was doing both sides at the same time.
@@coulombicdistortion1814 Here's the thing about multiplying by zero: anything multiplied by zero is zero. So 0(1/0=2/0) (1*0)/(0*0)=(2*0)/(0*0) 0/0=0/0
Travis Ryno I have been stuck on this since last evening. My 13 yr old told me exactly this, and then used Banach-Tarski model to say that 1+1=1 is mathematically possible. I don’t know right now whether to believe his hypothesis or continue to say x/0 is undefined.
Now, I had always been taught that X/0 was "undefined", while 0/0 was "indeterminate". The logic behind this is that the denominator (or "divisor") should always be able to be made equal to the numerator, by multiplication with some factor. So, for example, 1/2 = .5, thus 2 can be made equal to 1 by multiplication with.5. However, in the case of X/0, there is no factor that can make 0 = X, since 0 times ANYthing is always 0. So, there is no correct answer, therefore, the problem is "undefned". On the other hand, in the case of 0/0, literally ANY factor will make 0 equal to itself, so there is no INcorrect answer. Thus, in essence, any value is equal to any OTHER value, which is impossible. Therefore, the problem is called "indeterminate", since one cannot determine what value best solves the problem.
I know you said this is what you were taught, but it bears mentioning that this is just incorrect. There is no such a thing as "indeterminate" in mathematics, and people need to stop using this word forever. 0/0 does not exist. Period. That is all there is to it. And there is a very simple reason it does not, but it just has to do with what division itself is. Division is just multiplication: multiplication by the reciprocal, to be exact. 0 has no reciprocal. So one cannot divide by 0.
@@angelmendez-rivera351 because in calculus, 0 is not exactly 0, 0 can be 0.0002 or -0.00001, numbers are not exactly their values. That’s why there is indeterminate
@deaf I fail to see how those points connect. 0*x = 0 for all values of x is a true statement. I don't see how this implies that 0/0 = 1 any more than it does any arbitrary number.
@@angelmendez-rivera351 Yes, in terms of numerical value, indeterminate forms are considered undefined. But they are very useful in calculus because of how they affect limits. (f(x+h) - f(x))/h = 0/0 when h=0, so it's undefined. But the limit as h approaches 0 is very much defined (when f(x) is continuous), and is in fact the definition of the derivative. If 0/0 is just undefined, derivatives don't exist, and calculus doesn't work. That's why we have indeterminate forms, at least when working with limits
There's some great footage on RU-vid of mechanical calculators, oldschool ones, dividing by zero. No programmed-in "Math Error" there, the things just spin forever making a racket, they're probably subtracting zero over and over but maybe some of them are failing in a more clever way.
The computer is actually taught to not divide by zero. There are many situations in software where dividing by zero is caught and protected against. My brother used to work in a hardware store and he had a computer that gave a 'divided by 0' blue screen. According to the story, he laughed insanely laud at that blue screen. Usually that doesn't happen but the computer had a defect RAM which fed corrupted data into the processor as it fetched the information to execute the micro programs. The processor actually had a build in protection to prevent dividing by zero, it stopped the operation and 'breached' away from its micro instruction to the error handling of windows which on its term showed the blue screen. In short, the computer doesn't even attempt to divide by zero. If you were to try and do it it would probably try to apply a form of implemented long devision which would obviously fail and I have no clue what it would return.
Robert sorry RU-vid isn't letting me post my own comments one thing I would note for the people at number phile is it's as easy as defining 0*y=0 y in the complex plane but not =0 so 0 divided by 0 makes no sense since you can turn it into y*0/y1*0 the 0's can be seen to cancel and then you get y/y1 for any values y,y1 and therefore can take on any of any infinite values.
