I Finally Found Out What 0/0 Should Be
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- Опубликовано: 1 авг 2021
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0 divided by 0 is sort of a mystery. Does 0/0=1? Does 0/0=2? Is 0 over 0 infinity?
If we are talking about a 0 over 0 limit, we say 0 over 0 is indeterminate. However, in this video we won't focus on any 0/0 limit rule. Instead, we will try to figure out what 0 over 0 equals. Hopefully, we can have 0/0 explained.
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#math #brithemathguy #0/0
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Wait 3 days ago? This was seconds ago when this was uploaded
How is your comment 3 days ago??😶
@@arjuns.3752 I think it was uploaded early for channel members so it was originally uploaded for members but made public for subscribers today
@@p_square oh cool
This is the type of guy that doesn't take (0/0, or no) as an answer!
Also, if we take (0+0)/0 instead (0-0)/0 = 0/0 = x-x= 0 but 0/0 is also x. So 0/0=x=0. So we reach a contradiction: the *only* solution for 0/0 is 0, but also based on the sol presented in the vid, 0 *AND* infinity.
If you don't have a pizza, and you don't slice it, you will end up with no pizza. infinity pizza doesn't exist, 0 pizza also doesn't exist. so infinity pizza = no pizza. I need pizza.
That mean my stomach is empty :(
lol wtf
hold on isn’t this just 0/1
@@flatouttroll5932 yeah I think it's 0/1
Infinite Pizza is UNACCEPTABLE!!!
Maybe 0/0 is the friends we made along the way…
But what are the friends we made along the way
@@supe4701 the quadratic formula
Which is 0 LmAO
@@443MoneyTrees why did u have to call me out like that
My first friend that I made in college
Was due to literally debating over 0/0
So yea..
0/0 is the "+ c" constant from integrations.
But 0/0 = 1 and we don't add "+1" to the end. Maybe +c(0)/0 so it cancels to c lol
@@badmath9099no cause when determining limits of functions that simplify to 0/0 it can equal any number, but only one is true for that case. Just like + C
Great idea.
0/0 is when you divide your friends on your math marks
I still feel like there is no definitive answer, but I really appreciate all of the topics and perspectives you brought up in your video. Thank you so much for sharing it!
0/0 might be kinda far fetched to be infinity, but anything else divided by 0 just makes perfect sense. How many 0s can fit into 1 (1/0), infinity. As after an infinite amount of nothing is crammed into something, it will fill up. Zero is so unimaginably small, and infinity is so unimaginably big, so it just makes sense idk
@@santo8813 Ah I see! So I suppose that an infinite amount of zeros could fit into zero as well...
if you keep dividing 0 it goes up. rounding is up so zero is neither negative or positive. but when you start going up ALOT, infinity has value and 0 will start to be the same number as infinity. that's as simple as I can say it.
@@idigulay1274 Sorry I'm not quite sure I understand, and google translate couldn't help me out either. What do you mean?
@@PunmasterSTP I'm sorry, my phone in my pocket pressed random keys
Meanwhile, at the IEEE: Yeah, we thought long and hard about what algebraic properties are least likely to cause problems, and so we decided 0/0 should be unequal to itself.
Meanwhile at ECMA: Yeah so make a 1^100 equal 1^100+1. It's faster that way
Are they tripping on ecluidicin?
@@franchufranchu119 I know it's a joke, but almost every programming language uses the IEEE 754 "binary64" floating point standard format. This format (like every other fixed-precision floating-point formats) has precision limits which render some calculations into no-ops (they do nothing, even though data has actually been processed). Each format has its own specific limit, both for large absolute values and tiny abs values. "binary64" reaches its limit when the abs value reaches 2^53, because the mantissa has 52 real bits and 1 "ghost" (implicit) bit. Every other number after that limit must be an integer with 1 or more binary trailing zeros (mathematically, but not in memory) to preserve the magnitude (exponent) of the number
@@Rudxain I wish I could understand what any of this means 🤣
@@mikehenry9672 IEEE 754 is just a weird codename. "binary64" is almost never used as name because it's VERY ambiguous (anything can be a 64bit binary value). Floating-point is the opposite of Fixed-point, it means the "decimal" (actually binary) point can be moved freely left or right, you can represent a wide range of magnitudes (like the size of a solar system measured in centimeters, or the size of an atom measured in meters, although not with 100% accuracy). Fixed-precision means the memory use is constant, never grows, never shrinks, so there's a limit to both the magnitude (exponent) and mantissa (significant digits). It's just scientific notation but with specific (non-arbitrary) constant limits. The "ghost bit" is a bit whose value is always "1", so it doesn't need to be written in memory, therefore making it implicit, and squeezing more precision out of the limited memory. Trailing zeros are the zeros to the right, big numbers require them to preserve the magnitude if the mantissa is filled to the brim, these zeroes are also "ghosts" but in a different way, since their existence is purely mathematical (not in memory). The exponent is also responsible for "adding" these trailing zeros
3:26 Please note that in this step, you cannot convert the coefficient before 1/1, which is 1, to 0/0 to reduce to common denominator and get this because 0/0 still has an unknown value at this time so you don't know whether 0/0 is equal to 1 or not
I still think undefined is the best way to go about this and my main reason is through physics. Mass is equal to force divided by acceleration. However, if an object has no resultant force applied to it and is not accelerating, then it's mass could be calculated by 0/0. We know this mass could be anything, meaning the mass is undefined by this equation. This doesn't make the object have infinite mass or every object that wasn't accelerating would become a black hole of infinite density and destroy anything around it, which isn't the case. This isn't a solution derived from actual mathematical approaches so there are probably a lot of counter examples to this point but I felt this could be an interesting talking point.
Hmmmm wt about sayin that 0/0=0 but the pnt is definding this will always make the anser hve no sense and i was like thiss one 0/0is acctualy can be evry number but this wrong i think and i dnt like to say undefined bcz it just like i avoid to anser so uhm 0/0=0 is good for math or just sayin is equal infiniti . But well u have good pnt and now is 23:43 and m tooooo slmy so i tink that all dat i say is wrong and to many wrinting fault so uhm heh i home u see this and anser
Bud Hear me out
We all know that 0/anything= 0
So acc to rules of algebra
0/0= anything
So 0/3=0 so 0/0= 3
@@lalaommprakashray8499that's the weird part, though. 3 isn't "anything", it's something. It's like having infinite potential but never being able to use it.
It's like having a wish but never being able to ask the genie to grant it.
I agree because it says mathematically you cannot divide by zero in the form of a fraction
@@navsha2 "it's impossible because the rules say so" absolutely braindead take
0 sometimes acts as a number and sometimes acts like an identity
You can create an extension field defining 1/0 = ∞ and 0/0 = * is special element.
Getting something analogous to projective geometry
Indeed. This is wheel theory in a nutshell.
Wait, in set theory, 0 is just the cardinality of null. So, dividing null by null would be undefined as it is an empty set.
For instance, you can say that the difference of Set A-B would be A, but when would {}-{} make any sense?
Wow! you almost got it! 0/0=(1/0)*0 and 1/0=infinity.And infinity*0=1 so 0/0=1!😄
@@findystonerush9339 It doesn't appear justified that 0/0 = (1/0)*0 or that 1/0 = infinity, or that infinity * 0 = 1, nor have you even defined what you mean by infinity as a number.
