It's a concept which intrigues mathematicians, but scientists aren't so keen on it. More at www.sixtysymbols.com/ With Professor Laurence Eaves and Mike Merrfield
Once when he was still quite small my son asked me what the largest number was. I of course replied that there wasn't one. Upon his insistence that there must be I told to imagine that he take the largest number, whatever it was, and then add one to it. His eyes went wide and his jaw literally dropped. The look of someone who has just seen over the edge. Priceless.
I think the concept of 0 is even more profound than that of infinity. It's hard to imagine something going on forever, but even harder for me to imagine nothingness. No matter, no energy, no space, no anything. Profound.
To work out the answer to an integer divided by zero, you need to first define the number set that you are talking about. For natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), real numbers (R) and complex numbers (C), the answer is undefined, because there is no element of the set which satisfies the problem. However, when you start using limits and extended real and complex number systems, infinity is defined and can be a solution to the problem. (also see alephs!).
A great example of the math/physics divide is when you're making a Taylor series approximation. Once you write out a few terms, a physicist would say that the remainder is negligible, while a mathematician would worry very much about whether those remainder terms do in fact vanish.
tavi921 Mathematically, no, 3/0 does not equal infinity because, as you say, it leads to various paradoxes which break maths. That, however, is a problem with algebraic maths, rather than 3/0 not actually being equal to infinity. The problem is that algebraic maths cannot describe infinity adequately. Its like saying 3/0 = banana, and 7/0 = banana so 3 = 7. Thats just what happens when you use a term in algebraic maths that has no mathematical value. Obviously, you can divide 3 by 0 an infinite number of times. This is all that statement means. It has no mathematical value, so you don't do it, but fundamentally its demonstrably true. Mathematically, 3/0 = undefined. And the reason it is undefined is because no one knows how to define it.
***** What I meant is that infinity is a term that describes a concept, not a number. There is no number infinity. So "=" is understood in a logical sense instead of a mathematical sense. Actually I think you were making the same point.
When he was about three years old my child started talking about "the end of the number". I soon realised that he was talking about the concept of infinity. And I was surprised because I was planning to introduce this concept to him a bit later. He didn't have the right word then, so he made up his own.
The limit as x approaches 0+ of 3/x is infinity, but the limit as it approaches 0- is negative infinity, therefore you cannot associate a value to a number divided by zero, hence the term "undefined".
There's just one room cleaner, despite the size of the hotel. You'd think that would be a problem, and that the hotel is ridiculously understaffed, but the room cleaner has a clever way of getting around this issue. He cleans each room in half the time as he cleaned the last. He spends one hour on the first room, half an hour on the second, one quarter of an hour on the third, ect. Eventually, he gets all rooms cleaned within 2 hours.
I spent some time learning catastrophe theory at University because it helped removed irritating singularities from my calculations in theoretical Chemistry. It used to be the hip theory in the 1970s but by the time I was at University it was a great tool for dealing with infinities.
@MarkArandjus The problem with that theory is that it doesn't take into account negative numbers- which are smaller than 0.0001. Rather than being a slope down, it actually is a parabola that stops at infinity and goes to -3 and slopes upward.
There are cardinal numbers, ordinal numbers, hyper-real numbers, numbers from smooth analysis, and surreal numbers. Those are structures that can be used to discuss infinities within mathematics.
I was about to post the same comment (7 years later though). Indeed, 3 divided by 0 is not INFINITY !!!!!!!!!!!!!!!! It may have been wiser to ask the question to mathematicians.
@9hello123 A graph of a non-continous function approaches infinity near the asymptote, the asymptote itself is a point sized hole in the graph, which is why 3/0 is an undefined constant.
3/0 is NOT infinity. You can't say something is equal to infinity. Furthermore, if you say that 3/0 is infinity, then so is 7/0 which means that 3=7, which is obviously not true.
I think there are different kinds of infinity, let 3/0 = inf_3, and 7/0 = inf_7, then inf_3 * 7/3 = inf_7 In the case of the hotel, let the total number of friends to be inf_x, then in order to fit them in the said fashion, the hotel must have at least inf_2x rooms. Of course, this has little influence on our daily life, unless we are mathematicians.
DA oliHVAR No, I am not, although my formal education is yet to be finished I do believe that I possess sufficient rights to declare my understandings. You, on the other side, is being un reasonably presumptuous and insulting, only because my understanding may appear foolish to you, whatever level of education you may have, you do not appear to me as a modest, professional intellect, and for that reason, I refuse to hear anything else you would say.
I don't know what you mean when you say nothing can be equal to infinity. I mean isn't it obvious that ∞=∞? But anyway, you're right that 3/0 cannot be ∞. It would be like saying "infinity, zero times, is 3." Which is of course jibberish.
@itsmanofpopsicle Yes it does, If we're talking about f(Y) = 3/X, then the asymptote is still 0, it would take an infinite value of Y to reach X. Even if Y is a negative number. In fact if you go to Wikipedia and go to the asymptote article, there's a graph of such a function drawn out.
@kristijanadrian No, undefined is a synonym for unknown. ie: either there is no known answer or the question is illogical. Zero and infinity are opposite mathematical concepts, the only thing they have in common is that neither fits neatly into the human mind because of our lack of physical experience with them.
As soon as we use numbers in association with Geometry we soon come up against the problem of Infinity. The Natural Numbers 1, 2, 3, 4,., are used to count discrete objects. In the two-dimensional plane we assume that any point can be represented by a pair of numbers (a, b). Considering a straight line of length one unit, we assume that it consists of points. However, a point has zero width. One then has the problem of how many points you need to form the line. Dedekind and Cantor considered this problem, and the different types of Infinity. However does this Cartesian System of Geometry correctly model the space in which we live? (The Cartesian System assumes that space is continuous). Leibniz considered this problem, and felt that space was NOT continuous; he developed the concept of Infinitesmals and his Differential Calculus from this different concept of space. Perhaps Leibniz had stumbled upon a vague sense of the Planck length.