***** Actually, I think that algorithm doesn't quite simulate a division by zero because, for any value you insert as a divisor (if you swapped "int(n) - 0" for ,say, "int(n) - 3", for example), you'd still have an infinite loop (because the condition for the while loop will always be true and there is no condition for it to actually stop). A true general algorithm for a division of integers would be something like that: ----------------------------------------------------------------------------------------------- n = int(raw_input("Insert the dividend: ")) m = int(raw_input("Insert the divisor: ")) c = 0 result = n while True: result -= m if result < 0: remainder = result + m break c += 1 print c print "%d/%d = %d with a remainder of %d"%(n,m,c,remainder) -------------------------------------------------------------------------------------------------- If you insert 0 as the divisor, the "c" values will explode into infinity on the screen until you hit that close button, however, inserting other positive integer values would return normal division results. :D (also written in Python, because screw it, i'm on that lazy train too \o/)
Robert Dividng is reverse of multiplyin so: 4/3 is 4 * (1/3) and a proof of this is that: (4/3) * 3 = 4 (a/b) * b = a so: if b = 0 and a is any N then a =/= a which as answer is not in set N because any a in this set is equal to itself (4/0) * 0 = 0 ==> 4=0 The result of this nonsene came from the set. Any result on the corpus of N must result in the corpus of N and 0 is not in even in the set of N.
Robert Fennis Set dosent include result. Need a larger set with algebra over biger corpus with a diferent ring and more dimensions. And of course problem is solved, directX is working perfectly without gimbal lock on this wee issue of dividing by zero.
And that all the while glossing over X^X for negative X looking really strange (it's jumping all over the complex plane and is basically discontinous everywhere). That is not a function for which you want to find a limit. The complex version must be just as bizarre.
There's a video around of an old mechanical calculator which gets stuck in a loop when trying to divide by zero, and the operator has to press the abort button to stop it running. Nothing bad happens - it just keeps subtracting zero and counting how many times it subtracts zero and it never finishes.
'Maths genius?' Ohnhon this is stuff which, if you ever paid attention in school, should be logical. I don't understand why people believe that worked out concepts are so hard. You just gotta puzzle over it until you understand.
The word "everybody" is actually conjugated in the singular, as in "everybody was Kung Fu fighting", and not "everybody were". Genii is acceptable, however.
I met this channel a while ago, when i was in highschool and used to watch every video. Now, as i'm graduating in mathematics i come back and rewatch the same videos, but now in a different perspective. Numberphile was one of the main reasons i decided to study math in college, despite all flaws.
I did not say that infinity is any number.I said 0x(infinity) is an undefined number so you dont know whether it is equal to 1 or not , therefore you can not say that 0x(infinity)=1 is an impossible equation, it is an undefined equation. And you can do operations on infinity. For example: (1/infinity)=0 and: (+infinity)(-infinity)=-infinity
Oh c'mon, it's usually the otherway round. Unlike mathematicians, engineers are too boarged down with deadlines and budget constraints that they hardly have any luxury to play with theories and concept. Otherwise the boss would show them the door 😅
But all division does is count the subtractions that took place to reach the number. Therefore, it isn't infinity or the number you started with. It's 0. 20 / 4 = 5 (Five Subtractions) 20 - 0 = 20. No subtraction took place. 20 / 0 = 0 (Zero Subtractions)
@@Vespyr_ No, that is horribly incorrect. Firstly, that is not how division actually works: division is not repated subtraction, and multiplication is not repeated addition. Secondly, even if division did work that way, your answer is still wrong, becaue 20/0 would be equal, by your definition, to the number of times you have to subtract 0 from 20 to achieve 0. The problem is that, even if you subtract 0 an infinite amount of times from 20, you still do not achieve 0. The answer is not 0, nor is it an infinite number. It is just impossible to achieve 0 via such repeated subtractions, hence 20/0 is undefined. Nevermind this, because as I explained firstly, division is not repeated subtraction. The reason division by 0 is problematic is because, in order for division by a quantity A to be possible, you need to have the following property: if A·x = A·y, then x = y. This does not occur with 0. 0·1 = 0·(1 + 1), but 1 = 1 + 1 is false, in general. So division by 0 is hopeless.