@@maxv7323 first two are trivial from examples from the video.
third one idk in what context he got it to be 1, but we need to define more stuff to tackle these questions, current maths lack the transformations from logics and abstractions, we haven't defined what they mean and how they work, we don't understand this field because our axioms don't apply to it. It's an entirely new type of protomathematics that's only useful in highly specific cases and we can't figure out how to prove anything in it in a way everyone can interpret the same thing, because it touches on complex abstractions which need a ton of context, without any, anyone interpret what they want to interpret, it's a rabbit hole, but I do think there's a light in the end of the tunnel and it's like a wormhole to the answer, once someone flips the final switch and figures out how to properly map and how the rules work in this field
Thank you, you are helping us all.It’s always easier when you have the good teacher
Very interesting !
I've always liked to investigate areas of glitches in mathematics. It's like gateways to a whole new other dimension. The elements there behave strangely and don't seem to conform to the known laws of mathematics. We must investigate these like scientists and see what we might uncover, maybe the underlying structure or mechanism of mathematics and maybe reality to which this new mathematics will be telling.
There's no glitch in the expression:[0/0].
The issue is that the term "what is" has no meaning in mathematics, especially in algebra... unless you're describing which categorical numerical set certain numbers exist within. For example, that 1 is a real number, or that π is irrational.
That's completely different from asking what x "is" in the equation y=x/0. "Is" is a linguistic artifact, spoken by amateur mathematicians who aren't interest in solving equations the correct way to begin with. That's where the aberration is introduced; it's a cultural mistake, not a systematic variety.
Here is the answer:[ [0/0] ⇒ [0=0] ]
In English, that says that zero over zero is zero equals zero. Next, let's make it x/0 to see what happens.
[ [x/0] ⇒ [x=0] ]
Put this on a graph to see what happens.
www.google.com/search?q=0
That's just a graph of zero. However, the Zero Property of Multiplication states that 0=0*y for any real number y. The Multiplicative Property of Equality states that we can divide both sides by any real number to derive an equivalent statement. So that's actually a graph of y=0/0.
And y=x/0 is also that graph, because:[ [x/0] ⇒ [x=0] ]
There's no mathematical errors here. x/0=y doesn't define y. So we say that x/0 _"is"_ undefined. That doesn't mean that we broke math, or this is an undiscovered operation. We know what it means. It means that y can be any real number. That has a meaning in mathematics. We call it 'undefined,' not because we don't know what it is, but because we do know what it is, and what it is is any real number. Defined means that out of many, we've selected some of them, which is what x is in this case. x is the number that's defined in the equation y=x/0.
What is x's definition in this linear graph? It's 0. I'll prove it.
[ [y=x/0] ⇒ [y*0=x] ]
[ [y*0=0] ⇒ [0=x] ]
There it is forwards, and backwards. Sorry, but this is already a solved case. If you learned the basic properties of algebra you would know this already.
@@ladymercy5275
[seemingly] glitches. Sorry, if I was not clear. Of course, I wouldn't know if it's a real glitch or not unless I have investigated it, but if you did, good for you.
Math, though how practically powerful it is, still is incomplete. I guess you'd know that by Kurt Godel's theorem.
@@ladymercy5275 i don't really like thinking that infinity + 1 is still equal to infinity it's a long story but I suggest checking out a video made by veritasium on infinity
but it basically goes like this
you have a row from 1 to infinity each number serving as an index number
and then you have a column with A and B in an going to infinity in any order
corresponding to the index
so it would like like
1-AABABBABABABAA(so on till infinity)
2-ABABABABABABAA(so on till infinity)
3-ABAABABABABABB(so on till infinity)
(so on till infinity)
so we would have every possible sequence of string of infinite A's and B's since there are an infinite number of real numbers and each number is acting as an index for the string of A's and B's for the infinite sequence of A's and B's but if we move diagonally in AB column
and change the letter (if it's A change it to B vice versa)
then we will have a string of A's and B's present nowhere in the infinte sequence of A's and B's since the new string will be different from the first letter of the first row by 1 letter (so A turns B)
from the 2nd letter of the second row(B turns A)
and so on till infinity proving that an infinite sequence of A's and B's in an infinte combinations is greater than the infinte real numbers
hence some infinity's are greater than others
we could also write the the A's and B's as infinity² since it's infinte in both rows and columns while the real numbers are infinte only in rows
btw I still think 0/0 = infinity since let's say 25 divided by 5 is a representation of how many times I can subtract 5 from 25 until it's 0 or I am left with a number smaller than 5 which will become the remainder incase of 25 by 5 I can subtract 5 five times from 25 so 25/5= 5
therefore incase of 0/0 since you can subtract 0 and infinte amount of times from 0 it can be said that 0/0= infinity
I hope someone finds it and takes the patience to read this if you do please like it so I know my time was not wasted writing this huge essay thing
@@lelouch6457 this is actually very similar to the proof that there are more real numbers between 0 and 1 than there are natural numbers. It also goes that way, generating real random numbers of infinite length and then generating a new number by taking the digits along the diagonals of each number, thus generating a new number not yet present on the list.
@@lelouch6457 the time clearly wasn't wasted and i feel this perfectly encapsulates the idea of "infinity" as a concept, rather than another arithmetic number. my previous comments also shows how some infinities are bigger than other infinities. I suggest seeing veritasium's video on Gödel's incompleteness theorem, which highlights this proof given by cantor
It cannot be said enough: infinity is not a number! So I like that 0/0 can only be encompassed by infinity.
Right?!
It’s true that Infinity isn’t a Real Number, but it is a number in other number systems like the Hyperreals. (Although even in the Hyperreals 0/0 is left undefined.)
infinity is quite a few different numbers, actually. 0/0 is not encompassed by infinity, because infinity is valid mathematics, 0/0 is not. Bri got it wrong by assuming that 0/0 obeys mathematical principles, and thus concluded that it should be infinity according to that. but 0/0 lies outside of the scope of mathematics, so that assumption is wrong, and thus the conclusion that 0/0 has a relation to infinity is also wrong.
it's trivial in fact to get division of any number at all by zero to appear to be any number. for instance, you can drop a hole into y=x at absolutely any point by simply multiplying both sides by 1, as shown below:
- for any n != 0, n/n = 1
- y*1 = y, x*1 = 1, thus y = x is identical to y = x*1
- given both of the above y = x * n/n
- to get division by zero at x-value m set n = x - m
- now at x = m, y = x * (x-m)/(x-m) is identical to y = x * 0/0
- since x * 0 = 0, this means that we have y = 0/0
- if we take the limit here we will get that y = x, and thus at x = m it is necessarily true that 0/0 = x
- since we can set m to absolutely any value we want 0/0 is thus also equal to absolutely anything and everything simultaneously
if we look at other functions, like tangent, we can see that division by zero also shows up where solutions are impossible, like for tan(pi/2), where the limit from the left is positive infinity and the limit from the right is negative infinity. it's not possible to have two values more different from each other, yet division by zero yields both simultaneously an infinite number of times with just this one function.
@@Bodyknock *It's true that infinity isn't a Real Number, but it is a number in other number systems like the Hyperreals.*
No, this is false. "Infinity" is not a number in the hyperreal numbers. There are many numbers in the hyperreal numbers that satisfy the property of infinity, because that is what infinity is: a property of sets, not a number. There is no number in the hyperreal number system called "infinity", so to say that there is such a number is a lie.
@@sumdumbmick *infinity is quite a few different numbers, actually.*
No, it is not. This is a nonsensical statement. Infinity is a property of sets. Definitionally, we say that a set S is infinite if and only there exists an injection from the set N of natural numbers to the set S. Every number system is a set, as is every object in mathematics, and some number systems have infinite elements, and those elements are numbers that are infinite. There is no number called "infinity", though, because "infinity" is a property of sets, not a number.