@Haiguise1 Take the example of 1/f noise (i.e, signal). It exist in the physiological data such as heart rate, EEG, eye movements. At what scale you take measurements you still see self similarity.
A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of a/0, where a ≠ 0, as being Infinity. This infinity can be either positive, negative, or unsigned, depending on context.
@LamaPaj That's correct. ∞ is actually the limit of y=3/x when x approaches 0 (if x approaches from the left, if it is negative, then the limit is - ∞). He just wrote it that way to make it more easily understandable. I'm pretty sure he knows that ;)
for instantaneous velocity it is valid to say that it is the velocity an object will have if from the moment we are interested in it performs uniform linear motion (ΣF=0). the same for acceleration (obviously in this case ΣF=constant)
@Lunarsight Graph f(x)=1/x. The reason a constant divided by 0 is undefined is entirely dependent upon which side the number is coming from--either the positive or the negative. For example, the limit as x approaches 0 from the positive side is +infinity, but the limit as x approaches 0 from the negative side is -infinity. The reason it's undefined is because the two points do not meet.
@THERaikami1 The natural numbers are infinite, but countable - 1, 2, 3, etc. But there are an infinite number of numbers in between 1 and 2, which are uncountable. This is because there are a finite number of natural numbers between any two natural numbers (if x and y are natural numbers and x < y, then there are (y-x)-1 natural numbers in between them), but an infinite number of numbers between any two numbers in general.
I've written this also as a comment to another Brady's video: Physical world can be described with rational complex numbers. There's no need for irrational numbers as using Plank's measurements all measurements are quantified. But you still need complex numbers to easily describe for example elecro-magnetic fields. On the other hand as a mathematician I'm fascinated by the worlds that are created by imagining infinities that are between aleph0 and aleph1 or lager than aleph1 but not part of the power series of alephs. And that there is such thing as Banch-Tarski paradox which is impossible in nature as one never can cut anything the way the paradox cuts things up.
Me: Oh hey I've seen a Numberphile video on this already 0:25 - "You shouldn't really be talking to a scientist you should be talking to a mathematician" Me: ...I knew it
@madadane123 i think its actually undefined if you want to get technical, because if you plot it on a graph the gradient shoots off towards infinity and infinitesimal
@123Fusselbirne the range is how far the effect is able to reach. and as no matter how far away you get from an object, gravity still interacts. that means it's range is infinite
@immrgrunty1363: there is no such thing as dividing a number endless times. You can define infinity as the limit of 3/x as x approaches 0 from the right, but then you are dealing with limits and not regular division. Also, you must explicitly state whether you are coming from the left or right, as in the two cases you end up with + or - infinity (in this case those are different things). If you define 3/0 in regular division, then regular algebra breaks and you cannot solve anything.
Perhaps a good analogy would be an infinitely long tape measure - the number of ticks is infinite, since the tape measure has infinite length, but you can count the number of ticks along any given length of the tape. There are also an infinite number of places where you *could* draw ticks along the edge of the tape, and even between two ticks, there is still an infinite number of positions you could draw more ticks. That would be an uncountable infinity.
@TriMiro8107 3/0 is undefined because the in that circumstance, infinity is treated as '0' dividing '3' an endless number of times. so yes, 'infinity,' in a sense, works in that circumstance.
@MrSuednym Other way around. Mathematicians are applied Physicists. That is a little unfair because both are independent or can be, but they both need each other to get anywhere.
ö. . , Yeah, and how do you fill a hotel with infinite amount of rooms in the first place? And if you do, and this scenario of moving from room to room happens it takes an eternity - like literally. And did they finish building of the hotel in the first place - I hear the construction is still going on there... last night when I checked in:)
No, Brady Haran is the same person as Brady Haran, they're not just two people randomly having the same name. Also, most of the people in the sixtysymbols videos have already been in a few numberphile videos, and the same goes the other way around.
the definition of the base unit, ampere "the constant current that will produce an attractive force of 2 × 10-7 newton per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum" which contains infinity and can we can think of negligible cross section as an infinitecimal?
By definition a countably infinite set is a set that can be put into one-to-one correspondence with the set of natural numbers. For example, you can put the natural numbers and the odd naturals into such a correspondence.
As far as I know, there is at least one physical effect where true infinities become physical. It's the Casimir's effect: the force in vacuum making two parallel plates, spread by a small gap, attract to each other. To derive the magnitude of this force, physicists have to write an equation balancing the infinite number of virtual light waves outside the gap and another infinite number of similar waves inside the gap. The difference between these two infinities exactly predicts the actual force, which can be measured in the lab. But the most interesting fact in this story is that the maths behind this derivation, which seems utterly surreal (more surreal than just infinities), is in fact physical and very real.
There are no exceptions in mathematics. A number divided by a number equals a number. The idea that a number divided by itself is defined to be equal to one is flawed due to how trends work, but regardless a number divided by itself equals one. 0/0 = R, which contains 1, so consistency is preserved. Also: 0/∞ = 0. Notice that commutatively, that means 0/0 = ∞, which is consistent with 0/0 equaling R. R is all real numbers.
So, the distinction between Mathematicians and Physicists is wether you are concerned with axial-tangential sync-duration limits or compound relative-timing duration wave-particle type measurements, and philosophically, of course it's the same indeterminate qa principle of the perceived meaning of Actuality.., the degree/proportions of Cultural Relativism inherent in "truth in labeling, lawful-ness".
Countable and uncountable infinity.. now it makes sense.. i had been wondering how Chuck Norris could've counted to infinity twice. Honestly, i understand about .01% of what i see and hear on this channel, but i can't stop watching.