@@angelmendez-rivera351 I think you missed the point of glorified subtraction but that idea does work, 28 divided by 4 is just 28 minus 4 over and over till its 0, which is when it's been subracted 7 times
@@thefloormat3297 No, dude, I literally addressed it within my first sentence. Maybe you do not know how to read. Also, I already explained how subtraction does not work. You cannot subtract 0 over and over from 20 until you get 0. It is impossible.
I still remember that day when I was in the middle school. Our math teacher, let us use 1 divide some positive numbers smaller and smaller, than we found the results bigger and bigger. Then we use negative numbers bigger and bigger, and the results were smaller and smaller. On that day all of us remembered we cannot use some numbers simply to divide 0.
From the software engineering perspective I'd say that I highly doubt that any commonly used calculator uses iterative process to get an answer for X/0. It's just a check in the code: if operation is division and second argument is 0 then print "Error". So, the first guess is much closer to reality
Definitely more likely. Calculators only really do addition and subtraction so if you tried to divide by zero it would keep subtracting by zero an infinite amount of times, just like they demonstrated in the video. Its gotta be programmed to check for a non zero number to keep it from entering an infinite loop, that seems like the best solution
It's normally an exception, stopping your execution. If you have a divide by zero in your equation and you don't stop, you're in la-la land. CPUs handle integer division (which give division by zero and overflow), languages have standard libraries for floating point. The standard is to have zero, NaN (Not a number), Inf, and -Inf as distinct results. Most calculators now have this as well (processors are very cheap). NaN is different than infinity. Infinity normally means it encountered a number which exceeded the maximum value (e.g. 300 factorial), and infinity times zero is zero. Any math operation using NaN gives you NaN as a result. You also can have exceptions for overflow/infinity, and there may be cases where you want to know when you underflow (if you have x and y, which are not zero, but you get zero because the number is too small.. that's not normally one you worry about). A difficult problem in programming is when you have a one-off problem, where it goes into la-la land, and takes a few steps before it dies. Math is one of those things.
Unless someone forgets to tell the computer not to attempt the subtraction, in which case the computer may crash, which happened to an American warship, computers down for a day
I recently learned what the actual name for 0/0 is in Calculus. It's called an indeterminant, because it can give any answer. If we want to solve it, we need to know the function that created the 0/0, as they show. Then we take the derivative of the top and the bottom (separately), and try to divide again. We repeat until we don't get a 0/0
That’s not quite accurate. An indeterminate form like 0/0 is entirely meaningless on its own and fundamentally can not be “solved” unless you’re talking about in terms of a limit. Furthermore, 0/0 isn’t the only such form, so your use of L’hopital often isn’t applicable.
@@cpotisch Fair enough. Most of the time, the indeterminate form can be converted into a form usable with L'H. Although, you can just use the fact that e^x grows faster than x to get a quick answer
"You"? Which "you"? There's 2 people we see in the video, which one are you talking about? ...Or do you mean the plural "you" and combined both of them?
Jay Jeckel Come to think of it I think he does it on purpose. Because in the end it serves the same purpose and it's just a formality, he probably does it for fun to get under his mathematician friends skin.
Ivo Wilson I always assumed it was because he is from the UK and they do some things weird over there, like calling math 'maths' and sports 'sport'. Could be easier drawing on a chalk board or what you said. Either way, he does some great videos and his channel is worth checking out if you haven't already.
Lots of people do this. You pick up handwriting habits like this so you can more easily distinguish between symbols that look the same. x and the symbol for cross products, for example, look similar and will confuse people unless you draw the letter x as half circles.
This is a relatively common convention tbh. It was specifically adopted so that 'x' would be more readily distinguishable from the multiplication symbol in mathematical proofs and textbooks. A common alternative was to use * as the multiplication symbol, as most scientific calculators do.
@@ForeverStill_Fan1 start with C or C++ Understand the basics of programming... Don't directly jump to python... Python is a high level language doesn't help too much for building logic... For frameworks- depends on your interest
I think James' first description is pretty much perfect. You just keep subtracting a number until you get to zero. If you did 20 - 0 an infinite number of times, you'd still end up with 20, because every step leaves you with 20. So infinity essentially has no effect on subtracting (or dividing by) zero.