*0/0 is not encompassed by infinity, because infinity is valid mathematics, 0/0 is not.*
This much is true, though I get the impression we will strongly disagree as to _why_ this is true.
*it's trivial in fact to get division of any number at all by zero to be any number.*
This is a bold claim, and I take this to be the thesis of your comment, so I will be deconstructing the rest of your comment in context of this thesis.
*now at x = m, y = x·(x - m)/(x - m) is identical to y = x·0/0. Since x·0 = 0, this means that we have y = 0/0.*
Not so fast there. In order to conclude that x·0/0 = 0/0, you must necessarily assume associativity of ·, which is not at all warranted here.
*if we take the limit here we will get that y = x,...*
No, this is a nonsensical claim. In the equation y = 0/0, there is nothing to take the limit with respect to, we just have two constants. Also, limits are irrelevant to questions of evaluating arithmetic expressions.
*if we look at other functions, like tangent, we can see that division by zero also shows up where solutions are impossible, like for tan(π/2), where the limit from the left is +♾ and the limit from the right is -♾.*
Yes, it is true that lim sin(x)/cos(x) (x < π/2, x -> π/2) = +♾, and lim sin(x)/cos(x) (x > π/2, x -> π/2) = -♾. However, this has nothing to do with the topic of division by 0, since the denominator is never equal to 0 in these expressions. What you have proven is that lim tan(x) (x -> π/2) does not exist, which does not itself prove tan(π/2) is undefined. In fact, I have an easy counter-example to your claim. Let f : R -> R with f(x) = 0 if x = π/2 + n·π, where n is an integer, f(x) = tan(x) otherwise. Then here we have lim f(x) (x -> π/2) does not exist, yet f(π/2) = 0. This disproves your claim that lim f(x) (x -> π/2) not existing proves f(π/2) is undefined.
This question haunted me for a while. Thanks
Ohh thank you sm. !
My belief is that dividing by zero gives you the list of every number ever and will be.
Changing a scalar to an infinite set is quite an interesting property.
So you're saying that nothing over nothing is very large. Ok man. Ok.
I always said that 0/0 is equal to infinity! My understanding of division at the time was "How many times does the numerator go into the denominator?" and zero goes into zero infinite times, because zero doesn't add anything!
Yeah, but then it could also be negative infinity.
yeah but it doesn't only go infinite times. it also goes 1 and 2 and 3 and 4 times and literally every number of times since 0x always equals 0. no matter what the x is it is always equal to zero whether it's 73517390 or infinity or any other number. so that's why it isn't exactly equal to infinity and is undefined.
0/0 being equal to infinity (or 0) is semantically and mathematically impossible as
A) infinity isn't a number per se: infinity is not a value. Its a name given to a "boundless limit". Nothing ever equals infinity, things can only approach infinity as you change a variable. For example, x/y approaches infinity for x>0 as y tends to zero from the right. Whenever you hear a mathematician say something equals infinity it's shorthand for a limit of some kind.
B) let's use expression x/0
x/0 isnt infinity as the rules of algebra say that, if x/0 = infinity, then infinity times zero equals x, for any x you choose.
IF x/0 = ∞, THEN ∞ * 0 = x
However, this is obviously wrong as any number multiplied by zero is zero.
And infinity is not a number, it's an idea.
C) assuming the expression x/0 and x is 10 apples, you can't add an infinite amount of zero groups together and end up with say 10 apples
also you can rationalise 0/0 much simpler using the idea that division is the inverse of multiplication:
0/0 = 0 is absurd and incorrect because it would allow for the proving of 1=2 (which obviously is absurd)
0x1 = 0
0x2 = 0
lets say we allow the division of 0:
0x1/0 = 0x2/0 (both divided by 0, so equivalent to previous lines)
(0x1/0) x 1 = (0x2/0) x 2 (both times by 1, so equivalent to previous lines)
cancel out the zero on both sides and you get:
(0/0) x 1 = (0/0) x 2
cancel out the expression 0/0 on both sides and you are left with 1 = 2, which is obviously incorrect meaning division by 0 is impossible
Yep. I was thinking the same
Hi can you make more videos on integration? I am having a hard time keeping up with it.
As always the video was great!!
I will try!
Make sure to formally define and use Riemann integrability. Better yet compare it to Lebesgue integrals.
@@BriTheMathGuy differentiation too 😵
That's a really good interoperation. The only problem is that -♾ also satisfies the equalities above.
Okay that checks out
What if 0/0 definition can make us time travel
We could set it as an imaginary thing, like sqrt(-1) = i
That does not work.
You can't "really" (pun intended) say that, since that number would have to be equal to any number, it would practically be useless
I always imagine it as the set of all numbers.
Man I became a huge fan of you and your channel.
Keep up the good the work pal
But i would love it more, if you made your videos a bit longer
Cheers
Thanks so much!
Good job on this sir , Really interesting.
Glad you enjoyed it!
3:26 This formula can be thought of as multiplying both the numerator and denominator of the first fraction by the denominator of the second fraction and vice versa and then adding them. In this case, you can just say that the rule that you can multiply both the numerator and the denominator by the same number and get the same result doesn't apply to multiplying by zero
Exactly every number should be answer of the question
Quite interesting
I like to think that the derivative is 0/0 with context. For that to make sense, 0/0 must be context sensitive. I know that limits are involved with derivatives, but i find this line of thinking kinda neat.
Your line of thinking is not only not neat, but also wrong, since there is no division by 0 involved in the evaluation of a derivative.
@@angelmendez-rivera351 I'm fully aware that there isn't any division of 0/0. It's a limit, dy and dx aren't zero. But, this limiting process was created to make such computations possible. For example, if we want the velocity of an object, we divide the distance traveled Δx by the time Δt it took. When we want the instantaneous velocity, we would, using the above procedure, we would end up with 0/0. The derivative is computing 0/0 without actually having to compute 0/0. That's what it was created for.
@@victorscarpes *I'm fully aware that there isn't any division of 0/0. It's a limit, dy and dx aren't zero.*
dy and dx are also not quantities in themselves, even though the notation unfortunately suggests otherwise.
*But, this limiting process was created to make such computations possible. For example, if we want the velocity of an object, we divide the distance traveled Δx by the time Δt of travel. When we want the instantaneous velocity, we would, using the above procedure, we would end up with 0/0. The derivative is computing without actually having to compute 0/0. That's what it was created for.*
No, this is false. Historically, when calculus was rediscovered by Newton and Leibniz (I say rediscovered, because it is now well-known at this point that techniques of calculus has been used millennia before), they formulated it by appealing to infinitesimal quantities, and they called it infinitesimal calculus. The concepts of the derivative, the integral, and the method of exhaustion, were then understood as special applications of this infinitesimal calculus. The primary notion in this infinitesimal calculus was that there existed infinitesimal nonzero quantities ε that were taken to have the property that ε^2 = 0. This method, though, was extremely nonrigorous and very heavily criticized, it being widely seen as apparently inconsistent and leaving too much ambiguity. This created problems for calculus as a mathematical application. Later, when real analysis was invented, calculus was reformulated in terms of topological ideas and ε-δ arguments from real analysis. What this allowed was for a consistently rigorous mathematical theory that allowed us to do everything that calculus was invented to do, all without ever needing to appeal to infinitesimal quantities at all, instead, relying solely on the properties of the real numbers. Limits were invented not to allow calculations that involved division by 0. They were invented to set calculus on a foundation that did not rely on ill-defined infinitesimal quantities, but instead only on the properties of the already known system of the real numbers.
