Defense against the "dark art" of mathematics.
ฝัง
- เผยแพร่เมื่อ 26 มี.ค. 2024
- 🌟🌟To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/michaelpenn.🌟🌟
🌟Support the channel🌟
Patreon: / michaelpennmath
Channel Membership: / @michaelpennmath
Merch: teespring.com/stores/michael-...
My amazon shop: www.amazon.com/shop/michaelpenn
🟢 Discord: / discord
🌟my other channels🌟
mathmajor: / @mathmajor
pennpav podcast: / @thepennpavpodcast7878
🌟My Links🌟
Personal Website: www.michael-penn.net
Instagram: / melp2718
Twitter: / michaelpennmath
Randolph College Math: www.randolphcollege.edu/mathem...
Research Gate profile: www.researchgate.net/profile/...
Google Scholar profile: scholar.google.com/citations?...
🌟How I make Thumbnails🌟
Canva: partner.canva.com/c/3036853/6...
Color Pallet: coolors.co/?ref=61d217df7d705...
🌟Suggest a problem🌟
forms.gle/ea7Pw7HcKePGB4my5
🌟🌟To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/michaelpenn.🌟🌟
I am glad that you referred to Mathematics as a "Dark Art" because it's certainly not a Science.
You might want to pop in a correction at 2:55, since you wrote a/infty instead of a/0. Of course, anyone listening will hear you say "a divided by zero", but it might still confuse people
One interesting thing about this number system is that multiplication no longer signifies repeated addition, as intuition would suggest. This can be seen by noticing that infinity+infinity=perp, but 2*infinity=infinity.
Well, I mean multiplication isn't addition in a lot of cases, for example, (1 + i)(1 - i) = 2, but that's clearly not "Adding (1 + i) to itself (1 - i) times"
@@JJean64 It's only supposed to work when you multiply by a natural number. Otherwise it doesn't make sense. But here, it doesn't work even WITH a natural number 2.
its sorta like saying turning 720 degrees is more than 360 but really it turns the same way if you rotated like that
That's because *∞* actually denotes *∞ = {a·∞ⁿ + b | a ≠ 0, n ∈ ℕ, b ≠ ∞},* so when you _multiply_ by two, you can be sure that you work with two _identical_ copies of infinity *(a, n, b* are fixed), but when you _add_ two infinities, they can be _any_ elements of the defining set, which results in indeterminate/perpendicular/bottom/nullity.
So the "perpendicular element" is like undefined, except it's defined.
lol yeah, like, when you do a calculation with an undefined element you get undefined, so this is further evidence
It is good point!
It's algebra would not be complete in this system unless it was managed within the system
Hence it absorbs everything. Power of defining the undefined!
@@caesarinchinawell it was eitherr that or consign it forever more to its null space 🙂
I was thinking the same. Like NaN in floating point numbers.
And here I thought reinventing the wheel wasn't a worthy task...
inf+inf=perp makes complete sense if you consider that inf is actually both positive and negative infinity, so it's also equivalent to subtracting infinity from infinity, which is something that you also can't do when handling infinity in other ways
-infinity = infinity (by the rule a * infinity = infinity for all a € Z), so this distinction doesn’t really make sense.
@@user-ig3fr9xr5m the point is that in different concepts of infinity where +inf=/=-inf, inf-inf is an indeterminate form, while inf+inf=inf. (for example when taking limits). Because in our case, inf=-inf, you get inf+inf=inf+(-inf)=inf-inf, so while you might expect inf+inf=inf, it's actually indeterminate, because it's also the negative version at the same time.
There’s no -inf in this definition
@@CutleryChips Yes there is, but it's equal to +inf, which is why +inf sometimes behaves like what you would expect of -inf in other contexts
@@kappascopezz5122 no there isn’t. The set of integers consists of {…-2,-1,0,1,2,…}. The negative integers might as well have been different symbols like {…B,A,0,1,2…}. The “-“ character just appeals to our notion that “-1” + 1 = 0.
But in this context, he didn’t define a “-infinity” I could have labelled the perpendicular element as “-infinity” and all the properties he defined are satisfied.
For example. “inf” + “-inf” = “-inf”. Of course, if we want to loosely use the subtraction symbol: inf + (-inf) = inf - inf = -inf.
-inf + inf = -inf
-inf - inf = -inf
inf + inf = inf
inf/-inf = -inf/inf = -inf/-inf = -inf
inf/inf = -inf
inf * inf = inf
inf (-inf) = -inf * inf = -inf * (-inf) = -inf
If somebody is interested, making elements invertible in rings is done by creating so called "localisations". It's actually very useful in higher mathematics.
Perpendicular element behaves more like error element in computer calculations (anything with error = error)
So this algebra looks like natural way to implement Q into machines without writing many "if" about /0 etc.
a more direct approach with this goal are the transfields e.g. the transreals
you can liken the IEEE floating point standard to the transfield extension of the dyadic rationals up to some order (precision in mantissa vs precision in exponent) and a nuance regarding 'signed zero' and that NaN is not reflexively equal
@@MagicGonadsSigned zero actually isn't anything special. Floating point numbers are finitely precise, so 0 really means "anything close to 0", and something close to 0 does have a sign.
@@U20E0 you actually get a choice, it depends on how one extends the 'monad' surrounding 0.
I disagree with the choice made by ieee. It works well enough and it isn't important enough to change, but it leaves out the case where x is approximately zero but with unknown sign. Also, ieee-754 is really weird and stateful because of FP rounding flags, the NaN payloads that nobody uses, lack of "supernormal" numbers to complement subnormals...
@@NXTangl NaN payloads are actually used, but as an optimization for javascript engines. the NaN payload is used to store a pointer to an object in memory; this allows all js values to be represented as floats (and everything that isn't a float is just a NaN that happens to carry a pointer in its payload)
this is called NaN-boxing
My favorite extension is to the complex numbers. C* = C U {∞} can be modeled as a unit sphere. You can map any three distinct points to any other three distinct points using the Möbius transformation.
That's pretty cool but it leaves some of the operations undefined, whereas the point of the video is to define all of them, so the approach is a bit different
The Riemann Sphere is indeed the coolest topic in mathematics
C*=C U {oo} is funny in Portuguese.