If I understand my computer science right, computers' physical arithmetic processing units throw errors when they're ordered to divide by zero, which would cause horrible breakage. In practice, though, the command to divide by zero is intercepted by stuff like the operating system long before it actually manages to reach the hardware.
Ricky Bobby I mean, I get your point. But no matter how much we elaborate about it, we still had to deal with renormalization/regularisation in pretty sloppy ways for many years, and that's just an example. We also dealt with infinitesimals in a way in which it was not clear that they were even defined for many decades. Actually, an axiomatic description of many of todays QFTs is not known. So yeah, we basically handle infinity as a number until things go wrong and then go back and see why things went wrong. So the history of physics would still disagree with you :D
If we subtract 20 infinity times then the answer wont be zero so we also cant say infinity according to me ,if we divide zero with a number,the answer is nothing because if we subtract we wont get zero and in case 0/0 ,if we subtract it by any nomber the a.swer is already zero so it has set of infinity answers That means 0/0=all numbers¿¿
TomYale Physics builds mathematical models of the physical world, which don't necessarily perfectly describe it, only approximately. So infinities are commonplace in physics, they just mean "very very big, according to the scale of your model". Or sometimes they mean "you included terms in your sum that are not well described by your model", or it can have many other meanings aswell!
To get 1/0 = 2/0 into 1 = 2, you multiply by 0 on both sides, which we currently say is well defined. To get 1/0 = 2/0 into 1 = 2, you divide by 0 on both sides, which we see we cannot define unless we throw out multiplication by 0. A lot of things seem like nonsense if you don't pay attention in class.
And this is simply because infinity and 0 are both concepts, not numbers. It is present in mathematics because it is quite useful, such as algebra but to an extent. The very first number systems didn't include a zero at all, some argue that this is why the Roman Empire has fallen.
Usually the way I explain it to people is that almost the entirety of calculus is an attempt to simulate dividing by zero. There's that entire branch of mathematics (which most people find too complicated to be worth learning) that is pretty much just answering this question, and it's still a fuzzy and imperfect answer. So if you wanna see why you can't divide by zero, a basic overview of calculus often will do the job if it's being explained well.
I'm not a mathematician (more of a moral philosopher) and my favourite part is at 7:11... "people argue (to be honest) different ways depending on what they need..."
I have a few thoughts, though I may be wrong about a few things here. Why is it that when you approach 0 from the negative side of the function 1/x, it goes to negative infinity? Doesn't that imply that 0 is a positive number, since a negative number multiplied/divided by a positive number equals a negative number? Even if we have -0, it would just be "corrected" to just 0. What gives? :P
(1) Because as you approach 0 from the left, x will always be less than 0 (definition of negative). Since 1 is positive, a positive divided by negative is a negative. Since you're taking a limit, you never actually get to 0, but the closer you get, the more negative the number becomes, so you say it approaches negative infinity. (2) No, because by the definition of limits, you don't actually reach the point you're approaching. Limits are used when there are singularities in math (values for which something undefined or impossible happens). You're absolutely not allowed to plug that number into your expression and get a value, so you take a limit...you get really close and see if it's approaching a value. (3) Zero is neither positive nor negative, so the rules of positives and negatives don't apply. In fact, by definition, it couldn't be positive or negative even if you wanted because it's part of the definition of *both* positive (greater than 0) *and* negative (less than 0). The reason that +0 = 0 = -0 is because of how you can turn a unary operator into a binary operator. For example, -0 (unary because there's only one term) is the same operation as 0 - 0 (binary, two terms), which of course is 0. +0 is the same operation as 0 + 0, which is still 0. Neither unary operator (when converted into an equivalent binary operator) had any affect, so 0 must be special in that respect.
0 is not negative nor positive, it is a point. Saying '-0 and +0' is like saying '-axis and +axis.' The reason X (of 1/X) approaches -∞ from the left is because it is being divided by infinitely smaller negative numbers, not 0. The definitions of positive and negative are literally 'greater than and less than 0' respectively; there is no definition for '0 = 0' other than, well, 0.
when you say it approaches to 0 from the negative side. It means that x is not 0 but very close to 0 so in this case x= -0.00000......1 so you can't correct the negative sign.