@@angelmendez-rivera351 I must admit i mixed up a bit about the limit. Do infinitesimals create a bunch of weirdness and inconsistencies? Absolutely. Is analysis the mathematical rigorous way of defining this stuff? Absolutely. But, altough i love mathematics, i'm an engineer. Thinking of dx and dy as actual quantities of infinitly small size is pretty useful.
@@victorscarpes *Do infinitesimals create a bunch of weirdness and inconsistencies? Absolutely.*
They _used_ to. This is why they were replaced by limits. However, infinitesimals did not stay defeated, and they have made a comeback. In the mid 20th century, a mathematician by the name of Abraham Robinson develop a rigorous system for dealing with infinitesimal quantities and infinite quantities, a system that was dubbed "hyperreal numbers". These numbers are the basis for nonstandard analysis, which can serve as an alternative foundation for reformulating calculus in a simpler way. In this reformulation, limits are replaced by the standard part function. The standard part function is a function that gves you the real number closest to the finite hyperreal number you input. So for example, if I have a hyperreal number 7 + ε^2, where ε is infinitesimal, then st(7 + ε^2) = 7. If I have -3 - ε, then st(-3 - ε) = -3. When defining the derivative, you can simply define it as st([f:(x + ε) - f(x)]/ε), where f: denotes the natural extension of f to the hyperreal numbers. However, since this system is recent, relative to the history of mathematics, not many textbooks have been written implementing this system for educational purposes and it is not yet part of curricula in most countries. It is likely it will become common in the future, though.
Congrats on getting a sponsor bro! You totally deserve it.
Thanks a ton!
@@BriTheMathGuy at first glance, I read it as "thanks son"
I was like: well, that can't be right
Nothing divide nothing is still nothing
no because the nothing was divided out of it so u get left with 1
@@md-sl1io so how do you divide nothing ? its just nothing. You can't devide nothing by nothing, the answer is nothing.
@@md-sl1iowhere did the 1 come from
seems pretty solid
So this is why anything / 0 = complex infinity.
@@rahulkhatwani548 no bro try it 1/0 is the same as 0/0
0/0 = φ
Where φ = 0/0 😎
Mathematician can't even do that
only the "official" yt account of Euclid could do that
@@p_square it's just for rising star math RUclipsr challenge of Blackpenredpen , but people are liking it🤣
Pamiętaj cholero, nie dziel przez zero 😁
I think we could create a new set of numbers with the properties mentioned in this video, and call it "null numbers" or "abstract numbers" or something like that. This set would have its unity called "null unity" or "nonentity" and put it the symbol of "Ω"
As a programmer, I like the idea of 'null entity'. I understand how it can be hard to differentiate from 'zero' for everyone else, but 'undefined' is a different concept to 'zero'. If you have zero apples, you know how many apples you have. If you have 'undefined' apples, you don't know if you have zero, or a hundred, 0.25 apples, or infinite apples. But you still know they are apples.
@@rich1051414 "undefined" is not a number, so it doesn't answer the question "how many X are there". What is undefined is not the amount, but the calculation. So it just means maths stops working for a while and you just have to ask the question again. In other words, 0 / 0 = makes no sense
I'm supportive of this. You can make as many alternate systems as you want without threatening the originals.
The sad thing is that depending on the situation, mathematics bends the rules of 0/0 to fit whatever situation they want to put it in.
No
As I'm learning more about mathematics it occurs to me that it's less about making sense of the real world and more about definitions and semantics. But at the same time I still know pretty much nothing about mathematics, so...
It's easy. 0/0 is undefined and it's all. Why should be 0/0 defined if it's totaly useless?
We don’t know if it’s totally useless yet. If defined, it could become the way to evaluate x/0 which, at minimum that I can think of, would let us figure out what happens at the middle of a black hole where the volume is 0 but the mass is a number, creating x/0 density.
I’m 99% sure it will never work though, since you can cause a bunch of shenanigans by defining x/0. One example is the 1=2 proofs
@@kepler6873 No. I wrote that 0/0 is useless. This does not mean that X
/0 is also useless. But X/0 is undefined because it's nonsense. Nothing can be divided into zero parts. Therefore, division by zero is not defined. And the black hole has not zero volume. It has a volume that is equal to an infinitely small number, so this is a limit approaching zero. So even here there is no need to divide by zero to find that the density of a black hole is infinite.
@@kepler6873 It's undefinable. You can't just give it whatever definition you want. That's not what UNDEFINED means in the mathematics.
@@herbie_the_hillbillie_goat I made the comment at like 3AM, I know yeah. Sorry for blanking out on it at the time.
@@jancermak1988 Your understanding of division is not very good if you are thinking of division as "breaking into parts". Try dividing π by -e using that method, and you will quickly see why that is not a definition of division.
Anyway, you are wrong. Having a value for 0/0 and x/0 is useful, and this is why wheel theory and the theory of involution monoids in general was developed in the first place.
Same as i expected
I knew how to do it so for it lets take an example that you have to distribute 0 slices of pizza among your 0 friends so it means that you don't have any friends or any pizza to eat
since the real numbers form a field and which implies that arithmetic operations always yield a real number, then your definition leads directly to contradiction since you define 0/0 to be infinity, but infinity is NOT a real number. best to leave 0/0 undefined... at least within the real numbers. maybe an extension of the real numbers, like the hyperreals.
How did you miss the point of the video this badly? Obviously, he knows division by 0 is not possible within the real numbers. He is not trying to go against the mathematical consensus. He is exploring a new idea. He even said this in the video.
I typically define 0/0 as indeterminate (any value) and nonzero/0 as undefined (no value), but it really depends on what's most useful in the context you're working in.
I like how some people still say the rules of mathematics are made up when there are things you simply CANNOT do in it.
If you divide nothing to nothing they will get nothing which is also be written as 0/0=0
In pure mathematics it is undefined and should stay that way. But if you apply it in a context, it can be whatever you need it to be to make sense. For example: If you have an application where the graph of f(x)=x/x needs to be defined for any x, set 0/0=1 and you have a continous function.
0/0 is actually 0x = 0 which can be any real number so it means infinite solutions. therefore 0/0 can technically equal 0, 1 and infinity, but it stays undefined because basic expressions don't allow more than one solution and thus it's wrong to write one out of the infinite solutions. again it's only solvable when it comes to equations with one or more variable
This might sound dumb but why should basic expressions have to just have one solution? There are multiple expressions which don't just have one answer, for example √25 = ±5, meaning √25 = 5 and -5 at the same time because they both work for x²=25. In a similar way all square roots are equal to two numbers at once, all cube roots equal to 3 numbers, all fourth roots equal to 4 numbers etc. So, I don't see why 0/0 can't be infinitely many numbers at once just because it's a division
@@thewierdragonbaby4843 expressions mean you have different types of math operations, like 3x + 2. thats an algebraic expression and it means that there's no equal sign to something specific. you just simplify. also square root is considered to give one solution depending on the sign for example
sqrt(9) = 3 etc. you just don't write ±3 if you see the root only. same applies for logs, exponents, absolute values etc. and x² = 9 is different, you basically get 2 solutions because you in fact square root both sides and get |x| = 3 if you simplify it. and you get x = ±3
I mean you can get cases where in expressions you'd have more than 1 solution like using the quadratic formula for x > 0 or having x inside absolute value bars
@@XBGamerX20 oh okay, I guess that kinda makes sense, but why would you only have 1 answer for roots?
also on a completely unrelated note wouldn't |x| = 3 have infinite solutions if you consider the complex plane?
