Most definitions of division that I have seen use q * d + r = n if q is unique and abs(r)Additionally, if inverses are defined as separate objects then what is 2 plus the inverse of 2? It doesn't simplify to 2.5 because there's no addition axiom for numbers and multiplicative inverses, or for that matter any rules for inverses with inverses. So you might have 1/2 and 5/10 but they're not equal and can't be multiplied together.
1/0 = 0 (2018)
61–70 of 245 posts
Re: 1/0 = 0 (2018)
#62This article invents a new binary operation, calls it "division" and uses the "/" operator to denote it. But the article repeats multiple times that this new operation isn't a multiplicative inverse, so it's not actually division. For example, (a/b)*b=a isn't true for this new operation.
Under what definition of division is (a/b)*b=a true for all values?
Re: 1/0 = 0 (2018)
#63I've always wondered what would happen if we defined /0 as a new symbol, for example 'z'. The same as we define sqrt(-1) as 'i'. So if you can do 4*sqrt(-1)=4i, you could also do 4/0 = 4z. These two seems similar, as in taking something that should not exist, and just letting it exists in a totally different and orthogonal domain. I tried once to investigate the implications, but it quickly became far more complex th…
1,000,000 grains of sand is a heap of sand (Premise 1)
A heap of sand minus one grain is still a heap. (Premise 2)
- https://en.wikipedia.org/wiki/Sorites_paradox
So one grain of sand is a heap and then when you remove that grain the heap disappears, but you only removed one grain from a heap so this is impossible because it is discontinuous. One solution is to wrap the problem in fuzzy logic with a 'heapness' measure.
Generalizing this type of solution we have a practice of wrapping paradoxes in other forms of logic. You would define an interface between these logics. For example in Trits (0,1,UNKNOWN) you could define an interface where you can change the type of NOT-UNKNOWN from Trit to Boolean. This would return at least some part of the entropy to the original domain, preserving a continuity. Wave Function Collapse is another example of translating from one logical domain to another.
Re: 1/0 = 0 (2018)
#64I've always wondered what would happen if we defined /0 as a new symbol, for example 'z'. The same as we define sqrt(-1) as 'i'. So if you can do 4*sqrt(-1)=4i, you could also do 4/0 = 4z. These two seems similar, as in taking something that should not exist, and just letting it exists in a totally different and orthogonal domain. I tried once to investigate the implications, but it quickly became far more complex th…
In SQL, if you divide by zero, you get a NULL. If you divide by NULL, you get NULL (any operation involving a NULL yields NULL, even GROUP BY). I call it "a black hole zero", if it touches anything, that thing becomes a black hole zero. Some languages will wrap division by zero in a special type, a NaN (not a number). You can then reason on top if that NaN if you want to. So, in a sense, there are some people already…
Does it? Under SQL standard or what engine? At least Postgres raises when dividing by zero.
Re: 1/0 = 0 (2018)
#65Earlier quoted context omitted.
In SQL, if you divide by zero, you get a NULL. If you divide by NULL, you get NULL (any operation involving a NULL yields NULL, even GROUP BY). I call it "a black hole zero", if it touches anything, that thing becomes a black hole zero. Some languages will wrap division by zero in a special type, a NaN (not a number). You can then reason on top if that NaN if you want to. So, in a sense, there are some people already…
Does it? Under SQL standard or what engine? At least Postgres raises when dividing by zero.
You're right about division by zero. From SQL92:
4) The dyadic arithmetic operators , , , and (+, -, *, and /, respectively) specify
addition, subtraction, multiplication, and division, respec-
tively. If the value of a divisor is zero, then an exception
condition is raised: data exception-division by zero.
1) If the value of any simply contained in a
is the null value, then the result of
the is the null value.
So, dividing by NULL is allowed and yields NULL. Dividing by zero yielding NULL is non-standard (I used it though).
Re: 1/0 = 0 (2018)
#66Wouldn’t the logical value when dividing by zero be infinity, because zero can go into any number an infinite number of times?
Re: 1/0 = 0 (2018)
#67Earlier quoted context omitted.
That's nonsense. a/b is float in Python 3, and even in other languages a/b gets closer to it's actual value as a and b get bigger (the "limit", which is the basis of Algebra). So four operations in programming generally do agree with foundations of Algebra. But a/0=0 is %100 against Algebra. And it's very unintuitive. It's basically saying zero is the same as infinity, and therefore all numbers are the same, so why b…
Floats don't have multiplicative inverses, and the floating point operations don't give us any of the mathematical structures we expect of numbers. Floating point division already abandons algebra for the sake of usefulness.
Knuth vol 2 has a nice discussion of floating point operations and shows how to reason about them. Wilkinson's classic "Rounding Errors in Algebraic Processes" (1966) also has a good discussion.
Re: 1/0 = 0 (2018)
#68This article invents a new binary operation, calls it "division" and uses the "/" operator to denote it. But the article repeats multiple times that this new operation isn't a multiplicative inverse, so it's not actually division. For example, (a/b)*b=a isn't true for this new operation.
Under what definition of division is (a/b)*b=a true for all values?
If 0 is not an allowable value for b is necessary but not generally sufficient.
Re: 1/0 = 0 (2018)
#69Wouldn’t the logical value when dividing by zero be infinity, because zero can go into any number an infinite number of times?
Saying 1/0=∞ means creating a new number system with ∞ as a number. Now you have to figure out all operations with ∞, like -1*∞, 0*∞, ∞*∞, ∞/∞, or ∞-∞.
Making wrong definitions creates contradictions. With 1*x=x, ∞/∞=1, the associative property x*(y/z)=(x*y)/z, and ∞*∞=∞:
∞ = ∞*1 = ∞*(∞/∞) = (∞*∞)/∞ = ∞/∞ = 1
Re: 1/0 = 0 (2018)
#70> It’s saying that Pony is mathematically wrong. This is objectively false. Pff. The author wants to show off their knowledge of fields by defining a "division" operator where 1/0 = 0. Absolutely fine. I could define "addition" where 1 + 2 = 7. Totally fine. What I can't do is write a programming language where I use the universally recognised "+" symbols for this operation, call it "addition" and claim that it's tot…
It's a question of usefulness. If in your problem domain "1+2=7" is the most useful definition, then by all means do that. Why does the semicolon terminate statements and not the universally agreed upon period? Why does the period denote member access? Why is multiplication not denoted by the universally agreed [middle dot / cross character] (strike out the one that is not universally agreed in your country). The design and semantics of a programming language ought to be in service of the programs we wish to express, and informed by our decades of experience in human ergonomics. Blind reverence to religions of yore does us no good. Mathematical notation itself has gone through centuries of development and is not universal, with papers within the same field using different notation depending on what strikes the author's fancy. To treat it as sacred and immutable is to behave most un-mathematically. Hell, you can still get into a nice hours-long argument about whether or not the set of natural numbers includes zero or not (neither side will accept defeat, even though there is clearly a right answer)!