But it never does reach negative infinity. You CANNOT divide by zero. As you get closer to zero from the positive direction, it goes positive infinity, and you get closer from the negative direction, it goes negative infinity, but at 0, there is no value (and no it doesn't cancel out to 0 either! That'd be equally as troublesome as it equalling infinity).
6:25 totally agree with that. I know that when I do a really intense calculation on Desmos, the calculator displays to me a message saying "definitions are nested too deeply"
Well, I understood most things, but I think you need more than basic logic to understand the bit about the imaginary number axis and graphing the crazy squiggly lines because of it.
I always hated math and no one I know likes it, but it is nice to see people with a real passion and love for maths/numbers to make it more interesting
Here's a simpler explanation of why n/0 is impossible and 0/0 is undefined. I use it in my 6th grade classes in Italy using numbers instead of letters. 1) division is the inverse operation of multiplication. Calculating a/b means finding a number (let's call it "c") that multiplied by b gives a (in other words, c*b=a is basically the same thing as a/b=c, you're just reading it backwards. For example: 6/3=2 and 2*3=6 are basically the same thing, you're just reading it backwards). 2) if b=0, then you get a/0, which is impossible because there's no number that multiplied by 0 gives a. In fact, a/0=c means c*0=a, and there's no number "c" that multiplied by 0 gives a. Since the operation a/0 doesn't have a result, it's impossible. 3) if a=0 and b=0, then you get 0/0, which is undefined because any number multiplied by 0 always gives 0. In fact 0/0=c means c*0=0, which is always true, no matter what number "c" you choose. Since the operation 0/0 doesn't have one single result (it has infinite results), it is undefined. I hope this can be helpful.
If you look at zero in terms of inversion, zero is essentially "neutral infinity", in a way. If you take the inversion of zero with any base radius, you will get a variation of infinity as the answer.
I think you're confused in that you are making weird calculations based on our imperfect, artificially created interpretation of mathematics. Doing a bunch of maneuvers isn't proof that nothing is equal to everything, it's just a flaw in the math we created.
When he said, "For all we know, this line may wrap around the entire universe and connect" (paraphrased a bit), that got me thinking. It can't, and here's why. What mathematicians do to prove certain postulates or theorems occasionally is that they assume the end as an axiom, so let's assume that the number line does, indeed, wrap around the universe and connect end to end. Well, we also assume by this and the limit equation at about 5:00 that infinity is on one end and negative infinity is on the other. If this line wraps all the way around, end to end, and treating the respective infinities as the points on the line where it terminates, this would mean that infinity and negative infinity are adjacent to each other on this line. Not only are they adjacent, but there are also no numbers between them. There is no value that is greater than infinity (because x + infinity = infinity, whereas x = all real number) and no value is less than negative infinity (because negative infinity - x = negative infinity, whereas x = all real numbers). Because there is no value greater than infinity, it can't be possible to count up from infinity to negative infinity. And you can't count down to get to positive infinity. Another issue would be that the function 1/x approaches positive infinity from the right and negative infinity from the left. This sentence treats infinity as a location, not a number, just as they say in the video. Infinity is not a number and cannot be treated as such. But saying that it is at the end of the number line and connects both sides means that there is a terminal location for infinity, which there isn't. (Terminal meaning simultaneously that there is a singular value that we can point to and say that is this number and that it is at the end of the line.) Infinity has no conclusive end to it because there are many ways to represent infinity, such as: 1+2+3+4+5+6....ad infinitum *also* 2+4+6+8+10+12....ad infinitum *therefore* 1+2+3+4+5+6....ad infinitum = 2+4+6+8+10+12....ad infinitum Both of these number sets add to infinity. But one number set is clearly twice as large. If you wish to simplify it, this can also be represented by infinity = 2 x infinity. How can a number, cardinal or ordinal, be multiplied by any number other than 1 and still be equal to itself? Short answer, it can't. Long answer, it caaaaaaaaaaaaaaaaaaaaaaaaaaaan't. *In conclusion* There is no way to mathematically represent the fact that the number line wraps around. These problems have to be addressed for that to be true, and so far that's not possible, if it ever could be. Also, due to the number line wrapping around, it would also be true that negative infinity and infinity are equal, but I don't know how to prove that. I just intuitively know it. Thank you for coming to my TedTalk.