0/0 = 0 is absurd and incorrect because it would allow for the proving of 1=2 (which obviously is absurd)
0x1 = 0
0x2 = 0
lets say we allow the division of 0:
0x1/0 = 0x2/0 (both divided by 0, so equivalent to previous lines)
(0x1/0) x 1 = (0x2/0) x 2 (both times by 1, so equivalent to previous lines)
cancel out the zero on both sides and you get:
(0/0) x 1 = (0/0) x 2
cancel out the expression 0/0 on both sides and you are left with 1 = 2, which is obviously incorrect meaning division by 0 is impossible
0 apples given to 0 childeren = no one gets anything because there is no one to receive anything
It is definable as 1, and for really good reasons as well. In fact, it follows really from the arithmetic rules and sets.
Using set powers, cardinalities, os, Os and alephs is useful
Yes, but no, not really. It is true that infinity is in the realm of set theory, as you describe, but the topic of division by 0 is not a set-theoretic topic, it is a group-theoretic topic.
@@angelmendez-rivera351 you are correct
-♾️ satisfies your equations too.
It's better to stay as undefined.
Maybe it could be contextual, like 0/0=6 for f(x)=(x^2-9)/(x-3) to keep the continuity of the function, rather than marking it as discontinuous because of the undefinition.
how much is -♾️+1 though?
@@LorenzoF06 It's -♾️.
-♾️+x=-♾️, for any real number x.
@@LorenzoF06 similarly: -inf * a = -inf (assuming a is a positive real number).
@@solsystem1342 even -inf * -1 ?
just define it differently depending on your scenario
Zero doesn’t even feel like a real word now.
With my theory of 0, 0/0=+0 (the reason I made the “theory of 0” is because 0 is actually a placeholder that original mathematicians used because they didn’t know what went there yet, so I tried to figure it out, and my theory kinda makes sense, if anyone want to know it, just ask)
Please, tell me
I actually agree 0/0 should equal 0
So the only thing that changes is 0/x=0 for all real values of x, *including 0* . x/x=1 must remain for all real values of x except for 0, 'cause that would break the whole fabric of maths, and x/0 should of course stay undefined/not accepted.
I hate theoretical mathematics. I like practical mathematics. If you have 0 pizza and you decide to cut it into 0 pieces, you still have 0 pizza. AFAIC, a numerator of 0 negates anything that could possibly be in the denominator. Even 0.
@@GrndAdmiralThrawn What you're saying makes no sense.
Obviously, if you slice 0 pizza in any number of pieces, even 0, you still have 0 pizza.
But, what you did is not division : If you have 1 pizza, and slice it in 2 pieces, how many pizzas do you have ? Well, still 1 pizza. What you have however are 2 pizza halves, hence 1 divided by 2 equals a half.
So, using pizza slicing as a definition for division (which is perfectly fine), we arrive at : given A pizzas, and B pieces, we call the size of the pieces A/B
Exemple : 15/3 is 15 pizzas "sliced" into 3 pieces of 5 pizzas each. 15/3=5
So, what would be 0/0 ? 0 pizzas sliced into 0 pieces of pizza. What size is my no piece of pizza ? Well, I have 0 piece of 50000 pizzas, so should 0/0=50000 ?
No, since any piece size would work here, by practical maths, there just can't be any answer to question. And that's fine, since in practice, there are many questions where we don't have an answer to.
But then
0 = 3*0
0/0 = 3
0 = 3
@@mhmd-mc113 So here you're assuming 0/0 = 1 which is incorrect and you got 0/0 = 3, and I don't even know why that concludes to 0 = 3
@@DeJay7 i wasn't proofing it
I wad showing the other guy that not always algebra rules apply
Sometimes you simply cannot move stuff or devide or subtract infinity for example
Cause if you do youll get results like 0=1
And 0/0=3
And 1+2+3...=-1/12
me trying too explain to my sir. My sir made me stand out
I would instinctively say that things like 0/0, infinity - infinity, 1/0 etc don't need defining as anything other than themselves - but it is useful to examine them and define what properties they have. Then, if two or more turn out to share properties, you can advance the thesis that they are the same mathematical entity, or related to each other by a mathematical entity that is somehow involved with each of them.
When you login to a new game and your K/D ratio is 0/0 and showed as 0, then you get 1 kill so it's 1/0 and it will show your K/D as 1. *Problem solved, diving by 0 outputs the same thing you would get from dividing by 1*
Ah, so it's like a factorial!
Also, I know you're joking, but this probably causes way more problems than most other definitions
gamer
@@mrsharpie7899 TBH tho I'd rather salvage the indetermination in the context of the problem I'm solving. Since you can make the case for 0/0 = anything, giving it any one value would make it pretty inconvenient for many applications.
0/0 is equal to the set of all numbers since if 0 = 0x, then x can be any number.
which works quiet well with the infinit different limits you can get with 0/0. Like yx/x will always be y so we get any limit for 0/0. If we add 1/x to it we also get ±∞ and thus the extended real numbers. It will work with complex numbers aswell. Possibly for quarternions as well. So our current rules for any field will lead to limits of 0/0 sitations to be anything. Which is also a good reason to leave 0/0 undefined.
No, because 0/0 is not defined to be the solution multiset of 0 = 0·x, it is merely defined as an arithmetic function.
@@angelmendez-rivera351 0/0 is not undefined; it's indeterminate. There is a difference.
@@cubicinfinity No. It is not indeterminate. I have explained this elsewhere more than once, but there is no such a thing as indeterminate.
@@angelmendez-rivera351 This is the definition I am talking about: en.wikipedia.org/wiki/Indeterminate_form How can I learn about what you are referring to?
It should be everything. That is to say its not a 1 to 1 statement, any number meets the criteria.
0/0 can only be defined when 4th-dimesion is possible
Means the space exist in 0 having infinite space in 0
Last year I had an exam with a question asking to simplify an expression as much as possible. I simplified it down to something like "x + x/x", and when I got there, I thought that if I would replace "x/x" with "1", they would no longer be the same expressions since "x + x/x" isn't defined for x=0 (0/0) while "x + 1" is. Unfortunately, I lost a mark for this but never really understood why I was wrong. Do you agree with me or was I wrong?
Maybe you can deduce from the initial statement that x ≠ 0. Otherwise you can just say x + 1 (x ≠ 0)
@@h-a-y-k4149 Yeah I agree that "x + 1 (x ≠ 0)" should've been the correct answer, but stupid me thought that that was the same thing as "x + x/x". However, the correct answer was "x + 1", which is what's bothering me, especially because the teachers couldn't explain to me why.
@@baralike8206 depends on the beginning problem. If you could just substitute in 0 for the initial x and the whole thing works out properly, then 0 is a part of solution as well. So even if you somehow got x/x, solution still might be x = 0, depending on the initial conditions
You must look equation as finding a solution set. If zero is possible, you should take care of this case accordingly during the process of calculations. If zero is not possible, you should prove that It must be the case. For example, If
(x^2-1)/(x+1)=-2, then x must be different of -1, because It would say that we are deviding by zero which is not possible, hence, for x not equal to -1, we have ((x+1)(x-1))/(x+1)=-2 hence x-1=-2, hence x=-1. Which is a contradition with the hypothesis. This means that there is no solution for this problem. Usually, contradiction comes from this type of equation. The basic tip is to redo the calculations with care when some contradiction comes to appear. Logically thinking, this is because we have left behing some piece of information during the calculation that would show the problem in the computations. Can you imagine what would happen if I hadn't realized that x=-1 is not a solution? A simple substituition of x=-1 would solve that problem and let we know that x=-1 can't be a solution, but in exams It is not easy to predict what would happen.