Indeterminate is undefined, just that it is possible to assign it any value, though it isn't very helpful. When it comes to something that is undefined, there just isn't a value that can be used to represent it yet. For a time SQRT(-1) was undefined, but now we use i or j. Same thing applies to 1/0, which you can prove a value for if anyone ever payed attention to factorials.
@bohngat But they are mathematical constructions, If you take a snowflake for example there still is a minimum 'resolution' for the fractal which is at the size of the water molecules.
@123Fusselbirne exactly. it's kinda funny because you don't really use the symbol for infinity in physics because if you did then F=0 and then the range wouldn't be truly infinite. but like you mention you can have an unimaginably large number for the distance and no matter how much larger you make it there will still be a force. mathematicaly you just can't use the symbol infinite ^^
The problem with dividing by zero is that dividing by something has two meanings: 1) how many times does the bottom number fit in the top one? the answer to this would indeed be infinity, but also 2) the result * the bottom number makes the top one, and this is where you get problems, because 0*x = 0, so even infinity*0 would still equal 0, and not 3. Therefore, dividing by zero is impossible. (also, 0/0 is technically 1, this fits every description of division)
I had a comment before this explaining that (apart from -1^2 actually being -1 and (-1)^2 being 1), the opposite of squaring x is actually taking the positive or negative square root of x, and therefore your conclusion is incorrect, and not just generally taking the square root, whereas the opposite of dividing something by y is just multiplying it by y.
You can't have an 'Infinity Hotel' that's full up. The infinite rooms never fill up and the infinite guests never run out. The concept of infinity can only be accurately be described by the 'present continuous'. It's an imperfective rather than perfective aspect. It's like saying; I read a book with an infinite number of pages; it's a conceptual paradox. All one can actually say is - I am reading a book with an infinite number of pages.
I am kinda of a maths noob, but here it comes: From the hotel example they push guests to the cliff of infinity, just as they thought the world was flat, if you get to the end of the world you would fall off. Then they figured out the world was round and you could sail around it (Thanks to mister magellan). So imagen the number series was also round, and zero is the exact opposite of infinity. This means the guest at the hotel were shoved towards infinity, pop out in negative infinity and go towards zero. which means that if guest 1 is moved to room 2, guest 2 to room 3, and so fort, and in the end, guest minus 1 to room 1 again (assuming there is no room zero or infinity). That makes more sense to me than pushing them of the cliff of infinity and claiming they are still there, not? For the maths bit, this means infinity plus 1 is not also infinity (as the hotel example proclaimes), infinity plus 1 is minus infinity plus 1. Just as zero plus 1 is minus zero plus 1. And the circle is round. Then again, the number series circle would be infinitly huge, so 4th dimension or something? Perhaps land a little Riemann sphere on that one? Anyone?
6:18 I'm as far to a mathematician as you can think (I'm a sociologist) but I too hate when physicists and astronomers misuse the term infinity. It's one of the few complains I have anyway. Otherwise I love them and admire them deeply :-)
i think you're talking about the limit of 1/x, which is undefined as x approaches 0 for the reasons you stated - the left and right sides of the limit aren't equal to each other. you could say 1/x^2 is defined as infinity as x approaches 0 because both sides of the limit equal each other, but you must remember we are talking about a limiting function and not simple division such as 1/0 (with limits we are talking about 1/a number really really really really close to 0) 1/0 is undefined
Infinity, more or less (un)countability is often used in computation theory (the mathematical part of computer science) and used to describe problems and their computable complexity (won't go into details). It's not useless and not just something of the mind as it helps us in many ways (like knowing what to put effort into in algorithmics) and can be shown to be there for instance in the proof of the halting problem with use of diagonalisaton (look it up).
The Hotel problem seems trivial, but I do not think I ever thought about infinity properly until I learned of the problem and its solutions. I think it is fundamental to actually understanding what infinity is.
Hilbert’s Hotel sounds like a logical sleight of hand. What is the mathematical definition of “full”? If it means that every room has been assigned a positive integer then there are never any rooms available. The only way to check in is to find a non-Integer room such as room number pi.
Infinity as a concept is easy to understand, it's the implications in the real world that's the hard part. As evidenced by all the armchair mathematicians in the comments.
You are right that stars are about the same density at any point in space (the same number of stars per any specific volume) what they were saying is within the observable universe, as far as light has travelled since the big bang, there is a very large but finite number of stars. We can't tell how far it extends beyond this.
In my mind, infinity is such an alien concept, that the moment I hear someone talking about "infinity this" or "infinity that" or, speaking as if they have a grasp of it; I find it almost arrogant in a way. It is a purely abstract concept. I really don't think something infinite can exist, but I'm an agnostic there. At least we don't know of anything infinite yet. What I mean is, yeah, the hotel problem seems trivial, but in essence, it because we can't understand it, and might never do. Agree?
The term is indeterminate. And 0/0 = R. It is said to be indeterminate not because the answer isn't a single number but because the limit is indeterminate. Indeterminate is used only for limits. Indeterminate isn't what 0/0 actually equals; it means the limit calculation is indeterminate because it results in R, so you need to do something to find the real limit. And you cannot say 2=3 via 0/0. You could say 1=2=3=4=5=6=7 (etc.) but it is really R = R. 0/R = 0, so 0/0 = R. Do the long division.
Acctually, that would be 3/0*0=2/0*0 so that still doesn't make sense. if you say 2/3=2/3 and you multiply both sides by 3, the /3 goes away. 2=2. So you could say 2/0=3/0 meaning 2=3. But yeah, my explaination had some flaws, but allowing division by zero creates a lot of incredible complicated problems.
Ups looks like I said that the size of a singularity is also infinite. What I actually meant was that the formulas give an answer of zero or "infinitesimally small" (which again doesn't make sense and is impossible to grasp).