Doesn’t the real number line pretty much wrap around in a real-number cross-section of the Riemann Sphere? Also, 0 can be multiplied by any number and still equal itself. In fact, it can’t be otherwise. Or are you saying that 0 is not a cardinal or ordinal number?
@@paulchapman8023 I don't know anything about the Reimann Sphere, so I can't argue it's importance (though that doesn't mean it has none). As for zero, I genuinely hadn't considered it, despite the fact that it is cardinal and ordinal. It feels like a cop out of an answer to call it a special case, but i can't think of it any other way. Given its other exceptions (can't divide by it, exponential is 1 always, etc), I don't think its unfair to label it special and consider it separately in some other proof. That was a great question, for a long ago i made this comment I still can tell I hadn't considered zero at all, thank you for bringing it up!
Because zero has no value, working only with zeros makes no sense. 0^0 is the same as saying 0/0 which is, in actuality, I have nothing and I don’t divide it what’s the answer. See? That’s totally nonsensical. Lol
_This Comment is cross-posted!_ 1 is a more consistent answer. The *Taylor expansion* for e⁰ will be *0⁰/0! + 0¹/1! + 0²/2! + 0³/3! + ... = e⁰ = 1.* The only term that is not 0 is *0⁰/0!.* There is also the *Taylor expansion* for *cosine.* If *n* objects each have *k* states, then the equation for the number of the set's positions is *n^k.* Think about the number of positions that [a set of 0 objects each with 0 states] has. This is philosophical, but it is one state. As for *0^x = 0,* that is only true for _positive_ exponents of 0. The Binomial Theorem also relies on the *0⁰ = 1* statement. As for limits, those are only accurate to the true value for continuous functions. Take the piecewise equation *y = x if x ≠ 5, y = 1 if x = 5.* The limit of y as x approaches 5 is 5, but *y = 1* AT *x = 5.* As for the *Product&Quotient Rules* of exponents, under certain circumstances, those are false for 0. I hope this makes sense.
CompSci here: there is a standard called IEEE which defines mathematics for computers it says that for any positive, non-zero 'x': x/0 = inf, -x/0 = -inf and that 0/0 = nan where inf is not a number, just a representation of infinity (and it also defines operations such as inf + x = inf and so on...) and nan is just 'not a number' I must add that this is just the standard most programming languages uses for mathematical operations, not how it works in actual mathematics computers have hardware limits where mathematics doesn't
Mathologer said, "in higher levels of calculus it actually makes sense to treat infinity like a number and to actually write equations like three divided by zero is equal to infinity".
Negative and complex numbers are not being ignored in some of these contexts. Rather, you identify all different "directions" of infinity to be the same infinity. Think of it like folding a plane into a ball and adding a point called "infinity" at the top. This is the basic idea behind the Riemann sphere (you can look up Riemann sphere on Wikipedia if you want to know a bit more), which is an incredibly useful tool in Complex Analysis (basically, where you're doing calculus over the complex plane instead of over the real line).
0:16 ⛔ Division by 0 and 0^0 are problematic. 0:31 🔍 Why division by 0 isn't simply infinity. 2:00 🛑 1/0 ≠ infinity; it leads to mathematical contradictions. 2:56 📈 Limits don't make 1/0 equal infinity; they differ from different directions. 5:43 ↔ Approaching 0 from different sides leads to different results (±infinity). 5:59 🖩 Devices can't handle 1/0; it's an unresolved calculation. 6:45 🧮 0^0 is contentious; arguments for both 0 and 1, but undefined due to limits. 10:21 ❓ Undefined result for 0/0 depends on the approach angle.
