5 ms·
Some people say "oh, that's easy, 1/0 is +Infinity". So the real fun is at 0/0. The limit of x/y as x and y go to zero depends on which path across the xy plan
by phyzome 8y ago
Some people say "oh, that's easy, 1/0 is +Infinity". So the real fun is at 0/0.
The limit of x/y as x and y go to zero depends on which path across the xy plane you take towards the singularity. Along one approach, the limit is 0, along another approach the limit diverges to infinity, along yet another the limit is 17.
I'm not kidding! Go to https://www.geogebra.org/3d https://www.geogebra.org/3d and enter "x/y" and spin the graph around. The "value" at x=0 y=0 is the entirety of the z-axis.
For another perspective, try http://www.wolframalpha.com/input/?i=z%3Dx%2Fy http://www.wolframalpha.com/input/?i=z%3Dx%2Fy and turn on contour mode for the 3D plot. Notice how the contour lines radiate out from the z-axis; each of those is an "approach" to the singularity at a different z-value, and taking the limit along each line leads to a different value.
- foota 8y agoThat's a wonderful graphing website from the look of it!
- wodenokoto 8y agoI always thought division by zero was, at best, + and - infinity, depending on the path, which is why we leave it undefined. How would a path lead to 17?
- KenoFischer 8y agoTake the path (x,y)=(17a,a) as a goes to zero one dimensionally.
- gpm 8y ago24 / 2 17 / 1 1.7 / 0.1 0.17 / 0.01 0.017 / 0.001 ... 0 / 0
- MereInterest 8y agoTake the limit as x goes to 0 of (17*x)/(x). Both numerator and denominator go to 0, so this is a representation of 0/0. The limit is then equal to 17.
- tigershark 8y agoThis just means that 0/0 = 1. It doesn’t mean that any number / 0 tends to 17.
- phyzome 8y agoRight, it just means the limit of that path tends to 17. It's not a well-defined limit. In a function of one-variable, there's a distinction between one-sided and two-sided limits. I don't know what the terminology would be for multivariable functions, but this is closer in nature to a one-sided limit.
- zormino 8y agoBut think about what a limit is. Think of X and Y as their own functions. so something more like X(g) / Y(g). As g increases, X and Y can take any path across the plane that they want. We get to arbitrarily choose what X and Y are. All we are doing is describe how our limit behaves as we slide along the scale towards infinity. Just because g increased by went from 10 to 11 doesn't mean that X and Y behaved the same way. We can arbitrarily choose X and Y such the limit as g approaches infinity is 0 for both, but the way that X and Y are changing is different. Like the example above, if we choose X and Y such that X(g)/Y(g) is 17 for all g, but the X and Y both approach 0 as g approaches infinuty, then as g approaches infinity X and Y go to 0, but X / Y (0 / 0), goes to 17. It's not a true 0/0, because the functions we chose got to 0 at different rates.
- phyzome 8y agox=170, y=10 x=17, y=1 x=1.7, y=0.1 x=0.17, y=0.01 The answer keeps being 17, even as x and y both get vanishingly close to zero. I encourage you to play around with a 3D graphing calculator and see all the different paths you can take along that surface to reach the singularity. They all "reach" it at different heights.
- DoctorOetker 8y agoyou are explaining why 0 / 0 can approach 17, this is much less controversial, and is often considered the bottom element...
- Jach 8y agoIn calculus you're supposed to learn that y/0 with y != 0 is "undefined" (±Inf with limit expressions), while 0/0 is "indeterminate" (can construct any value using a limit expression that contains x/x with x -> 0).
- legohead 8y agoI would argue 1/0 is infinity. Not with mathematics, just logically. By using the wording "how many times does 0 go into 1?" You bring up 0/0. But 0/<anything> is already defined, it's 0. So 0/infinity = 0 Programmatically, I think I've always wanted X/0 to be 0. For example: progress bars, currency, or damage in a video game. It wouldn't be very helpful to have infinity be an answer there.
- banachtarski 8y agoYour argument is flawed because your zero is positively biased. Why infinity and why not negative infinity.
- deleted 8y ago[deleted]
- phyzome 8y agoI encourage you to look at the graphs. Yes, the value along the y-axis is 0, and the value along the x-axis approaches positive and negative infinity. It's path-dependent. Take a look.
- rurban 8y agoIn float yes. In int there's no Inf. There's only MAX_INT or 0 or throw an exception. Normal langs throw. That what the checked int div will do. But the unchecked int div has no return type which allows throwing an error. And pony is strongly typed, unlike most other langs. That's why it can guarantee much more safeties and performance than all other languages.
- anoplus 8y agoI like your point here. I think 1/0 = Infinity+ is a satisfying expression. Its a clear concept which can be visualized in a simple graph. I think 0/0 is a different concept than 1/0. It's a different expression. 0/0 doesn't explicitly express a particular "path" on the graph. We can call it (0/0) different names if we want. They can say 0/0 = "undefined". I am currently satisfied with 0/0 simply equals to 0/0, or simply "undefined". If we all agree it is "undefined" we ironically defined it.
- ummonk 8y ago>I like your point here. I think 1/0 = Infinity+ is a satisfying expression. Its a clear concept which can be visualized in a simple graph. Except it isn't. It still depends on which side you take the path from - from the positive denominator or the negative denominator side. (This is why IEEE 754 has a single signless infinity).
- anoplus 8y agoOpps, I forgot the negative side. Thanks.
- jibal 8y ago> IEEE 754 has a single signless infinity No it doesn't. I don't know where you got that idea, but it can't be from ever writing any floating point code, or looking at the IEEE-754 standard, or the floating point format. Please see https://www.h-schmidt.net/FloatConverter/IEEE754.html https://www.h-schmidt.net/FloatConverter/IEEE754.html and stop spreading radically wrong misinformation. > It still depends on which side you take the path from - from the positive denominator or the negative denominator side. In IEEE-754, 1/0 == Infinity and 1/-0 == -Infinity.
- ummonk 8y agoDon't know why you're getting downvoted, but thanks for the clarification. Looks like I misremembered. Apparently it also has +0 and -0...
- 8y ago
- yoz-y 8y agoSince the "1/0 = 0" is only defined for integer division then I'd argue that the chosen constant should be MAX_INT. (or MIN_INT in case of signe values and negative numerator)
- vmchale 8y agoLimits aren't the only thing broken by division by zero, though. For integers it's mainly algebraic stuff.