No, there are different "sizes" of infinity. It really should be considered different "types". It is basically having an infinite number of things (countable) and being able to divide something into infinite pieces (uncountable). It is might be a misconception to say one is "bigger" than the other, but they do have different properties and "uncountable" does seem "bigger" because it contains more things (infinitely more things) than "countable".
@megaelliott That's not quite true either. lim 3/x, x-> 0+ = ∞, yes, but lim 3/x, x->0- = -∞. Technically the limit doesn't actually exist because the left- and right-hand limits are disjoint.
Alright. You make a good point. The inifity that we "understand" is our interpretation of infinity since no real infinity can exist, and hence, it is trivial and we can understand it. I think I just have to accept that when mathematicians, and physicists to a lesser degree, talk about infinity, they are talking about purely maths and the concept of infinity and not the real thing. Hmm. I must ponder upon this some more. Peace!
I thought the gravitational force that acts between two masses has a range from 0 to infinity 1/r^2 law, it is similar with electric fields so there are some uses of infinity in physics.
Infinity isn't a number, it's just an idea or concept, so we can't write something like 1/0= infinity anymore than we can write it equals apple pie or pogo stick. We can't treat it like a normal number because it isn't one. The most we can say is that something tends to infinity.
Saying 3/0 = infinity is just a simplified explanation of the mathematical concept, which is lim x->0 3/x = inf (the limit as x approaches 0 of 3/x is infinity) That is to say that 3/0 itself has no value, but the behavior on either side is to stretch unrestrainedly large in the positive direction with no upper bound. So, yes, saying "3/0 = infinity" is technically incorrect, but it's close enough to get the concept across.
Heh, glad to see I'm not the only math major who was shaking his head sadly at the 3/0 equals infinity bit. And of course the hotel had a countably infinite number of rooms so if an uncountably infinite number of new guests showed up (ie one for each real number) the hotel couldn't fit them in.
because the limit as x approaches 0 from the left of 3/0 equals -inf, which does not equal inf and therefore does not equal the limit from the left. Of course since this is a science video, not a maths video, we could very well state that 3/0 has a superstate of being inf and -inf at the same time. Of course it really depends on what you're trying to do with 3/0 and whichever would be most proctical
Not at all - the main problem being that of time dilation. Given an infinite amount of time, the matter will be condensed into a single point (although Heisenberg uncertainty principle says you can't actually have a point), but generally speaking the centre point is never actually reached, matter just continues to get closer and closer to it. What actually happens is a matter of speculation, as spacetime is screwy in a black hole and we're talking Planck distances, but basically size > 0.
@kristijanadrian Actually an infinte pound bill would have the interesting property that no matter what you bought with it you would always get an infinite pound note as change. The problem with issuing infinite pound notes is social, not mathematical.
@Phoboskomboa ...and simultaneously collapse the entire universe into an Infinite black hole.Damn that's a nice scary interpretation of it. Infinity is a yawning chasm of utter oblivion which has driven brilliant men insane. I try to give it a wide berth.
If you had infinity of any substance it would not only fill all gaps in the universe, but it would cram all space in the universe to an infinite density. Then it would cause the universe to expand to an infinite size.
The day I learned about Cantor's diagonalization technique to prove that the integers are a lesser order of infinity than the real numbers, I knew that I was destined to be a nerd.
3 / 0 Is not infinity for several reasons: First, when a / b = c this means that if you take c multiples of b, you get a. It does not matter how many multiples of 0 you take, you will always get 0, ALSO when done infinitely many times. So '3 / 0 = infinity' would create a contradiction. Second, when a / 0 = infinity, then a = infinity * 0, so 3 = infinity * 0 and 7 = infinity * 0. This gives 3 = 7, which is a contradiction.
@ 2:11 please note the Egyptian symbol he shows for infinity is partially the same as the glyph/symbol for KA or the body double, your shadow that follows you around for an eternity. The ka as a spiritual double was born with every man and lived on after he died as long as it had a place to live. After death the ka became supreme. Kings thus claimed to have multiple kas. Rameses II announced that he had over 20. science is a religion >> U do know that?
Once when he was still quite small my son asked me what the largest number was. I of course replied that there wasn't one. Upon his insistence that there must be I told to imagine that he take the largest number, whatever it was, and then add one to it.
His eyes went wide and his jaw literally dropped. The look of someone who has just seen over the edge. Priceless.
SlideRulePirate Until you have aleph-null...
Or until you get a bit 8-bit about that. :)
The largest infinity might be the number of numbers in total, not
including that number itself (all the ordinals, cardinals, surreal
numbers, etc.)
I am that son.
1:20 *matt parker voice* pfft physicists
What,,, there is no such voice
I heard it.
I think the concept of 0 is even more profound than that of infinity. It's hard to imagine something going on forever, but even harder for me to imagine nothingness. No matter, no energy, no space, no anything. Profound.
To work out the answer to an integer divided by zero, you need to first define the number set that you are talking about. For natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), real numbers (R) and complex numbers (C), the answer is undefined, because there is no element of the set which satisfies the problem. However, when you start using limits and extended real and complex number systems, infinity is defined and can be a solution to the problem. (also see alephs!).
A great example of the math/physics divide is when you're making a Taylor series approximation. Once you write out a few terms, a physicist would say that the remainder is negligible, while a mathematician would worry very much about whether those remainder terms do in fact vanish.
tavi921 Mathematically, no, 3/0 does not equal infinity because, as you say, it leads to various paradoxes which break maths. That, however, is a problem with algebraic maths, rather than 3/0 not actually being equal to infinity. The problem is that algebraic maths cannot describe infinity adequately. Its like saying 3/0 = banana, and 7/0 = banana so 3 = 7. Thats just what happens when you use a term in algebraic maths that has no mathematical value.