@@baralike8206 imo the main reason is that generally the answer must be as simple as possible. For example, if the answer is just 1, saying 8/8 (which is still 1) is still a correct result but the teachers may consider it wrong. Also leaving it like x + x/x still doesn't mean that x can't be 0 (see the answer of @Me That is). You need to explicitly say that x ≠ 0. One can easily deduce this condition, though, but it's just a formality in my opinion and not crucial (especially if the problem has already stated that x≠0)
Let's appreciate this guy for solving problems which are beyond math rules.
zero stands for lack of value therefore nothing devided by nothing is still nothing
I'd rather not trouble my self with these kind of problems
EDIT: This is one of the comments I randomly leave behind without thinking and then later regret it. Please disregard it.
I think that 0/0 is NaN, because when programming, languages like javascript say that NaN != NaN and any operation used returns NaN
Another thing I would like to note: In JS, Infinity - Infinity = NaN because it's impossible for it to say whether one is greater than the other.
isnt it only nan because its not defined? and that languages are programmed to, well, treat 0/0 as undefined
NaN stands for Not a Number
You can’t actually have infinity in programming. Computers can’t store infinite digits, so that is a completely invalid point. And as the commenter above me already stated, NaN stands for Not a Number, which is the same thing as undefined.
NaN! = NaN factorial = NaN*(NaN-1)*....*(NaN-∞) = NaN*NaN*...*NaN = NaN^∞. And by the definition of a factorial (ln(n!)=n*ln(n)-n+1), you have that n!= e^(n*ln(n)-n+1) = n^n*e^(1-n). By setting n=NaN ==> NaN!= NaN^NaN*e(1-NaN)
NaN means "Not a Number", so it's the same thing as saying "undefined".
4:11 negative infinity also works?
I KNEW it.
I KNEW that dividing by 0 would have to equal Infinity.
When I was little I tried to devise a system where 0/0 = 1. I Said 1/0 is some constant called Omega, and 0 times Omega is 1. To resolve the issue where 0 + 0 = 0 implies 2 = 1, I asserted 0 + 0 > 0 instead. I let 0 be defined as 1 - 1. Therefore 2 - 2 > 1 - 1. This also means for example 3 - 2 > 1. Zero squared is 0 - 0. 0^2 - 0^2 = 0^3, etc. I constructed an infinite sum k = 1 + 0 + 0^2 + 0^3 etc, and noticed that 1 - k behaves like the new "zero," breaking division and such. To resolve the new issue of division by 1 - k, I devised a system where (1-k)/(1-k) = 1, and 1/(1-k) = Omega_1, wherein a new constant k_1 such that (1 - k_1) caused new division problems until the creation of the new constant Omega_2, etc.
"When I was little" So you knew advanced mathematics when you were "little"? (Whatever age that is)
@@nikkiofthevalley "Little" in this case means like 14 years old
your system succs
that's not advanced math, that's just LOGIC
at the time of this reply, I'm 12, and I've literally
found a better way to approximate the area under
a curve (using right triangles)
I don't really know advanced math
I just play around with numbers a lot
I USE LOGIC
Can you define k_1 and prove that 1/(1-k_1) can't be defined as Omega_1?
Well, if you view division as the reverse of multiplication, so that say 1/2 = .5 because .5 × 2 = 1 and any number over itself equals one because anything times 1 equals itself, then the question becomes what times 0 equals 0, which is obviously 0. This view also explains why you can't divide by 0 since nothing times 0 will ever equal the dividend.
Nope! 0/0=(1/0)*0.And 1/0=infinity.And infinity*0=1 soooo! 0/0=1 proof!
@@findystonerush9339 lmao no. not even trying to be formal but that is ass proof.
you cant just say 0/0=(1/0)*0
if you pull equations out of your ass, things go wrong
0/0 is "nothing". not zero, nothing. 0/0 cant be defined logically. but how we should treat it is dependent on the situation. sometimes it is nice for 0/0 to be one. but sometimes its better to treat it as 0. or just forbidden.
@@findystonerush9339 0/0 being equal to infinity or 1 is semantically and mathematically impossible as
A) infinity isn't a number per se: infinity is not a value. Its a name given to a "boundless limit". Nothing ever equals infinity, things can only approach infinity as you change a variable. For example, x/y approaches infinity for x>0 as y tends to zero from the right. Whenever you hear a mathematician say something equals infinity it's shorthand for a limit of some kind.
B) let's use expression x/0
x/0 isnt infinity as the rules of algebra say that, if x/0 = infinity, then infinity times zero equals x, for any x you choose.
IF x/0 = ∞, THEN ∞ * 0 = x
However, this is obviously wrong as any number multiplied by zero is zero.
And infinity is not a number, it's an idea.
C) assuming the expression x/0 and x is 10 apples, you can't add an infinite amount of zero groups together and end up with say 10 apples
also you can rationalise 0/0 much simpler using the idea that division is the inverse of multiplication:
0/0 = 0 is absurd and incorrect because it would allow for the proving of 1=2 (which obviously is absurd)
0x1 = 0
0x2 = 0
lets say we allow the division of 0:
0x1/0 = 0x2/0 (both divided by 0, so equivalent to previous lines)
(0x1/0) x 1 = (0x2/0) x 2 (both times by 1, so equivalent to previous lines)
cancel out the zero on both sides and you get:
(0/0) x 1 = (0/0) x 2
cancel out the expression 0/0 on both sides and you are left with 1 = 2, which is obviously incorrect meaning division by 0 is impossible
D) if it were equal to infinity, it would be equal to negative infinity, as infinity is a direction. So to find out the difference, we'd simply do infinity - negative infinity, right? By doing this, we run into the Hibbert's paradox of the grand hotel.
Suppose Hilbert’s hotel has an infinite number of rooms and infinite number of guests are booked into the hotel. By common sense, it seems like the hotel is fully booked right? Wrong. Infinite sets just defy logic. Suppose there was another guest who wanted to book into the hotel, all the hotel staff have to do is just shift guest in room number 1 to the next, the guest in room number two to the third and so on… So by this logic ∞+1=∞
Similarly ∞−1=∞
Just remove the guest from room number 1 and shift the remaining guests to the predecessor of their room numbers. You still have an infinite number of guests.
Let’s apply this logic then Seemingly ∞−∞=0
But suppose we remove the guests which are present in the rooms having an odd number(1,3,5…..) we still have infinite number of guests. So we get ∞−∞=∞
Let’s remove all the guests except the ones present in the first 50 rooms. So ∞−∞=50
You see where I’m going with this? Simply that ∞−∞ is indeterminable.
So as a result, x/0 is indeterminable as assuming it leads to infinity, the natural convention is that it leads to negative infinity, and attempting to calculate the origin by subtracting them from each other leads to a paradox
I feel like sometime people will assign a value to 0/0 with a letter just like “i”
Secondary School: _"You cannot divide by zero"_
Baccalaureate: *YOU FOOL 😎* **Proceeds with Functions Limits**
So I guess the best answer is to define 0/0 as an axiom for "F*CK THAT I'M OUTTA HERE".
Logically it's impossible
0 represents the absence of things/numbers
"There's 0 pens on the table = no pen on the table"
In division...when we try to devide nothing it will always give us nothing "0"
Therefor
We cannot devide nothingness by nothingness
Nothing happened/happens/will happen we do so, therefore it's Mathematically Invalid to try to find a solutions to an nonexistent problem
That's my take on it, what do you guys think?
False.
0/0 is not logically impossible. If you get 0/0 in your math, then no errors have occurred so far. If you get 0/0 in your math, then what that tells you is that 0=0. There's not an error.