Negative infinity plus positive infinity is zero. The graph made perfect sense. They are symmetric mirrors of each other and cancel out. both exist at once to make the curves cancel.
If negative infinity plus positive infinity would always equal zero, the following would hold: (+inf) + (-inf) = 0 lim(x) + lim(-2x) = 0 lim(x-2x) = 0 -inf = 0 You get different results depending on what you put in. That's why you say it is undefined.
5:53 From a programming perspective - it's almost always going to be hardcoded on a divide function to throw an error if the divisor is zero. Having a step counter (detecting the "exploding" in a direction) on an iterative process is so prone to fatal mistakes that most people wouldn't code it that way as the sole way to catch a common problem. That said, there is a third option. You can have your iteration record its result and then have the next iteration compare against that result. Even if it's not truly a non-terminating iteration, it WOULD be one if those match because it would mean the precision of your value prevents you from getting anything out of further iterations. (Identical input in a deterministic function gets the identical output every time.) This is a more robust way of quickly finding the limit of your calculation ability on a given problem and it happens to catch "divide by zero" while it's at it. There's also the 4th check, verifying if the precision limit has been reached. Most code you see now will use a large signed integer to store values while it's calculating while displaying FAR less. For example, the Windows calculator has 36 characters on its output screen so they're likely using a 64 bit "double precision" value to store digits. This is a standard, useful so you don't hit errors with your custom "38 bit double" but you can decide within your loop if you've calculated far enough that the user won't see it. Frankly, what most programmers would do is all 4. - The divide function will generally already be built into the code for any compiler with the "divide by zero" error in place, but if you're coding at a more basic level you would do that yourself. You always call the same divide function for the actual division of any 2 numbers, so the 0 is always caught even if it shows up later in the process. - The iteration loop is given an array for the previous iterations results. At the end of the iteration it calls a separate function to iteratively compare the latest result against the array. The comparing function returns a value and the loop uses that value to determine if it needs to stop.- Any time an iterative process kicks off, you give it a number of iterations it's allowed and the loop can either give an error or a partial answer once the iteration limit is reached. You can also make this smarter with better calculators so a complex loop is given fewer iterations or the whole process is given a set number of iterations to allocate. - The iteration loop is given an array for the previous iterations results. At the end of the iteration it calls a separate function to iteratively compare the latest result against the array. The comparing function returns a value and the loop uses that value to determine if it needs to stop. - Finally check if the value change is of low enough precision to matter. Essentially, you compartmentalize your functions so that they catch the sorts of problems they each introduce. The top level, division, fast-ejects known problems like 0 so you're not burning through limited resources on a calculator powered by a cheap solar cell or a watch battery (simple calculators) or taking a long time deep in an operation of a more complex calculator. The loop counter then catches ANY iterative process that takes too long, as you mentioned and it calls a halt to the loop, either giving out an approximation if it can, or an error if it can't. Finally, inside the loop it has a faster way to catch repeating processes, whether that's something like 1/3 where it repeats fast, or 1/81 where it takes a while to repeat. Finally, it figures out where it can stop caring, so something like 1/9973, it would loop through 30+ times, realize it hit its precision limit, and return the bits it has so far rather than continuing out 9966 times and realizing from the remainder and last calculated value that it's repeating.
A long time ago Mr. Talbot, my maths teacher, said that we can't accept dividing by zero because if we do we can 'prove' false things. I can't remember exactly how he did it but he proved that 1 = 0. He then said that if you can 'prove' one false thing you can 'prove' any thing that is in fact false. A quick google found this; x and y are 2 non-zero numbers where x=y. So x^2 = xy. Subtract y^2 from both sides, X^2 - y^2 = xy - y^2. Divide both sides by x-y we get x + y = y which is clearly false. Dividing by x-y when x=y is dividing by zero. which is why this goes wrong.
In my grade 12 history class I did a paper about the origins of zero and the teacher had to get a math teacher to grade it. I got a 90% because my writing is dry and transactional like an instruction manual and apparently that counts.