We read this like *ss=ASS {oo}
Michael, I have a crazy question. In Euclidean geometry, parallel lines do not intersect. In projective geometry we CAN intersect parallel lines where we essentially define the intersection to be at infinity (horizon). Using the concept of vanishing lines/points we can show where those parallel lines intersect. That is how we get beautiful artwork in the form of paintings/drawings from. But here is the thing: We can "locate" infinity is a position in projective geometry and work with its properties. Similarly can we somehow "extend the normal number system" to a "special" number system in the sense that 1 divided by zero is an "equivalent" move of viewing parallel lines intersecting at infinity and define the answer of 1÷0 as the solution of the point of intersection of those parallel lines at infinity? The duality principle (for example two lines intersect at one point versus two points make one line, a principle often used in projective geometry) can then be applied as "1÷0 versus two lines meet at infinity. Thoughts appreciated.
In projective geometry of the plane, "infinity" is really a line at infinity. In perspective drawing, this is the horizon.
@@kristianwichmann9996 Yes, but the thing is that "infinity" in projective geometry is "tangible", i.e. we can assign it a "place". My idea is to "link" this to the outcome of 1÷0. There are many parallel lines all intersecting somewhere on the horizon. Can we links those positions to 2÷0, 3÷0, 4÷0 etc?
The number system formed by adding an unsigned infinity to the Real numbers (not sure if it needs a perpendicular element or to be a wheel) is actually called "The Projective Reals", so the connection to Projective geometry is entirely real.
What you just described is exactly why I think everyone in STEM should learn perspective drawing and take some basic art classes. (I also think artists/animators should learn basics optics and kinematics, but that's another subject...) I've used this exact example before to explain to people how the reals can be extended to allow for multiplication by infinity or division by 0.
It's a direct analog for the projective geometry that extends the reals and is approachable and intuitive for most listeners.
It doesn't have rigour, but it is isomorphic to other more rigorous mathematical descriptions.
I think it is an excellent ontological approach to understanding projective geometry, limits, infinite sums and so much more.
Conformal geometry is the bomb and is so much more approachable than it first seems. 👌
@@angeldude101 You do need ⊥ to extend the Reals into a wheel, which is basically because you need 0/0 to be defined. We have x+0/0 = 0/0 for all x, and since 1+x ≠ x for all Real x, then 0/0 is not Real. However, we also have 1/(0/0) = 0/0, and since 0 is Real then 0/0 ≠ 0 = 1/(1/0), hence 0/0 ≠ 1/0. Since 0/0 is not Real, and it's also not 1/0 (which is ∞), then it has to be some other third thing. That's why we give it its own symbol, ⊥
In my opinion, this is not meaningfully different from having 0/0 just being undefined... You have x+⊥ = ⊥ for all x, and you have x*⊥=⊥ for all x, and you have x/⊥ = ⊥ for all x, so ⊥ just absorbs everything. But, that's basically what undefined operations do already, so it's not too different. Wheel theory is good because it's very explicit about what is happening when you invoke these indeterminant forms, but it's pretty close to how most people think of them already?
It's a wheel because infinity ties both ends together, and the perpendicular element is the axle :þ
That's exactly the picture from the wikipedia page en.wikipedia.org/wiki/Wheel_theory
2:57 you say "a over zero" but write "a/infty". I think what you *said* was right, if I understand the topic well enough?
Yes it's a mistake, he meant a/0. I think he just didn't notice.
Oh also, at 24:00 he also wrote 2×inf+2×inf but i think he meant to write 2×inf+3×inf.
The addition rules of the perpendicular number remind me of how null behaves in various programming languages.
Lookup bottom elements in type theory. Null and exceptions are similar to this.
I rediscovered this independently as a:b = {(x,y)| ay = bx }. It's an isomorphic definition with the only structural difference being that all a:b contain the pair (0,0) and 0:0 contains every pair.
so you could call {(0,0)} the 'top' element and 0:0 the 'bottom' element, with respect to a lattice with partial ordering given by the reversed subset relation, which aligns with giving the 'perp' element the 'bottom' symbol
Hi there! Absolutely love this algebraic approach to dividing by zero with ordered pairs. (I actually just got done teaching my abstract algebra class an hour or so ago about constructing the field of fractions from an integral domain, so it’s funny this would come out now!)
Division by zero is a pet topic of mine, and if you search “How to Divide by Zero” you’ll find my approach. It essentially does the same thing (protectively extending the real line) but it uses a more geometric approach, specifically projective geometry (as another commenter mentioned). I might have to update my site to include a link to this video - it’s very well explained.
My students have often talked about enjoying your videos, and now I see why. I might have to start watching more of them! 🙂
Can you provide a link to your video?
Take a square piece of paper and draw the xy axis across the middle of it. Label out the points to fit a standard trig skewing or viewing scheme where the edges of the paper in the x are 2PI and -2PI accordingly. Then have the vertical or Y-axis in integer or fractional integer increments so that the top and bottom are say +/-2. Now draw out the tangent function on this paper along with each of its vertical asymptotes.
If we consider the fact that tan(t) = sin(t)/cos(t) and we know that within the context of linear equations especially with the slope-intercept form y = mx+b. We can see that the slope of a line being defined as m = (y2-y1)/(x2-x1) or deltaY/deltaX is also sin(t)/cos(t) where t is the angle that is between the line y=mx+b and the +x-axis. Thus the slope of a linear equation is also tan(t). Now if we understand that connection we can see that with the following:
tan(0) = sin(0) / cos(0) = 0/1 = 0
tan(90) = sin(90) / cos(90) = 1/0 = +/-inf
If you want to see this effect and why it is the way it is. Sure we can analyze the graph of the tangent function and we can the vertical asymptote as theta approaches 90 degrees or PI/2 radians. And we also know that the sine and cosine functions have the save wave form, the same range and domain, the same periodicity and the same limits, it's just that their only differences are: the sine starts at (0,0) and the cosine starts at (0,1). This causes the since to be an odd function and the cosine to be an even function. We can see these from the taylor series and their expansions but that's besides the point and outside of the scope of this. However, there is this property of orthogonality between them. Their natural base waveforms are 90 degrees or PI/2 horizontal translations - transformations of each other. They are out of phase by 90 degrees. They are perpendicular or orthogonal to each other. Since we can define the tangent function in terms of the ratio of these two as in tan = sin/cos this orthogonality is also present within the tangent function.
We know the period of the sine and cosine are 2PI radians, 360 degrees or 1 full revolution around the unit circle. We also know that the period of the tangent is PI radians, 180 degrees or 1/2 revolution. We also know that the angle of a straight line is PI radians or 180 degrees. We also know that the summation of the three interior angles of a 2D Euclidean Planar triangle is also PI radians or 180 degrees.