Obviously, you can divide 3 by 0 an infinite number of times. This is all that statement means. It has no mathematical value, so you don't do it, but fundamentally its demonstrably true.
Mathematically, 3/0 = undefined. And the reason it is undefined is because no one knows how to define it.
We define it by the term "infinity".
oh_no_mrbill Define it mathematically so that you can do calculations with it.
***** What I meant is that infinity is a term that describes a concept, not a number. There is no number infinity. So "=" is understood in a logical sense instead of a mathematical sense. Actually I think you were making the same point.
oh_no_mrbill You literally just said what I explained in my original comment, yes.
***** I just wanted to clear it up, when you said "no one knows how to define" infinity it does have a very clear logical definition.
When he was about three years old my child started talking about "the end of the number". I soon realised that he was talking about the concept of infinity. And I was surprised because I was planning to introduce this concept to him a bit later. He didn't have the right word then, so he made up his own.
> "Very, very big" is as far from the infinity as you can get.
Oh, this is going to be one of my favourite quotes of all time, ever.
I love this zoom in on Wallace's face. It starts innocently, then quickly becomes "I will destroy you. Infinitely."
The limit as x approaches 0+ of 3/x is infinity, but the limit as it approaches 0- is negative infinity, therefore you cannot associate a value to a number divided by zero, hence the term "undefined".
infinite rooms? hate to be the roomcleaner at that hotel
There's just one room cleaner, despite the size of the hotel. You'd think that would be a problem, and that the hotel is ridiculously understaffed, but the room cleaner has a clever way of getting around this issue. He cleans each room in half the time as he cleaned the last. He spends one hour on the first room, half an hour on the second, one quarter of an hour on the third, ect. Eventually, he gets all rooms cleaned within 2 hours.
@@katiekatie6289 what
I spent some time learning catastrophe theory at University because it helped removed irritating singularities from my calculations in theoretical Chemistry.
It used to be the hip theory in the 1970s but by the time I was at University it was a great tool for dealing with infinities.
i like a lot the way this professor explains everything, very clear
@MarkArandjus The problem with that theory is that it doesn't take into account negative numbers- which are smaller than 0.0001. Rather than being a slope down, it actually is a parabola that stops at infinity and goes to -3 and slopes upward.
There are cardinal numbers, ordinal numbers, hyper-real numbers, numbers from smooth analysis, and surreal numbers. Those are structures that can be used to discuss infinities within mathematics.
1:28 NOOOOOOOOOOOOOOOOOOOOOOOO
I was about to post the same comment (7 years later though).
Indeed, 3 divided by 0 is not INFINITY !!!!!!!!!!!!!!!!
It may have been wiser to ask the question to mathematicians.
@9hello123 A graph of a non-continous function approaches infinity near the asymptote, the asymptote itself is a point sized hole in the graph, which is why 3/0 is an undefined constant.
3/0 is NOT infinity. You can't say something is equal to infinity. Furthermore, if you say that 3/0 is infinity, then so is 7/0 which means that 3=7, which is obviously not true.
I think there are different kinds of infinity, let 3/0 = inf_3, and 7/0 = inf_7, then
inf_3 * 7/3 = inf_7
In the case of the hotel, let the total number of friends to be inf_x, then in order to fit them in the said fashion, the hotel must have at least inf_2x rooms.
Of course, this has little influence on our daily life, unless we are mathematicians.
you are in the 6th grade aren't you
DA oliHVAR No, I am not, although my formal education is yet to be finished I do believe that I possess sufficient rights to declare my understandings. You, on the other side, is being un reasonably presumptuous and insulting, only because my understanding may appear foolish to you, whatever level of education you may have, you do not appear to me as a modest, professional intellect, and for that reason, I refuse to hear anything else you would say.
James Wang You are still wrong about the arithmetic though.
I don't know what you mean when you say nothing can be equal to infinity. I mean isn't it obvious that ∞=∞? But anyway, you're right that 3/0 cannot be ∞. It would be like saying "infinity, zero times, is 3." Which is of course jibberish.
@itsmanofpopsicle Yes it does, If we're talking about f(Y) = 3/X, then the asymptote is still 0, it would take an infinite value of Y to reach X. Even if Y is a negative number.
In fact if you go to Wikipedia and go to the asymptote article, there's a graph of such a function drawn out.
@kristijanadrian No, undefined is a synonym for unknown. ie: either there is no known answer or the question is illogical. Zero and infinity are opposite mathematical concepts, the only thing they have in common is that neither fits neatly into the human mind because of our lack of physical experience with them.
As soon as we use numbers in association with Geometry we soon come up against the problem of Infinity. The Natural Numbers 1, 2, 3, 4,., are used to count discrete objects. In the two-dimensional plane we assume that any point can be represented by a pair of numbers (a, b). Considering a straight line of length one unit, we assume that it consists of points.
However, a point has zero width. One then has the problem of how many points you need to form the line. Dedekind and Cantor considered this problem, and the different types of Infinity. However does this Cartesian System of Geometry correctly model the space in which we live? (The Cartesian System assumes that space is continuous).
Leibniz considered this problem, and felt that space was NOT continuous; he developed the concept of Infinitesmals and his Differential Calculus from this different concept of space. Perhaps Leibniz had stumbled upon a vague sense of the Planck length.
great videos, link and info, thanks
@Haiguise1
Take the example of 1/f noise (i.e, signal). It exist in the physiological data such as heart rate, EEG, eye movements. At what scale you take measurements you still see self similarity.
A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of a/0, where a ≠ 0, as being Infinity. This infinity can be either positive, negative, or unsigned, depending on context.