0=0 is correct math.
@@ladymercy5275 0/0 and 0=0 are two different things
4/4 is 1...but 0/0 doesn't give us 1 because of the reasons I've stated before.
First I was like in school you can't define 0/0 because you can't devide with 0. Now I question myself.
This video is the greatest proof why you should not divide by 0
Theorically (for pure maths) maybe you could give a meaning for 0/0 but when you put it on practice no one discovered yet a way to use 0/0 with aplicativable value (it's useless for physics and engeeniers) but It doesn't mean it will be impossible
I think it should be 0 because infinitely expanding nothing should give you nothing.
In more detail, in the expression a/b for a != 0, the limit of a/b as b approaches 0 is infinity or negative infinity (depending on which side you start from. That is, dividing by 0 is like infinitely expanding (in one way or another). Basically, when you divide a number by a value between 0 and 1, the smaller the denominator, the larger absolute value the resulting expression gets. So I see dividing by 0 as the ultimate expansion operator.
Great, so if you infinitely expand anything other than 0, the result is infinite. But I think things should be different with 0. If you infinitely expand 0, you still get 0. If you have nothing and then infinitely zoom into it, you will still see nothing! But if we accept that 0/0 should be infinite, then we are saying that 0 is indeed something that can be expanded, which to me contradicts the very notion of 0, which is pure nothingness.
So, infinitely expanding 0 should still be 0 from a philosophical point of view.
Except it often isn't. The whole idea of Calculus is actually based on taking the difference quotient formula (essentially a modified slope formula for curves) and applying what happens when you make the change in x and make it zero. This in turn makes the change in y also zero. With the exception of whole numbers (like an equation of y=7), the result is not zero, but instead a whole other function that describes the slope of the line tangent to the original equation.
I get what you are saying but when you look at in a particle sense. If you got a bucket of nothing and keep trying to fill it with nothing then you can constantly keep filling it and infinite amount of times thus inversely. You take as much nothing out as you wish. It is why personally I avoid the question by denouncing zero as a number in the first place and more of a concept
0/0 = 0 is absurd and incorrect because it would allow for the proving of 1=2 (which obviously is absurd)
0x1 = 0
0x2 = 0
lets say we allow the division of 0:
0x1/0 = 0x2/0 (both divided by 0, so equivalent to previous lines)
(0x1/0) x 1 = (0x2/0) x 2 (both times by 1, so equivalent to previous lines)
cancel out the zero on both sides and you get:
(0/0) x 1 = (0/0) x 2
cancel out the expression 0/0 on both sides and you are left with 1 = 2, which is obviously incorrect meaning division by 0 is impossible
as for the first part, say we have 20 oranges and want to distribute them amongst a table. if i wanted to divide them into 2 groups, the expression would be 20/2 meaning each person receives 10 oranges. if i wanted to divide it into 1 group, it'd be 20/1 --> each person receives 20 oranges
for dividing by zero, however, what is the number of oranges that each person receives when 20 cookies are evenly distributed among 0 people at a table? There is no way to distribute 20 oranges to nobody, so the resulting answer is undefined, not zero, because the parameters defining how the oranges are to be distributed are zero
If you had 0 cookies and 0 friends, there are 0 cookies left over since you never had any to begin with
I always thought of it as of dividing nothing on nothing. Logically, it will give you nothing. But then, dividing nothing on anything is nothing, and diving anything on nothing is impossible, so dividing nothing on nothing may give you an infinity...
0/0 is the brain cells you have
Yeah it can be infinity or zero ,its perspective you think :)
This is what Bri was referring to about his last video if anyone is wondering:
One workaround to this is to define any such indeterminate forms to equal the nullity, ⊥ . It’s essentially an absorbing element that is more “powerful” than 0 or ∞, so we define results like:
0/0 = ⊥
10/0 = ⊥ etc…
Same goes with indeterminate forms involving infinity. ‘⊥’ has the properties:
x + ⊥ = ⊥
⊥ + ⊥ = ⊥
x ⊥ = ⊥
x / ⊥ = ⊥
⊥ - ⊥ = ⊥ etc… for any x, including x = ⊥
The only exception to our standard maths rules are the properties 0*x = 0 and x / x = 1 for any non-zero number x, and also x - x = 0;
Since ⊥/⊥ = ⊥ and 0*⊥ = ⊥ etc…
So basically rather than just infinity, we create a new concept ‘⊥’ which we can treat like a number.
Again this is only a theoretical work-around to the problem, it is not official.
I always thought it was 0.
It’s a limit question if 0 by 0 form is formed.
hmm, I mostly encountered the 0/0=1 definition as it proves convienient in the Taylor formula for example.
In general I think it really should stay undefined because if you take 0/0 = x and multiply with 0 you get 0 = 0*x, which is an equation that *every number* fulfills.
If every number fulfils the equation then why can't it be equal to every number at once? We already have expressions such as √4 = ±2 and ∛1 = 1 and -0.5±0.8660254...i, which have multiple answers (in this case 2 and -2 are both answers for √4 because they both work for x²=4 and 1, -0.5+0.8660254...i and -0.5-0.8660254...i are all answers for ∛1 because they all work for x³=1), so I don't see why 0/0 can't be every number at the same time because every number works for 0*x=0.
@@thewierdragonbaby4843 The square root function is uniquely defined to be positive, so √4=2 and ∛1 = 1. If you don't believe me, search "Epic Math Time square root" on RUclips.
But I get your point, you want 0/0 to be a set of all real numbers. I think that is definitely interesting, though I don't know if that would actually be useful in any way.
@@BedrockBlocker The problem I have with only having one number be equal to stuff is that even though two numbers work for x²=4, you just pick one of them and use that with no real reason why. I mean, sure lots of the time in real life it's the positive one but this is maths, not science, so I don't think maths should change based off of what we have in real life. Of course if you try to pick one of the infinitely many numbers that you would get for 0/0 you don't know what to pick because there's no real life indication of which one to pick, but if you allow multiple answers there doesn't seem to be a problem, at least in my mind.
Also wouldn't non real numbers work for this as well? I was actually thinking of 0/0 being a set of all numbers, real or not
@@thewierdragonbaby4843 Looks like this has become more of a philosophical argument as of now :)
I am a maths student in my 7th semester so to me, I see numbers as what I can do with them, just as you said we have maths not science, and we can do whatever we want as long as we accept the consequences.
Having 0/0 be the set of real or complex numbers (what you mean by all numbers) just would not help out in any way, I could just write down the real numbers or the complex numbers if I'm referring to them. 0/0 having that definition is just not useful.
It's similar to dividing by 0.
Why would you want to do it except just "to do it"? Well, funnily enough there are actually applications where you allow infinity as a number and set 0/x = infinity (for x>0) and x/infinity = 0, this has applications i.e. in measure theory. But you then have to live with the consequences that these numbers then are not a field anymore.
@@BedrockBlocker yeah, I guess that's true
0:38 *WRONG*
0 / 0 is not undefined. It is indeterminate. There is a big difference between the two.
This channel does a good job of spewing nonsense about mathematics to appeal to a larger audience, and this video is no different.