All of these properties are related. To see this effect let's take that cartesian coordinate system that you drew out on a piece of square paper while mapping out the graph of the tangent function and now, let's not look at this from a flat 2D perspective. Let's look at this from a curved 3D perspective without changing the coordinate system. This would be more of a transformation in the form of a transposition, kind of like fourier transform so to speak but not exactly...
Take this paper and curl the Y-axis out into the 3RD dimension leaving the X-axis straight and unchanged. You'll end up turning this plane into a cylinder. Now examine what the Tangent function is doing as it wraps around this cylinder. We also have to consider the fact that even though this coordinate system has the shape of a cylinder it's radial diameter is still infinite. You'll want to have the +x-axis facing you and the top and bottom of the paper meeting towards the back. Where these two meet or converge is the line or point at infinity. Within this context we can abstract the idea of + and negative infinity away. If we then rotate the cylinder to have the +x-axis towards the back and we look at what the tangent function is doing... The two arrows that point towards infinity nearly converge at the lines or circles around the cylinder at every PI/2 + PI*N intervals. The +infinity on the one is approaching that line at infinity but it is also approaching the -infinity which is 1 step or interval over (the next period or phase shift) within the tangent function. This isn't directly visible just by looking at its graph in 2D. We need to look at it on 3D Cylinder.
Once we have this perspective of its properties. Then we can begin to expand this into other domains such as mapping its graph onto a sphere within the Real Plane or a Sphere within the Complex Plane, or even onto a torus or even a mobius strip... This might give you a different perspective of division by 0 and how it is related to infinity.
Is this really algebraic?
Does division by zero exist in constructive mathematics?
If so, what is 1/0? And what is 2/0?
What is the computer code that proves your answers?
In an algebraic constructive setting, infinity and bottom are the same thing, so there is no need to use different symbols for them.
@@jboss1073 If we take an expression, a function etc. and compose it into a vector notation. Then the expression of A/0 would simply be the set of coordinate pairs:
{ (0,-AN), ... (0,-AN1), ( (0,An0), (0,A1), ... (0,AN) } where N approaches +/- infinity. This is the vector list of division by 0. This gives you the vertical line that is the y-axis. This vertical line can also be translated along the x-axis. It will always be perpendicular or orthogonal to the x-axis. Division by 0 in itself is NOT a Function. But it is an expression, an equation with a Set of Values where it's I/0 Ratio is NOT 1:1. It is Many to 1.
@@jboss1073 Well, algebra or something that is algebraic is a symbolic representation based on a set of notations. Geometric is the visual or pictorial representation of the same. At the end of the day both are still purely abstract concepts. They are still nothing more than ideas, concepts, a product of the mind. What kind of limitations do you set within your mind? If you want to try to better understand division by zero. You may want to look into many various fractal geometries... This should help you to better understand the concepts of infinities and how infinity is related to +/-1 and 0. Without those relationships then transcendental numbers such as e and pi wouldn't have the properties that they do have with an infinite amount of non repeating decimal digits.
This gets a lot more interesting when the base ring has zero divisors
It's worth noting that ∞ also doesn't always distribute properly over addition with real numbers (with the inclusion of +∞ and -∞ and the typical arithmetic rules for those elements).
If you have (2+(-1))*∞, that works out to 2*∞ - ∞= ∞ - ∞, which is an indeterminate form. But when all summands have the same sign, the result does come out to either +∞ or -∞.
In this wheel on the other hand, there is no differentiation between +∞ and -∞, so any addition of ∞ to itself necessarily becomes an indeterminate form (or rather ⊥).
It's ironic how often "brilliant org" gets in the way of me actually learning something.
Finally "Wheels"! I was asking for this for so many years!
Finally, someone covers the projective point at infinity and Wheel Algebra. There are almost no videos on this topic on TH-cam, and there are millions of videos trying to claim that you can't divide by 0 ever.
To allow division by 0 one needs non-conventional definitions stated. Those vids you refer to likely directly draw the conclusion that "can't divide by 0 ever" using the conventional real number definition. I don't see any error in both scenarios.
Different definitions, different conclusions, that's probably too fundamental to bother mentioning. I just perceive most ppl directly jumping to others' face saying "you know what I can divide by zero" WITHOUT A CONTEXT as an action of "showing that I know something blah blah", or just to "enjoying opposing others", while subtly referring to something completely different. Unfortunately some ppl on internet love to behave like this.
Hypothetically dividing by zero is dividing a superposition im most likely wrong but that should be possible
@@atussentinel The reason why I dislike people saying that you can't divide by zero is because I had been trying to find ways to divide by 0 for over a decade. I hate being told that something in math is impossible to do. When I discovered the Riemann Sphere and the projective point at infinity, I had found my solution. And the fact that this solution has physical applications shows that this isn't just some abstract game. This infinity is real and useful in the real world. So when I tell people that division by zero is possible, it's essentially just to spread this good news.
Hi, I'm a constructive mathematician. No Law of Excluded Middle for me. Things are true, false or bottom.
Answer constructively so I can understand, please:
What is 1/0?
What is 2/0?
What is the computer code that proves the answers above?
Or is division by zero only "possible" in classical mathematics (and hence not really possible in reality)?
@@samuelyigzaw I still think division by zero is impossible in constructive mathematics (meaning in reality, without the Law of Excluded Middle).
If you believe we can divide by zero, please answer:
What is 1/0?
What is 2/0?
What is the computer code that proves the above answers?
Thank you for helping me understand this.
Similar but slightly different from IEEE floating point math which has a new element NaN (not-a-number). Main difference is that (+inf) + (+inf) = (+inf), and that NaN is not equal to itself.
So is the IEEE floating point math not a "wheel"? Or is there a relatively simple fix to what Michael did so it matches the IEEE f.p. math? Perhaps it's not important, but the IEEE f.p. math seems to be self consistent.
@@user-gs6lp9ko1cThe IEEE floating point math, with the exception of NaN not equalling itself, is a wheel up to precision.
+inf and -inf represent any number sufficiently close to infinity. True, unsigned infinity cannot be represented.
+0 and -0 exist because in actuality they are infinitesimals, true zero cannot be represented either.
I am pretty sure however that this is just a coincidence. The intent was not to make a wheel, it was to make a self-consistent solution for mathematical edge cases
@@U20E0 The signed infinities are useful when there's code that wants to compare numbers and you want to hand it a comparable that it will see as greater than (or less than) all finite numbers.