@LamaPaj That's correct. ∞ is actually the limit of y=3/x when x approaches 0 (if x approaches from the left, if it is negative, then the limit is - ∞). He just wrote it that way to make it more easily understandable. I'm pretty sure he knows that ;)
for instantaneous velocity it is valid to say that it is the velocity an object will have if from the moment we are interested in it performs uniform linear motion (ΣF=0). the same for acceleration (obviously in this case ΣF=constant)
@Lunarsight Graph f(x)=1/x. The reason a constant divided by 0 is undefined is entirely dependent upon which side the number is coming from--either the positive or the negative. For example, the limit as x approaches 0 from the positive side is +infinity, but the limit as x approaches 0 from the negative side is -infinity. The reason it's undefined is because the two points do not meet.
@THERaikami1 The natural numbers are infinite, but countable - 1, 2, 3, etc. But there are an infinite number of numbers in between 1 and 2, which are uncountable. This is because there are a finite number of natural numbers between any two natural numbers (if x and y are natural numbers and x < y, then there are (y-x)-1 natural numbers in between them), but an infinite number of numbers between any two numbers in general.
I've written this also as a comment to another Brady's video: Physical world can be described with rational complex numbers. There's no need for irrational numbers as using Plank's measurements all measurements are quantified. But you still need complex numbers to easily describe for example elecro-magnetic fields.
On the other hand as a mathematician I'm fascinated by the worlds that are created by imagining infinities that are between aleph0 and aleph1 or lager than aleph1 but not part of the power series of alephs. And that there is such thing as Banch-Tarski paradox which is impossible in nature as one never can cut anything the way the paradox cuts things up.
Me: Oh hey I've seen a Numberphile video on this already
0:25 - "You shouldn't really be talking to a scientist you should be talking to a mathematician"
Me: ...I knew it
@madadane123 i think its actually undefined if you want to get technical, because if you plot it on a graph the gradient shoots off towards infinity and infinitesimal
I love how at 1:18 he says "and divide it by zero" as if he had a choice...
You, Sir, made my day. :)
@123Fusselbirne the range is how far the effect is able to reach. and as no matter how far away you get from an object, gravity still interacts. that means it's range is infinite
Just a suggestion but what would be really good is a whole channel dedicated to mathematics :)!
@immrgrunty1363: there is no such thing as dividing a number endless times. You can define infinity as the limit of 3/x as x approaches 0 from the right, but then you are dealing with limits and not regular division. Also, you must explicitly state whether you are coming from the left or right, as in the two cases you end up with + or - infinity (in this case those are different things).
If you define 3/0 in regular division, then regular algebra breaks and you cannot solve anything.
Perhaps a good analogy would be an infinitely long tape measure - the number of ticks is infinite, since the tape measure has infinite length, but you can count the number of ticks along any given length of the tape. There are also an infinite number of places where you *could* draw ticks along the edge of the tape, and even between two ticks, there is still an infinite number of positions you could draw more ticks. That would be an uncountable infinity.
@TriMiro8107 3/0 is undefined because the in that circumstance, infinity is treated as '0' dividing '3' an endless number of times. so yes, 'infinity,' in a sense, works in that circumstance.
Infinity is a number that's always greater than 5, no matter how large a value of 5 you use.
@LamaPaj A better way to put it is that 3/t tends towards infinity as t tends to 0.
@MrSuednym
Other way around. Mathematicians are applied Physicists. That is a little unfair because both are independent or can be, but they both need each other to get anywhere.
@OhhSoulz I'm glad it helped.
Worst hotel ever. Everytime someone checks in you have to move... you won't get any sleep there.
ö. . , Yeah, and how do you fill a hotel with infinite amount of rooms in the first place? And if you do, and this scenario of moving from room to room happens it takes an eternity - like literally. And did they finish building of the hotel in the first place - I hear the construction is still going on there... last night when I checked in:)
No, Brady Haran is the same person as Brady Haran, they're not just two people randomly having the same name. Also, most of the people in the sixtysymbols videos have already been in a few numberphile videos, and the same goes the other way around.
the definition of the base unit, ampere "the constant current that will produce an attractive force of 2 × 10-7 newton per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum" which contains infinity and can we can think of negligible cross section as an infinitecimal?
By definition a countably infinite set is a set that can be put into one-to-one correspondence with the set of natural numbers. For example, you can put the natural numbers and the odd naturals into such a correspondence.
As far as I know, there is at least one physical effect where true infinities become physical. It's the Casimir's effect: the force in vacuum making two parallel plates, spread by a small gap, attract to each other. To derive the magnitude of this force, physicists have to write an equation balancing the infinite number of virtual light waves outside the gap and another infinite number of similar waves inside the gap. The difference between these two infinities exactly predicts the actual force, which can be measured in the lab. But the most interesting fact in this story is that the maths behind this derivation, which seems utterly surreal (more surreal than just infinities), is in fact physical and very real.
There are no exceptions in mathematics.
A number divided by a number equals a number.
The idea that a number divided by itself is defined to be equal to one is flawed due to how trends work, but regardless a number divided by itself equals one. 0/0 = R, which contains 1, so consistency is preserved.
Also: 0/∞ = 0. Notice that commutatively, that means 0/0 = ∞, which is consistent with 0/0 equaling R.
R is all real numbers.
So, the distinction between Mathematicians and Physicists is wether you are concerned with axial-tangential sync-duration limits or compound relative-timing duration wave-particle type measurements, and philosophically, of course it's the same indeterminate qa principle of the perceived meaning of Actuality.., the degree/proportions of Cultural Relativism inherent in "truth in labeling, lawful-ness".
Countable and uncountable infinity.. now it makes sense.. i had been wondering how Chuck Norris could've counted to infinity twice. Honestly, i understand about .01% of what i see and hear on this channel, but i can't stop watching.
Indeterminate is undefined, just that it is possible to assign it any value, though it isn't very helpful. When it comes to something that is undefined, there just isn't a value that can be used to represent it yet. For a time SQRT(-1) was undefined, but now we use i or j. Same thing applies to 1/0, which you can prove a value for if anyone ever payed attention to factorials.
