Actually, you CAN divide by zero.
254,164
Published 2023-11-03
You've probably heard that you "can't" divide by zero, but why not? As it turns out, adding in the inverse of a number is a well-defined process in math, similar to how you can add in the solution to x^2 = -1. The result is a new number system. In this video, we find out what happens when you apply this process to add division by zero. The result is a pleasant mix of surprising and completely expected.
Notes:
1. Normal rules of algebra means a ring.
2. Topology is important too, but algebra alone is enough for 1/0.
3. Cup would be more correct symbol for union than plus, but this is YouTube :).
4. We did not need to start with the reals, adding 1/0 in any ring results in the zero ring.
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Normal algebra rules: en.wikipedia.org/wiki/Ring_(mathematics)#Definitio…
Localization: en.wikipedia.org/wiki/Localization_(commutative_al…)
Dyadic rationals: en.wikipedia.org/wiki/Dyadic_rational
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CHAPTERS
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0:00 Intro
1:00 Localization
2:00 Zero invers
All Comments (21)
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Great video. I rate it zero of zero. Way to go!
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Saying "you can't do XYZ" in maths is really just a shorthand for saying "The systems of maths that arrises by expanding an existing one to include XYZ is not interesting / useful / non-trivial / connected to other branch of maths." This is probably obvious to anyone who has studied higher maths and is familiar with the idea of there being many different systems of maths (different number systems, different starting axioms, etc...) that we can choose between at will; but far more alien to those who haven't gone beyond high school maths and think of it as a single, rigid, god given, singular thing.
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After lots of hour I finally implemented a fully working calculator for the zero ring: def add_or_mul_or_div_or_sub(a, b): return 0 It was hard work but will be worth it for future calculations
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Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field.
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"If you divide by zero, all numbers are zero". That's a cruel punishment
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I was not expecting pure math from this channel, but I probably should've given that I learned about semigroups from a one-off comment in one of your Python videos. This is awesome.
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"It's not that you can't divide by zero, it just doesn't do anything useful to define" is what I gathered.
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Given the typical content of this channel, i was assuming the set of numbers we would arrive at would be blackboard F, for floating point as specified IEEE 754
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In a commutative algebra lecture, the professor gave the important proposition that the localisation at a (multiplicative) set is 0 if and only if the set contains 0 a very fitting name: If you divide by zero, everyone dies (when something becomes zero, people often call that „killing the element“).
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You can't divide by 0 until you invent a rule that you can.
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The main difference here is that including a square root of -1 is a field extension of R. In fact, it is a very special field extension. It is the splitting field of R (in many ways, it is better than R). But the ignoring that, the important thing is that C has R embedded in it; the natural homomorphism from R to C is injective, or in other words, the kernel is trivial. This means R is isomorphic to a subring (subfield) of C, so this extension doesn't lose you any of R. On the other hand, if you include 1/0, the new ring no longer has R embedded in it - it is not an extension of R. The natural (and only) homomorphism from R into the zero ring is as far from injective as it could be - the kernel is the entire set R. So there is nothing that looks like R inside the zero ring. This should be pretty obvious given that R is uncountable, while the zero ring only has 1 element.
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I thought you were going to talk about the projective real number line, which has an inverse of 0, so division is defined on everything but now addition/subtraction isn't.
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i guess the (number)universe does collapse if you try to divide by zero
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1:44 you say "if we also throw in[] inverses of every positive whole number" but that's somewhat redundant, right? Wouldn't it suffice to just use inverses of primes?
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2:22 is this really a true deduction? We just defined that 0* 1/0 = 1 so clearly NOT everything multiplied by 0 is 0 anymore.
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Great video, congratulations! Making these theoretical details of math visible to the regular user/student is a valuable way to promote math studying.
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I was expecting this to be a video on IEEE floating points, but this is interesting in its own right.
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eh, just make it a wheel. You get zero, you get infinity, you get any symbol [x, 0], and you get a special element [0,0] (where for any regular value [a, b], to translate it into the real numbers, is just a/b, though some values such as [x,0] can't be translated)
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Dude! I loved this. Can you tell us what you used to create these animations and share the source code for these as well?
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you can also tweak the rules slightly to create a useful system, like what was done with floats; 1 / 0 = infinity. 1/-0 = -infinity. 0 * infinity = NaN, NaN with most operators just produces NaN.