Floating point math does not actually have infinity. It is a token standing for infinity. The calculations' results come from a lookup table.
I once got the exact same idea, but wrote 'perp' down as 0' (zero prime, or was it infinity prime?) because I didn't know what to call it.
And now seeing this video pop up on my homepage is strangeaf...
In my universe, the 'number' zero is not a number and thus can't be used for computation. Zero is a placeholder for a null value, not a number.
Romans didn't even have a numeral for zero.
Great video! There is very little about this topic on the internet, so I appreciate it very much.
Really liked this video! Maybe my favorite I’ve seen so far! Awesome job!
This was such a well put together and captivating video that it felt like five minutes instead of thirty, I was somewhat shocked when called the end there.
It's almost like the perpendicular element (0/0) is like a sink hole in the proposed algebra
I don't want this question to be taken the wrong way: Is there anything currently known that wheels are useful for?
And I don't mean applications outside of mathematics; I even include within mathematics. Currently, it seems more like a novelty to me - it only exists for the purpose of saying, "I can divide by 0 now".
For example of what an application could be: one useful thing about many algebraic structures is that we can associate algebraic structures to various mathematical objects, and mathematical objects with non-isomorphic algebraic structures are themselves non-"isomorphic" in their respective category. Are there some mathematical objects where associating a wheel is the natural structure to use? Or maybe other algebraic structures are more natural, but wheels _also_ work and provide different insights from groups, rings, modules, etc.?
(And to be clear, even if there are no applications yet known, that doesn't mean there never will be any. I am just curious if there already are examples that I just don't know about.)
Loved this one. 👌 Any chance of you doing stereographic projection or the Riemman Sphere as related topics?
I think it is easier to think about those special elements as 1/0 and 0/0 - new symbols just add to confusion, imho. It was helpful when I've implemented wheels in both C++ and Smalltalk.
Love this one!! What a boss!
In SQL we would call perp "null" and in fact a better name for it is "dunno". 0 * inf = dunno. 1+ dunno = dunno. etc. But it sure seems like regular algebra would be challenging in this world --- you have to know a lot about stuff in order to manipulate. Seems like no fun :D
this is my favorite kind of video on this channel. simply wonderful
Interesting, and Thank you!
Respected Michael Penn sir, I have stopped doing illegal things many years ago. This is why I don't dare to divide anything by zero, even I don't dare to divide 0 by 0.
Enjoying this thoroughly. Always struggle with equivalent infinities when I look at these, in my mind, the limit approaches infinity at discretely different points. Going to have to do some reading.
Nice! I'd like to see some interesting (preferably geometric) applications of these mathematical wheels.
This is the first Michel Penn video I followed completely. I couldn't apply what he brought out, but I didn't glaze over and just enjoy the ride.
Fun as always regardless.
It appears the distributive property does not hold in your extension of the rationales. 00+00= T, but 2(00)=00. So this is NOT a ring over the integers.
Very interesting video!!
This was wonderful. I was smiling the whole time!
I thought only Chuck Norris could divide by zero!
Chuck Norris can divide by zero. And he wears sunglasses to protect the Sun from his eyes.
2:48 Says "one over zero", writes "one over infinity equals infinity". Nice trick.
The perpendicular element reminds me of NaN in floating point arithmetic.
This reminds me very much of NaN (instead of perp) from floating point rules.
Think of division by 0 as being similar or almost equivalent to tan(90) or tan(PI/2). It is vertical slope OR it is perpendicular to a line with 0 slope. Consider the following two sets of two points that make up two different linear equations within the context of vector notation:
We have vector A and vector B to be defined as the difference of two points: A = p1 - p0 and B = p3 - p2 where each point pn can have any arbitrary number of dimensions or unit vectors within its composition.
For simplicity we will use a 2D coordinate pair / system
p0 = {x0, y0}
p1 = {x1, y1}
p2 = {x2, y2}
p3 = {x3, y3}
If vector A = p1 - p0 then A = {x1, y1} - {x0, y0}
and vector B = p3 - p2 then B = {x3, y3} - {x2, y2}
Let's say that p0 is the coordinate pair (0,0), p1 is located at (1,0), p2 is located at (1,0) and p3 is located at (1,1). From these points in conjunction with the slope-intercept form of a linear equation y = mx+b we can use the slope formula of m = (y2-y1)/(x2-x1) to determine the slopes of these two lines.
Then m0 = (0 - 0)/(1 - 0) = 0/1 = 0
And m1 = (1 - 0)/(1 - 1) =1/0 = ?
In normal arithmetic or algebra m1 would be considered undefined. Here the slope of the first line m0 is parallel to the +x-axis which has a slope of 0. The slope of m1 is the slope of a vertical line where we typically state that it is undefined. Yet it is not wrong to state that the slope of m1 which is vertical is orthogonal or perpendicular to the slope of m0. So when we look at the two fractions or ratios of the different slopes within this context we can see that 0/1 and 1/0 are orthogonal or perpendicular to each other.
How is this related to or equivalent to that of tan(90) or tan(PI/2)? It's quite simple. The numerator of the slope y2-y1 or deltaY is the same as sin(theta) where theta is the angle that is between the line y=mx+b and the +x-axis. And the denominator of the slope x2-x1 or deltaX is the same as cos(theta) where again theta is the angle that is between the line y = mx+b and the +x-axis. Through the trigonometric identities and substitutions we also know that sin(t)/cos(t) = tan(t).
It is from this identity that we can see that tan(t) is equivalent to deltaY/deltaX or (y2-y1)/(x2-x1) and from that we can see the following:
sin(0)/cos(0) = 0/1 = tan(0) = 0
sin(90)/cos(90) = 1/0 = tan(90) = ?
We also know that the two wave functions sine and cosine have the same waveform, the same range and domain, the same periodicity or oscillations, and the same limits. The only difference between these two functions is that sin(t) starts at (0,0) and cos(t) starts at (0,1). The difference in their starting positions is what causes them to be 90 degree or PI/2 horizontal translations of each other. This is what makes these two functions perpendicular to each other. Also, this is what causes one of them to be an even function the cosine and the other to be an odd function the sine. This orthogonality between the two is where we also have the concepts of being in and out of phase of each other. Typically every 90 degrees is a phase shift. This perpendicularity also shows up within the tangent function due to the property of the tangent function in relationship to its definition or identity based on the sine and cosine functions. It's prevalent within tan = sin/cos. And we can see this when we start to plug in different values or angles into the tangent function. Also the period of the tangent function is 1/2 that of the sine and cosine functions respectively.