@bohngat But they are mathematical constructions, If you take a snowflake for example there still is a minimum 'resolution' for the fractal which is at the size of the water molecules.
You can however define 1/infinity because the limit as x approaches infinity of a/x is always equal to zero, there is no plus or minus zero.
@123Fusselbirne exactly. it's kinda funny because you don't really use the symbol for infinity in physics because if you did then F=0 and then the range wouldn't be truly infinite. but like you mention you can have an unimaginably large number for the distance and no matter how much larger you make it there will still be a force. mathematicaly you just can't use the symbol infinite ^^
is (1/0 = infinty) greater or (2/0 = infinty ) greater ?
another (infinty + 1 = infinty or greater number that have another special name)?
The problem with dividing by zero is that dividing by something has two meanings: 1) how many times does the bottom number fit in the top one? the answer to this would indeed be infinity, but also 2) the result * the bottom number makes the top one, and this is where you get problems, because 0*x = 0, so even infinity*0 would still equal 0, and not 3. Therefore, dividing by zero is impossible. (also, 0/0 is technically 1, this fits every description of division)
You're welcome, happy to help :)
I had a comment before this explaining that (apart from -1^2 actually being -1 and (-1)^2 being 1), the opposite of squaring x is actually taking the positive or negative square root of x, and therefore your conclusion is incorrect, and not just generally taking the square root, whereas the opposite of dividing something by y is just multiplying it by y.
You can't have an 'Infinity Hotel' that's full up. The infinite rooms never fill up and the infinite guests never run out.
The concept of infinity can only be accurately be described by the 'present continuous'. It's an imperfective rather than perfective aspect. It's like saying; I read a book with an infinite number of pages; it's a conceptual paradox. All one can actually say is - I am reading a book with an infinite number of pages.
I am kinda of a maths noob, but here it comes: From the hotel example they push guests to the cliff of infinity, just as they thought the world was flat, if you get to the end of the world you would fall off. Then they figured out the world was round and you could sail around it (Thanks to mister magellan). So imagen the number series was also round, and zero is the exact opposite of infinity. This means the guest at the hotel were shoved towards infinity, pop out in negative infinity and go towards zero. which means that if guest 1 is moved to room 2, guest 2 to room 3, and so fort, and in the end, guest minus 1 to room 1 again (assuming there is no room zero or infinity). That makes more sense to me than pushing them of the cliff of infinity and claiming they are still there, not?
For the maths bit, this means infinity plus 1 is not also infinity (as the hotel example proclaimes), infinity plus 1 is minus infinity plus 1. Just as zero plus 1 is minus zero plus 1. And the circle is round.
Then again, the number series circle would be infinitly huge, so 4th dimension or something? Perhaps land a little Riemann sphere on that one? Anyone?
Loving it .......superb.
6:18
I'm as far to a mathematician as you can think (I'm a sociologist) but I too hate when physicists and astronomers misuse the term infinity. It's one of the few complains I have anyway. Otherwise I love them and admire them deeply :-)
i think you're talking about the limit of 1/x, which is undefined as x approaches 0 for the reasons you stated - the left and right sides of the limit aren't equal to each other.
you could say 1/x^2 is defined as infinity as x approaches 0 because both sides of the limit equal each other, but you must remember we are talking about a limiting function and not simple division such as 1/0 (with limits we are talking about 1/a number really really really really close to 0)
1/0 is undefined
Infinity, more or less (un)countability is often used in computation theory (the mathematical part of computer science) and used to describe problems and their computable complexity (won't go into details). It's not useless and not just something of the mind as it helps us in many ways (like knowing what to put effort into in algorithmics) and can be shown to be there for instance in the proof of the halting problem with use of diagonalisaton (look it up).
3 divided by 0 isn't infinity, it's indeterminate.
The limit of 3/x when x approches 0 is plus or minus infinity.
The Hotel problem seems trivial, but I do not think I ever thought about infinity properly until I learned of the problem and its solutions. I think it is fundamental to actually understanding what infinity is.
Hilbert’s Hotel sounds like a logical sleight of hand. What is the mathematical definition of “full”? If it means that every room has been assigned a positive integer then there are never any rooms available. The only way to check in is to find a non-Integer room such as room number pi.
Infinity as a concept is easy to understand, it's the implications in the real world that's the hard part. As evidenced by all the armchair mathematicians in the comments.
You are right that stars are about the same density at any point in space (the same number of stars per any specific volume) what they were saying is within the observable universe, as far as light has travelled since the big bang, there is a very large but finite number of stars. We can't tell how far it extends beyond this.
In my mind, infinity is such an alien concept, that the moment I hear someone talking about "infinity this" or "infinity that" or, speaking as if they have a grasp of it; I find it almost arrogant in a way. It is a purely abstract concept. I really don't think something infinite can exist, but I'm an agnostic there. At least we don't know of anything infinite yet.
What I mean is, yeah, the hotel problem seems trivial, but in essence, it because we can't understand it, and might never do. Agree?
The term is indeterminate. And 0/0 = R. It is said to be indeterminate not because the answer isn't a single number but because the limit is indeterminate. Indeterminate is used only for limits. Indeterminate isn't what 0/0 actually equals; it means the limit calculation is indeterminate because it results in R, so you need to do something to find the real limit. And you cannot say 2=3 via 0/0. You could say 1=2=3=4=5=6=7 (etc.) but it is really R = R. 0/R = 0, so 0/0 = R. Do the long division.
Hm, interesting. I guess I didn't understand that bit about black holes at all. I'll have to look into it. Thanks for your insight!
Acctually, that would be 3/0*0=2/0*0 so that still doesn't make sense.
if you say 2/3=2/3 and you multiply both sides by 3, the /3 goes away. 2=2.