I don't really see a functional difference between the terms 'indeterminate' and 'undefined,' albeit I see undefined used interchangeably with 'erroneous' frequently. So I think I agree with what your point is. The expression [ x/0 ] for [ x=0 ] is not at all the same entity as it is for [ x>0 ] or [ x
@@ladymercy5275 From my personal interpretation, I always interpreted undefined as a value that can not be defined using out numerical system. For example, before i was invented as the square root of -1, the square root of -1 for some time could not be defined, thus considered undefined until we chose to define √-1 as i. Another example would be the limit as X ➡️ 0, for 1/x. The limit does not exist, so I would also say it is undefined. Indeterminate, on the other hand, from my personal interpretation, states that a value can not be determined to either exist as a value in our numerical system or not, as defined by the term undefined. In layman terms, undefined means "There is no value that can be attributed to this algebraic expression" while indeterminate means "it is impossible to determine whether this value exists in our finite complex numerical system or not."
A good analogy for this is whether a person is atheist or agonist. The atheist is confident that no god exists, this is the undefined form. The agonistic can not determine if God exists or not, this is the indeterminate form.
#/0 and tan90 are both equal to infinity
dividing nothing by nothing = nothing
Before watching, my first thought was "Illegal".
Really hope we're seeing by now how incomplete the math system is, and how many wordings teach it differently, with different implications in results
Math can never be complete. Any complete system that contains set theory must be inconsistent, and consistency is more important to most mathematicians than completeness.
@@varmituofm well considering we need math to be consistent, I'd have to say a few things about the inconsistency.
For instance:
0/0 cannot be undefined.
Because:
.1/0=0
(.1÷0)
2/0=0
(2÷0)
When you divide by a zero, the answer is 0.
So,
The term /0 will or should always result in 0, regardless of the numerator.
So 0/0 will, following the underlying operation (/0) always equal 0.
To check, we would take the result:0
And multiply it by the denominator:0
0×0=0.
Spot on result.
@@varmituofm the reason is because in the case of:
1/2
We say
(1÷2)
Which equals .5.
As part of our operations to check results, we always follow with:
.5×2
Which: =1 as mathmatical proof.
So the inverse of the operation in division is used to check that the operation is correct.
If multiplying 0 always results in 0, then so does dividing by zero.
@@Taime88 your argument assumes that the field requires multiplicative inverses for all elements in the field. It doesn't. It requires multiplicative inverses for non-zero elements. There is no requirement that 0*a have an inverse.
@@varmituofm there is when checking math.
If I say for instance 2/1=2 I check it by taking 2 and multiplying by 1.
In the expression 0/0=x in which I solve for x, I can rewrite it as:
X×0=0×0 using inverse multiplication to solve for x. 0×0=0
X×0 regardless of the value of x will lead to 0.
As well, if rewritten as a division (0÷0) how many times does 0 go into 0? 1 time, so 1 0.
In the example of 2÷1 1 goes into two 2 times, so two 1's (1+2=2) in 1÷1 how many times does 1 go into 1, one time (1).
In 100÷10, how many times does 10 go into 100? 10 (10+10+10+10+10+10+10+10+10+10)
So in 0÷0 how many times does 0 go into 0?
1 time (0)
1 time. This breaks down immediately when you go to check though: 1×0=0 but so does any other number multiplied by 0. So the answer is variable by definition, which is why it's better to use an x as the solution in an equation.
Thus:
0/0=x
0/0×0=x×0
0=0 (no matter what x equals, for this equation to be equal, this side must be 0.)
So inverse multiplication is the best approach for solving this problem
If we define division of two real numbers to be subsets of R, then we get 0/0=R, x/0=empty set for x not 0, and x/y to be the singleton set containing the number defined by usual division if y is not 0.
You could do this, but definitions of arithmetic operations as being subset-values instead of being real valued are useless.
@@angelmendez-rivera351 no they are useful. Consider the topological space Spec k[x] where k is an algebraically closed field. The points are the elements of k corresponding to the ideals generated by linear polynomials, together with a generic point, corresponding to the 0 ideal. Each non generic point forms a closed subset by itself.
If you localize a ring by a prime ideal, you will see that the 0/0 resembles the concept of the generic point.
@@Grassmpl I think you missed the part where I specified "arithmetic" operations. I didn't say anything about topology.
Start with a set(or several if necessary). Define your operation. Prove properties. Use them.
Or if we’re just playing around start with an existing such structure, extend the domain by a new element, insist rules of the previous structure should keep holding until you get confused. Patch up the result with glue.
very simple eplanation
I've always thought about multiplication and division as an organization of groups; for example X multiplied by Y is the same thing as X groups of Y. Similarly, X divided by Y is the same as turning X into a number of groups equal to Y. Explaining all of that probably seems redundant because most people might have already concluded all of that on their own, but most of the reason I'm doing it is just put things into words rather than symbols for the purposes of creating a more mentally tangible situation.
Anyways, apply this to what 0 represents: nothing.
If you take nothing and put it into no groups, that means there are no groups of nothing. By definition, having no groups of nothing would mean you have no groups in which nothing resides, as in, every group has something in it. Thus you effectively have everything.
I think one reason why all of this might be an issue on a computational level is because you can only describe the answer by what the input is not.
Hopefully all of that was understandable and logically sound.
Yeah I think what you said makes most sense.
I see a lot of people saying having no groups of nothing, for example someone commented that cutting 0 slices out of a non-existent pizza means there's no pizza hence 0/0 = 0, but that doesn't really make sense because no groups of nothing should mean that all groups have something.
1/0 = blue, so therefore 0/0 = 0 * blue. But wait, 0 * blue is just 1. So…
🤔
0/0 would be 1 if 1/0 could exist. Unfortunately, the existence of 1/0 contradicts the associativity of multiplication, and it also may contradict the distributivity of multiplication over addition, or the associativity of addition, depending on how one chooses to define the algebraic structure. Regardless, if multiplication is not associative, then even if every element has a multiplicative element, one cannot coherently define a division operation.
I agree that you can’t do much with my thesis here. Like with the nullity example, 0 would have to lose its famed zero property for this to work (0 * anything = 0) but of course we can’t really have that.
i think its everything at once if it’s not undefined. 1/0=inf. 2/0 = inf. So 0/0 is every fraction existing. also 0=0x anything
splitting nothing into no amount of pieces gives nothing
The best definition I can come up with is the Universal Set (-All possible values).
*===== PROOFS =====*
·> *0x = 0* is always true because of the *Zero Product Property.* If we solve for x here, by a *rule of true statements,* we get the *Universal Set,* but by algebra, we get *0/0.* We can use the *Reflexive Property* to define *0/0* as the *Universal Set.*
·> The 3-dimensional equation of *z = xy* has *2 perpendicular lines* at *z = 0.*
·> My Intersection Formulas (-On my One Portal) all result in *0/0* for *overlapping equations.*
·> Multiplying *non-zero values* by the Universal Set will result in the Universal Set. 0 is also an exception to the *Product&Quotient Rules of Exponents.* This avoids contradictions of *0A÷0 = 0B÷0* resulting in *A = B.*
If you add to the Universal Set, you will get the Universal Set.
*The best definition I can come up with is the universal set*
No, you cannot use this as a definition, because as I explained elsewhere, the universal set does not exist.
*0·x = 0 is always true because the Zero Product Property.*
No, it is not. 0·x = 0 is only true if (R, +, 0) is a group, (R, ·, 0) is a monoid, and · left-distributes over +.
*If we solve for x here, by a rule of true statements, we get the universal set,...*
No, we do not,.... because the universal set does not exist. See the axiom of regularity, for more information.
@@angelmendez-rivera351 :
What do you mean *0x = 0* can be false? I do not understand your explanation. Multiplying by zero will result in zero. Multiplying by one is trivial.
I am confused by "No it is not. 0·x = 0 is only true if [...]".
@@angelmendez-rivera351:
If you are saying *{n, n, n} ≠ n,* I could definitely see what you mean.