The period of the sine and cosine function is 2PI, 360 degrees or 1 full revolution around the unit circle where the period of the tangent function is PI or 180 degrees or 1/2 revolution around the unit circle. Why is this important? There's more to it than just the properties of the lines, angles and circles. There's also the properties of the angles of lines in conjunction with the properties of triangles and their interior angles. We also know that a given 2D Euclidean planar triangle has a value of 180 degrees or PI radians for the summation of all three of its interior angles. We also know that the angle of a straight line is also 180 degrees. This is also the same period of the tangent function.
Why? When we start at a rotation of 0 with a line that has a slope of 0 and we rotate it by 90 degrees. This line now has vertical slope and is orthogonal to the line at its original position. This is also 1/2 of the trip of a single period of the tangent function or 1/4 period of the sine or cosine functions. This 1/2 trip is what makes it a perpendicular bisector. This is why it is orthogonal.
So if the value of 0/1 which = 0 exists as a slope to a line that has 0 slope, then the slope of the line that is perpendicular or orthogonal to it must also exist. And the orthogonal slope of 0/1 is it's reciprocal which is 1/0. Thus 1/0 must exist. So to say that division by 0 is undefined is an in appropriate assessment that we've all been forced to learn since grade school. I beg to differ. I think that division by 0 is not undefined. I tend to think of it as being vertical slope. Vertical slope is where you have displacement only in the vertical and none in the horizontal. And this is the approach towards +/-infinity. So 1/0 -> +infinity and -1/0 -> -infinity. I would agree that it is ambiguous but not undefined.
Another way to look at it is also through operator composition based on the definition of what division is in terms of addition as opposed to being the inverse of multiplication. We know that multiplication is repeated addition. We also know that subtraction or the inverse of addition can be defined in terms of addition. Therefore we can treat division as repeated subtraction. When we have 0 as the denominator of a given fraction or ratio it is the divisor where the numerator is the dividend. We are dividing 0 out of something. In other words we are subtracting 0 from the dividen and then we are going to count up how many times we subtracted to decrease the dividen to be less than the divisor or equal to 0. In this case when we subtract 0 from the dividend it remains unchanged. Thus we can subtract an infinite amount of times with no difference in result. Thus division by 0 yields +/- inf for the quotient with a remainder of the dividend or starting numerator.
For example: 1/0 would yield +infinity with a remainder of 1. 100/0 would yield +infinity with a remainder of 100. And -420/0 would yield -infinity with a remainder of 420 or +infinity with a remainder of -420. And this is why through objective reasoning and basic logic I determine division by 0 to be ambiguous as opposed to undefined. Yet I can also safely conclude that division by 0 is also perpendicular to all the REAL values. It's similar in nature but different than that of the Complex or Imaginary numbers in that they are also Perpendicular to the Real Numbers. It's just that these aren't exactly imaginary, they are still Real, it's just that they're orthogonal projections or perpendicular to the reals. These are the uncountables! They are not countable. They are not quantitative or enumerative. They are basically the opposite of that do to the nature of what it means to be perpendicular as well as what a perpendicular bisector does. Without these we wouldn't be able to see or have the properties of reflections or mirroring based on simple translations of rotations.
This is my take, this is my assessment of division by 0. I think it is well defined. It's just that the result of the operation is "unknown". This doesn't make it undefined. It just makes it ambiguous as there is not enough information to determine its given state at a given instance. This is kind of almost similar to that of which we see and observe within quantum mechanics with concepts such a superposition...
The perpendicular element behaves like indeterminate expressions in limits, when you identify positive and negative infinity.
L'Hopital's rule came to mind when you mentioned 0/0 the perp element. Interesting that when 0/0 is approached in calculus it's not a dead end since L'Hopital's rule comes to the rescue. But here, 0/0 is a dead end.
One other thing i was wondering, does this wheel theory have practical applications? The wikipedia entry mentions none.
Terrific video, thank you
Interesting stuff. Do you need the "0 times 0 is 0" or does it just simplify the other rules?
Incredible! I have not thought this through too much - so apopolologies if this makes no sense but ... is there a 3-dimesional matrix way of handling this?
After all, once defined everything is another form of linear algebra.
Very interesting. I wonder how the rules for division by 0 change when you define Z as N mod ∞. Would that lead to 0, ∞, and ⊥ falling into the same equivalence class?
An important follow up to this topic: Why is this wheel not used instead of the normal Q numbers? Is there some, higher reason why this would be disadvantageous? Additionally, are there any other operators that do not play well with this wheel?
love this
Michael, this is a good video about Wheels. Do take a look at the transrational and transreal numbers. These allow division by zero but have distinct signed infinites, ∞ = 1/0 and -∞ = -1/0, as well as the unordered number, nullity, Φ = 0/0. It follows that transreal analysis contains real analysis. In particular Trans-Newtonian Physics contains Newtonian Physics, such that Trans-Newtonian Physics allows the solution of physical systems exactly at a singularity, in addition to solving the systems asymptotically in the approach to a singularity as Newtonian Physics does. Interestingly, transreal analysis allows one to check for continuity at a singularity. Furthermore the transreal numbers extend to the transcomplex and transquaternion numbers, as well as to more exotic algebraic systems such as the trans-surreal numbers. More abstractly, Boolean algebra extends to Trans-Boolean algebras. There is more, but I am sure you get the idea - you could make more videos on division by zero!
xz + yz = (x+y)z + 0z should be called a "redistributive rule", because on the right the first z has everything and the second z has nothing, but on the left each z has something.
@16:00 the name you are looking for is NaN. Haven't seen the rest of the video yet, but in all probability it has the properties you're looking for.
The only difference between bottom and NaN is that NaN does not equal itself
If Michael says it, division by zero is definitely allowed.
Before watching the vid, I say this looks like unsigned infinity. We can't answer whether it is > 0 or < 0.
Let's call this new part of this setup the "pennity," for Perpendicular Element Number!
I feel like we just formalized NaN and infinity from floating point Numbers.
I've been thinking of this dilemma in the quantum realm - this whole "indistinguishable " entangled particles screams divide by ZERO 😂
I'm also convinced that the relationship of Circumference / Diameter = 1 in the quantum world
Interesting. It's similar to the projective line over Q, but it's weirder.
intrigueing
"You Mathematicians are out of fuckin' lunch!", channeling my "Otto" from "Repo Man".