So you could say 2/0=3/0 meaning 2=3.
But yeah, my explaination had some flaws, but allowing division by zero creates a lot of incredible complicated problems.
Mathematicians are scientists.
Ups looks like I said that the size of a singularity is also infinite. What I actually meant was that the formulas give an answer of zero or "infinitesimally small" (which again doesn't make sense and is impossible to grasp).
About the hotel: If every occupant said: "Let me first check if my neighbor can move in the next room".... they could never free any room.
No, there are different "sizes" of infinity. It really should be considered different "types". It is basically having an infinite number of things (countable) and being able to divide something into infinite pieces (uncountable). It is might be a misconception to say one is "bigger" than the other, but they do have different properties and "uncountable" does seem "bigger" because it contains more things (infinitely more things) than "countable".
0:25 "You shouldn't really be talking to scientists...need to talk to mathematicians."
SNAP!
@megaelliott That's not quite true either. lim 3/x, x-> 0+ = ∞, yes, but lim 3/x, x->0- = -∞. Technically the limit doesn't actually exist because the left- and right-hand limits are disjoint.
Alright. You make a good point. The inifity that we "understand" is our interpretation of infinity since no real infinity can exist, and hence, it is trivial and we can understand it.
I think I just have to accept that when mathematicians, and physicists to a lesser degree, talk about infinity, they are talking about purely maths and the concept of infinity and not the real thing. Hmm. I must ponder upon this some more.
Peace!
the picture is called grafix cat
Very popular on image boards for editing, so to answer your questions, it's photoshopped
The funny thing is, on numberphile(one of Brady's other channels), there is a video explaining why 3/0=undefined, not infinity.
I thought the gravitational force that acts between two masses has a range from 0 to infinity 1/r^2 law, it is similar with electric fields so there are some uses of infinity in physics.
1:30 WRONG WRONG WRONG WRONG, IT IS UNDEFINED!!!! JEEEZ THIS GETS ON MY NURVES
Infinity isn't a number, it's just an idea or concept, so we can't write something like 1/0= infinity anymore than we can write it equals apple pie or pogo stick. We can't treat it like a normal number because it isn't one. The most we can say is that something tends to infinity.
Saying 3/0 = infinity is just a simplified explanation of the mathematical concept, which is
lim x->0 3/x = inf (the limit as x approaches 0 of 3/x is infinity)
That is to say that 3/0 itself has no value, but the behavior on either side is to stretch unrestrainedly large in the positive direction with no upper bound.
So, yes, saying "3/0 = infinity" is technically incorrect, but it's close enough to get the concept across.
Heh, glad to see I'm not the only math major who was shaking his head sadly at the 3/0 equals infinity bit. And of course the hotel had a countably infinite number of rooms so if an uncountably infinite number of new guests showed up (ie one for each real number) the hotel couldn't fit them in.
I think at 1:16 that he means to refer to is the right hand limit of 3/X as X->0. Otherwise you start to run into a lot of problems.
As a scientist, it's obvious what he means. For a general audience, however, he's explaining it entirely wrong.
because the limit as x approaches 0 from the left of 3/0 equals -inf, which does not equal inf and therefore does not equal the limit from the left. Of course since this is a science video, not a maths video, we could very well state that 3/0 has a superstate of being inf and -inf at the same time. Of course it really depends on what you're trying to do with 3/0 and whichever would be most proctical
Not at all - the main problem being that of time dilation. Given an infinite amount of time, the matter will be condensed into a single point (although Heisenberg uncertainty principle says you can't actually have a point), but generally speaking the centre point is never actually reached, matter just continues to get closer and closer to it. What actually happens is a matter of speculation, as spacetime is screwy in a black hole and we're talking Planck distances, but basically size > 0.
There is infinitely many more kinds of infinities than any kind of infinity you care to name.
@kristijanadrian Actually an infinte pound bill would have the interesting property that no matter what you bought with it you would always get an infinite pound note as change. The problem with issuing infinite pound notes is social, not mathematical.
@Phoboskomboa
...and simultaneously collapse the entire universe into an Infinite black hole.Damn that's a nice scary interpretation of it.
Infinity is a yawning chasm of utter oblivion which has driven brilliant men insane. I try to give it a wide berth.
If you had infinity of any substance it would not only fill all gaps in the universe, but it would cram all space in the universe to an infinite density. Then it would cause the universe to expand to an infinite size.
The day I learned about Cantor's diagonalization technique to prove that the integers are a lesser order of infinity than the real numbers, I knew that I was destined to be a nerd.
great videos
(:
2:29 the figure could represent infinity with his arms by having them 180 degrees apart
For those getting upset over him saying X/0 = ∞, calm down. He's a physicist; he doesn't use real maths.
And that's an excuse?
"Sorry for being factually incorrect; I'm a physicist, not a mathematician."
e = 2
Prove me wrong.
@@spacejunk2186 sqrt(2) = 1
3 / 0 Is not infinity for several reasons:
First, when a / b = c this means that if you take c multiples of b, you get a.
It does not matter how many multiples of 0 you take, you will always get 0, ALSO when done infinitely many times. So '3 / 0 = infinity' would create a contradiction.
Second, when a / 0 = infinity, then a = infinity * 0, so 3 = infinity * 0 and 7 = infinity * 0.
This gives 3 = 7, which is a contradiction.
He didn't mean it so literally, this video is meant for the layman to go "woah man!"
@ 2:11 please note the Egyptian symbol he shows for infinity is partially the same as the glyph/symbol for KA or the body double, your shadow that follows you around for an eternity.
The ka as a spiritual double was born with every man and lived on after he died as long as it had a place to live.
After death the ka became supreme. Kings thus claimed to have multiple kas. Rameses II announced that he had over 20.
science is a religion >> U do know that?