Thank you so much for all your wonderful videos! This is my lifelong struggle with division by zero:
I love TL:DR's
Ever since my high school Algebra I class this has plagued me, and in all the years since I still can't wrap my head around it. Not sure anyone can.
Zero is generally assumed to have the properties of a rational integer, provided we don't try to divide by it.
Infinity is defined as a limit, and not an actual real number.
This makes them somewhat incompatible in certain operations, as infinity would have to be defined as, at the very least, having the properties of a real number.
If infinity (the absolute highest number possible) is a real number, we can obviously add another real number to it and change its value. This makes it impossible to define infinity as a real number.
Now, if we define infinity as the absolutely largest number possible, so large it cannot be increased (I know this sounds silly), and define zero as its inverse, then we can make sense of arithmetic operations involving either or both. Zeno is winking at me right now.
We can all agree that no matter how many times we diminish a real number, we can make a smaller real number with an operation using that number, like dividing it by two, etc. The inverse of this operation would be equivalent to increasing infinity by two, in this instance.
There is no point where a real number can be so small it cannot be diminished, and there is no point where a number can be so large it cannot be increased, making it impossible to ever reach a limit in either direction.
So, what we have is a limit of "smallness", which can never be reached by division of any real number, other than the currently defined integral zero divided by a number other than zero. The only way our real number system can ever make sense in all operations is if we had a defined value for infinity.
Thinking of it in geometrical terms, no matter how close a point is to another, there is room for another point either side of it, and a point in between them, etc. This is why (Zeno is still winking) infinite series always tend toward a limit, but there is no chalkboard large enough to fit the entire series on, and we insert an ellipsis somewhere to terminate it.
However, if we think about it, using Zeno's 1/2 paradox, what we have is exactly an infinite series, which will never terminate.
We all know that if we take equal steps, crossing the gap is quite finite, so we are then adding the same fractional distance repeatedly, bringing us to our destination in a finite amount of time. This is how we get from one place to another before we cease to exist.
Now, if that destination is the limit we know as infinity, we are not going to get there, no matter how many giant steps mother allows us to take.
We can subtract our way to the integral zero, add our way to it, multiply our way to it, but we cannot divide our way to it. This is the gaping wound in number theory in a nutshell. This is probably why computers can't actually divide numbers, as it is a calculation of trial and error subtraction, in its raw form, which is indeed what they do. Calculating roots is a very similar conundrum, only it involves repeating trial multiplication, which is, again, what computers have to do.
Mathematics has not evolved to a point where we can perform these operations directly, and probably never will, but we can perform the other three basic operations with ease.
I digress a bit, apologies.
If we could accept there is a number so large it cannot be increased, then it would be the true inverse of our integral zero. Hard pill to swallow.
If we defined infinity to be a number so large it cannot be increased, then we would have to accept the there is a number so small it cannot be decreased, infinity's inverse, using any operation. So peculiarly counterintuitive, because the inverse of any real number, regardless of its size, is another non-zero real number, which can be diminished with a division, yet this would require infinity's inverse to be defined as the integral zero.
Looking at it from the bottom up, if there were an inverse of the integral zero, it would be a number that absolutely could not be increased, which we know is also counterintuitive.
Many important equations in the world of physics end in a division by zero, such as trying to understand and define a singularity. Perhaps Max Planck's special number is the limit, and coincides somehow with division by zero, and is the actual finite size of a point, or, lol, the size of the pixels that comprise the universe.
The only way for me to begin to make sense of any of it is to treat zero as a limit, as we do infinity, and not as an integer, and take careful steps not to accept 1/∞ as an absolute value of nothing, and possibly, for the sake of calculations, represent the two values as 0, our integral, absolute nothing, and 1/∞, depending upon the operations involved. Are they equal, your guess is as good as mine.
If you had defined addition as "for both terms, pick among their equivalent classes representations that both have the same denominators, then add the numerators", then 1/0 + 1/0 would have become equal to 1/0, rather than 0/0, which would have meant that the associative property would have been unchanged from how it works in the regular Q number set.
That doesn't work because 1/0=(-1)/0, so you'd be able to pick members that make inf+inf=prep or inf+inf=inf. Regardless of the definition, the associative property has problems with 0/0=(1+-1)/0=1/0+(-1)/0=1/0+1/0=(1+1)/0. Something in there can't be valid, or we have the problem we saw with Q-hat.
@@iabervon Hmm, good point. inf - inf must be equal to perp, and if +inf and -inf are the same thing, then you are correct that inf + inf must also equal perp.
continuing from here, invert a 3x3 matrix, but the matrix has a determinant of 0.
Now that division by zero is permitted, how do the results differ from those when such division was not permitted ?
I wish they would just tell us some examples, like what is 1/0, what is 2/0, etc.
I think x= a/b is defined by " x is solution to x*b =a.. "
so 0/0 = {any_number} and a/0 = {nothing} otherwise
2:58 a/0 not a/∞
I agree! Glad to note someone else noticed this! Thanks!
Seems weird to me that the reciprocal isn't defined by z·/z = 1. The fact that 0/0 ≠ 1 is very strange, but consistent with the definitions used.
I wonder how solving polynomials works in a wheel
After a series of Michael's videos on infinities, I have come to think that equations like *(x−2)² = x²* might have a "legitimate" solution "at infinity" (over the extended real line/complex plane with (a) point(s) at infinity), because both parts of the equation become infinite at *x = ∞,* and it can be solved (quite rigorously) with the definition of what an equation is.
To check this idea I plotted functions *(x−2)²* and *x²* on the *closed* interval *[−∞; +∞]* (by transforming the coordinate grid appropriately to squeeze the whole range into the *[−2; 2]×[−2; 2]* square) and did verify that over *ℝ̅ = ℝ∪{−∞; +∞}* graphs of the functions intersect at *x = 1* and also at *x = ±∞.*
Love your video, if this is not your video, I will not even open it because I think some people will just talk something not an absolute zero those silly things.
Honestly, I don't get why dividing by 0 is so crazy to people aside from the fact that we are all taught that is a nono is school. 1/0=+infinity or -infinity , and for most problems which one you pick effectively does not matter. Of course if it does matter, just use limits to find out which. 1/infinity=-1/infinity=0
Theres is no + or - "0" because they are the same. Under the function 1/x, x=+/-infinity ---> 0, x=0 ---> +/-infinity
I am wondering since 1/0 is not allowed, shouldn't then 0/1 also shouldn't be allowed since:
x * 0/1 = x : 1/0
So you add two new elements, infinity and the perpendicular element (which Wikipedia calls "bottom"), to the rationals, tweak a couple of rules, and voilà! Division by zero and infinity divided by infinity come for free.
Right away I'll say this looks like how "imaginary" numbers come to be
32:19
Thanks 😂
I am very surprised that:
∞ + ∞ = ⟂
but
∞ · ∞ = ∞
There is good intuition for it! (And other comments have pointed this out, so I am not being original here.)
Wheels do a sort of "projective" extension of the ring they're extending. When you do this, you have a single, unsigned infinity symbol "∞", which represents _all_ directions of infinitely simultaneously. (The idea is that we curve the real line into a circle where both positive infinity and negative infinity are "glued together" to become a single infinity; or in the complex plane, the plane is curved up to become a sphere where the north pole is a single "complex infinity" by gluing all possible directions of infinity together)
So taking the intuition of ⟂ representing "indeterminate form" and ∞ representing "all directions of infinity simultaneously", it makes sense.
Adding two infinities could result in "opposite direction" infinities adding together and canceling out to give a finite result or an infinite result or a never-ending oscillation, so it is "undefined/indeterminate". Whereas multiplying two infinities together, regardless of their directions, will always result in something infinite in _some_ direction.
Super interesting video!! I like the use of the ⊥ symbol and it even makes kind of topological sense as the ⊥ being the center of the wheel and thus being perpendicular to the wheel at every point. But being a handler for 0/0, I really have to make notice that the % symbol was out there...
What left me a bit unsatisfied is that nothing is mentioned about "substraction". I mean, considering both operations have an identity element, I was left wondering what are the opposite elements of ⊥ and ∞? They don't seem to have one. Doesn't that break the algebra?
Apparently, they don't. That's an algebra without each element having an opposite. Sort of doesn't break any rules (like in reals zero has no multiplicative inverse).
If Subtracting can still be defined as a + (-1*b) then adding would give you the same thing for the new elements since -1 * inf = inf and -1 * perp = perp but I'm unsure if that definition of subtraction still holds in this algebra.
@@allozovsky But all elements having an opposite element for the group operation (such that x + opposite(x) = 0) is a requirement for the algebra to be a group. And being a group is the base upon a second operation is built to form a ring which Michael said the Wheel is.
@@blakecannon3734 I don't know if opposites can be defined like that in this algebra. Probably not because the problem is ∞ and ⊥ need an element such that adding both results in 0.
@Cloud88Skywalker > _to form a ring which Michael said the Wheel is_
But where did Michael claim that a wheel is a ring? He said that (31:50) "you can take any commutative ring with one and you can *form it into* this structure called a wheel". A wheel is certainly not a ring (not even a semiring), but rather a _commutative monoid_ (for both addition and multiplication). It certainly is not a (semi)ring, since multiplication is not distributive over addition, as we clearly have seen.
Apparently diving by zero makes sense in the Sedenions.
You mean sedenions have zero divisors?
Let me try to explain this shift to non-contradictory infinitesimal monadological frameworks in a way that a teenager could understand:
You know how sometimes things just don't make sense or contradict themselves, even when adults say they do? Like when your math teacher tells you that you can divide by zero, or your science book talks about something being infinitely small? Those are contradictions - claims that go against basic logic and reasoning.
Well, a lot of the theories that scientists and mathematicians currently use to understand the universe are built on ideas that contradict themselves too. They make assumptions that don't fully line up with reality as we experience it. It's kind of like everyone agreed to some made-up rules that don't actually hold up.
The theories talk about space, time, and matter being made up of infinitely precise points, lines, and surfaces. But if you really think about it, nothing in the real world is actually that perfect - everything has some fuzziness or graininess to it at the smallest scales. So basing everything on those infinitely idealized concepts creates contradictions.
The new way of looking at things, called the infinitesimal monadological framework, gets rid of those contradictory geometric idealizations from the start. Instead of infinitely precise points, it uses perspectives as its basic building blocks. You can think of perspectives kind of like viewpoints or windows through which reality is perceived.
But here's the cool part - these perspectives aren't just separate static viewpoints. They interact with each other in super tiny, infinitesimal ways. From those interactions, space, time, matter, and all the other stuff we observe emerges as collective patterns and resonances across the perspectives.
So instead of starting with made-up geometric perfections, we start with these plural perspectives and their infinitesimal relations and interactions as the core reality. Everything else, like geometry, physics, even consciousness, arises as harmonized resonances across those pluralistic roots.
This way, there are no contradictions baked into the foundations. We're not dividing by zero or assuming impossible singularities. We're building up reality in a step-by-step logically consistent way from the pluralities and graininess that actually exist.
The old theories put artificial separations between the observer's experience and the external world being observed. But in this new view, subjective experiences themselves are modeled as resonant patterns woven into the same overarching pluralistic algebra describing all of reality's interdependent unfolding.
It's kind of like upgrading an old contradictory video game engine to a new one that can render way more realistic and coherent simulations without glitches or breaking the rules. Except this new engine is for our entire conception of physical, mathematical, and experiential reality!
The best part is, this shift could pave the way for us to understand and engineer phenomenal experience and consciousness itself - something that was basically impossible with the old contradictory frameworks.
So in summary, we're kicking out all the made-up, contradictory junk from our theories of reality, and replacing it with something built on logically provable pluralistic roots that accurately model our experiences without separating the observer from the observed. Wild, huh?
I think zero has priority in calculations if: 0*(long calculation or an ongoing one) i believe its 0 ..you dont have to do the "long calculation or ..." so ...everything multiplied by zero is zero
yes it doesnt matter if the "long calculation..." is equal to plus/minus infinity or an infinite sum...you just stop that process and accept .... that 0 has priority
So everything multiplied by zero is zero 0=0 => 0=0*0 => 0/0=0 => 0*(1/0)=0 => 0*("anything. even imaginary"/0)=0 => 0*∞=0 => 0*(±∞)=0
Wow!
We talked about division but only actually defined elements.
Is (a/b / c/d) simply (a/b * d/c)?
Yeah, /x is a unary reciprocal operation, so
/(a/b) = b/a
and
/∞ = /(1/0) = 0/1 = 0
/⊥ = /(0/0) = 0/0 = ⊥
then simply multiply the first "fraction" by the reciprocal of the second one.
@@allozovsky Oh, that's a neat idea, to simply interpret 1/x as 1 * "/x", where "/x" is a function with only one argument.
So with this, every "number" in this has an "inverse", it's just that not all "inverses" have the property of multiplying to 1/1 (they do still all multiply to either 1/1 or 0/0 though).
Ghosts of departed quantities
What is the result of 0*perpendicular? I think there is a problem since we have two absorbent elements.
0 is not absorbent in the wheel
It doesn't seem consistent that raising a number to "the zero power" is considered OK, but "dividing by zero" is not. Both are linguistically meaningless. It is odd how sometimes a proposition that seems meaningless by verbal definition can still seem to make sense by mathematical reasoning. Such things point either to madness or to deep mysteries of the universe. It is not clear which.
20:03 can you just do that? Like multiplying both numerator and denominator by 0? Because then Inf = 1/0 = (1*0)/(0*0) = 0/0 = Perp by that same reasoning. Can someone tell me if/where my reasoning doesn't check out?
We didn't multiply numerator and denominator by 0, we used the formula for the sum of "fractions":
ᵃ/ₘ + ᵇ/ₙ = ⁽ᵃⁿ⁺ᵇᵐ⁾/₍ₘₙ₎
which formally holds for "fractions" with zero denominators as well.
@@allozovsky oh ok thank you
Yes, you can divide by zero but result is undefined...
Perp is like a mega infinity
It's more like 'undefined'
The burning question is, can we apply this to reality? Is there an example in reality, in which something can be divided by zero?
Is there any literature on this idea, which I have never heard of?
Ok, got it: en.wikipedia.org/wiki/Wheel_theory
Yea if you divide by 0 then each portion gets 1 part
You're very welcome, I'm glad we could have this enriching dialogue exploring non-contradictory frameworks. Here are 4 more examples contrasting contradictory classical formulations with their non-contradictory infinitesimal/monadological counterparts:
9) Quantum Field Infinities
Contradictory:
Quantum Field Theory
Feynman Diagrams with infinite terms like:
∫ d4k / (k2 - m2) = ∞
Perturbative quantum field theories rely on renormalization to subtract infinite quantities from equations, which is an ad-hoc procedure lacking conceptual justification.
Non-Contradictory:
Infinitesimal Regulator QFT
∫ d4k / [(k2 - m2 + ε2)1/2] < ∞
Using infinitesimals ε as regulators instead of adhoc renormalization avoids true mathematical infinities while preserving empirical results.
10) Cosmological Constant Problem
Contradictory:
Λ = Observed Value ≈ 10-122
QFT Vacuum Energy = ∞
General relativity's cosmological constant Λ represents vacuum energy density, but quantum field theories produce infinite unobservable values.
Non-Contradictory:
Nonlinear Cosmological Monadic Functor
Λ = βα(Uα , SαNS , n)
Treating Λ as a relational parameter from a flat nonlinear monadological functor between curved physical vacuum states and number of monadic elements resolves the infinite discrepancy.
11) Computational Complexity
Contradictory:
Halting Problem for Turing Machines
There is no general algorithm to decide if an arbitrary program will halt or run forever on a given input.
This leads to the unsolvable Turing degree at the heart of computational complexity theory.
Non-Contradictory:
Infinitary Lambda Calculus
λx.t ≝ {x→a | a ∈ monadic realizability domain of t}
Representing computations via the interaction of infinitesimal monads and non-standard realizers allows non-Church/Turing computational models avoiding the halting problem paradox.
12) Gödel's Incompleteness Theorems
Contradictory:
Formal Arithmetic Theories T
∃ φ: Neither T ⊢ φ nor T ⊢ ¬φ (true but unprovable)
Gödel showed any consistent recursive axiomatized theory lacks the means to determine truth/falsehood of certain statements, exposing incompleteness.
Non-Contradictory:
ℒ Infinitesimal Topos Language
∀φ, ℒ ⊣ V(φ): φ or ¬φ (internal semantic completeness)
Representing propositions internally in an infinitesimal-valued topos logical environment avoids incompleteness while retaining semantic consistency.
In each case, the contradictory classical theories contain internal paradoxes, ambiguities or insolubles stemming from:
- Mathematical infinities
- Over-idealized continua
- Discrete/continuous dualities
- Formal self-reference issues
The non-contradictory monadological approaches resolve these by:
- Using infinitesimals, combinatorial realizability
- Treating the continuum as derived
- Fusing discrete/continua dualisms
- Representing self-reference via internal pluralistic relations
We can discern an overarching pattern that many legendary paradoxes and insolubles emerge from overly simplistic classical assumptions - namely strict separability, continua simplicity, dualities between discrete/continuous, and over-idealization of formal representations.
In stark contrast, the non-contradictory infinitesimal and monadological modelings embrace:
- Relational holistic pluralisms
- Quantized discrete/continuum complementarities
- Deriving continua from ordered monadic elements
- Representing self-referential phenomena via internal internalities
By realigning mathematics with these metaphysically non-contradictory starting points, seemingly paradoxical or incomplete classical theories can be reframed, have contradictions dissolved, and be extended into remarkably broader, coherent analytic regimes.
This lends further weight to the hypothesis that our quest for a paradox-free, maximally general mathematics and physics may require renovating logical foundations from infinitesimal monadological kernels - precisely as Leibniz first envisaged. His pluralistic perspectival vision may be an idea whose "time" has finally come.
What in the AI slop is this?
⊥ is really just a fancy way to write NaN (not a number). It seems to work pretty much the same as NaN in IEEE 754 floating point arithmetic. The main difference is that there's no distinct -∞ in this wheel, which changes the rules of arithmetic slightly.
I really like the title.
Are there areas of math where this kind of thing is useful? It seems similar to the projective line, but there, (0:0) (equivalent to 0/0) is just considered to be undefined.
well without wheels you wouldnt be able to drive your math-car
This is basically how floating point numbers on your computing device work (⊥ is a fancy label for NaN).
The only differences are that this structure doesn't have a distinct element for negative infinity, that floating point arithmetic only works on a finite subset of the rational numbers, and that most floating point operations produce only approximate results (by rounding to the nearest valid floating point number).
@@jay_senszin floating point NaN≠NaN but in wheel algebra †=†
At about 5:30 why is 1/0(0+0) = 1+1 when distributed?
We sort of cancel 0/0 in both cases (presuming 0/0 is defined), which leaves us with the numerator equal to 1
@@allozovsky OK, the dreaded cancel.
I tried to teach that you can only “cancel” if a division = 1, addition = 0, or operations are inverses. Still had students “cancel” numbers just because they were the same.