The problem is that infinity is neither a real nor a complex number, nor an element of any algebraic field, and the proposition that "x * 0 = 0" only holds if x is an element of some algebraic field. It is a theorem that depends on the field axioms.
The real numbers can be extended to include two special elements ∞ and -∞, but this extension does not constitute a field, and the range of expressions in which these symbols make sense is very strictly and narrowly defined (see Rudin's PMA, Definition 1.23):
(a) If x is real then
x + ∞ = +∞, x - ∞ = -∞, x / +∞ = x / -∞ = 0.
(b) If x > 0 then x * (+∞) = +∞, x * (-∞) = -∞.
(c) If x < 0 then x * (+∞) = -∞, x * (-∞) = +∞.
The extended real number system is most commonly used when dealing with limits of sequences, where you may also see such symbols appear:
3.15 Definition Let {sₙ} be a sequence of real numbers with the following property: For every real M there is an integer N such that n ≥ N implies sₙ ≥ M. We then write
sₙ ⟶ +∞.
In no other contexts do the symbols ∞ and -∞ make any sense. They only make sense according to the definitions given.
It's usually the case that when you see people discussing infinity that they are actually talking about sequences of numbers that are unbounded above (or below). The expression "sₙ ⟶ +∞" is meant to denote such a sequence, and the definitions that extend the real number line (as in Definition 1.23 above) are used to do some higher-level algebra on limits of sums and products of sequences (e.g. the limit of sₙ + tₙ as n becomes "very large" for two sequences {sₙ}, {tₙ}) to shortcut around the lower-level formalisms of epsilons and neighborhoods of limit points in some metric space, which is how the limits of sequences are rigorously defined.
In no case do the symbols ∞ and -∞ refer to actual numbers. They are used in expressions that refer to properties of certain sequences once you look far enough down the sequence, past its first, second, hundredth, umpteenth, "Nth" terms, and so on.
Thus when you see people informally and loosely use expressions such as "infinity times zero" they're not actually multiplying two numbers together, but rather talking about the behavior of the product of two sequences as you evaluate terms further down both sequences; one of which is unbounded, while the other can be brought arbitrarily close to (but not necessarily equal to) zero. You will notice that no conclusions can be drawn regarding the behavior of such a product in general, whether referencing the definitions comprising the extended real number system or the lower-level definitions in terms of epsilons and neighborhoods of limit points.
So much confusion today comes down to people confidently using words, symbols, and signs they don't understand the definitions nor meanings of. Sometimes I wonder if this is the real esoteric meaning of the ancient Tower of Babel mythos.
Infinity doesn't need to be in some "algebraic field" for it to be patently true that an infinite amount of nothing is still nothing, and that adding zero to itself over and over again for an infinitely long time will never give you a result other than zero. It's only impossible to define if you overthink it, and/or maintain a needlessly narrow definition of what a "number" is.
Or, if you really insist on speaking in mathematician-ese, an infinite series of zero is zero, and a zero-bounded summation is zero regardless of the summand:
x · y ≡ Σ(y, i = 1) x = y times { x + x + … + x } ≡ Σ(x,i=1) y = x times { y + y + … + y }
julia> i = 0
0
julia> while true
println(i)
global i += 0
end
0
0
0
0
0
0
0
0
0
...and on and on until the heat death of the universe or you hit Ctrl-C.
Either way, seems pretty straightforward to define if you have a clear definition of what multiplication is in the first place (and what either zero or infinite iterations of that definition will produce).
Ok, let's assume you are correct and that ∞ · 0 = 0. Consider then the two sequences sₙ = n, tₙ = 1/n.
By Definition 3.15 as provided in my last post, sₙ ⟶ +∞, and you will have to take it for granted that tₙ ⟶ 0 [0]. Intuitively we can see that the terms of {sₙ} are 1, 2, 3, ... tending to +∞; for {tₙ} we have 1, 1/2, 1/3, ... tending to zero, for progressively larger values of n.
Now I ask what happens if we multiply the "infinite'th" terms of both sequences together. The first few terms of this product would be 1 · 1, 2 · 1/2, 3 · 1/3, and so on; I ask what the value x is in the limit sₙ · tₙ ⟶ x as we evaluate further and further "nth" terms of both sequences.
You may have observed from the first three terms evaluated that sₙ · tₙ = n(1/n) = 1. Thus, as we continue to increase the value of n, it's always the case that sₙ · tₙ ⟶ 1 and the product tends to 1, because the product is constant and irrespective of n; we've "cancelled it out."
The limit of the product is the product of the limits [1]; that is, sₙ · tₙ ⟶ +∞ · 0, as we first established that sₙ ⟶ +∞, tₙ ⟶ 0.
If we thus take your supposition that +∞ · 0 = 0 for granted, we obtain sₙ · tₙ ⟶ 0, which contradicts our previous result that sₙ · tₙ ⟶ 1.
Thus we can either dispense with the cited established theorems of analysis used to deduce that sₙ · tₙ ⟶ 1, or conclude that the supposition +∞ · 0 = 0 must be false.
It might be the case that Σ(∞, i = 1) 0 = 0, but you can't extend this to conclude +∞ · 0 = 0 in general. Lots of intuitions from informal mathematics and even calculus start to break down once you examine the lower-level "machine code" of proof and analysis, especially once you start talking about concepts like infinity.
> Now I ask what happens if we multiply the "infinite'th" terms of both sequences together.
In that case, you would've reached their respective limits, and you're back to adding one of those limits into itself an other-limit number of times. If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.
> Thus we can either dispense with the cited established theorems of analysis used to deduce that sₙ · tₙ ⟶ 1
You don't need to do that. You just need to accept that zero is just as much of a mathematical special case as infinity - unsurprisingly, since it's the inverse of infinity and vice versa.
> It might be the case that Σ(∞, i = 1) 0 = 0, but you can't extend this to conclude +∞ · 0 = 0 in general.
Sure you can, unless you've got some other definition of multiplication that's impossible to express as self-summation.
Even if you go with the alternative definition of multiplication as a scaling operation (wherein you're computing m × n by taking the slope from (x=0,y=0) to (x=1,y=m) and then looking up y where x=n), if m is zero then the line being drawn never stops being vertical, and if n is zero then you never leave (0,0) in the first place. Doesn't matter if the other factor is infinitely far in either the x or y axis; you're still ending up with zero no matter how hard you try and fight it.
> Lots of intuitions from informal mathematics and even calculus start to break down once you examine the lower-level "machine code" of proof and analysis, especially once you start talking about concepts like infinity.
Sure, but in this case, it's the intuition that multiplying something by its inverse (a.k.a. dividing something by itself) is always 1 that breaks down, not the above-verifiable and inescapable fact that multiplying something by zero is always zero. 0 ÷ 0 = n looks like it should correct for any value of n (incl. n = 1), since multiplying both sides by zero to eliminate that divide-by-zero will always produce a correct equation, but since m ÷ n ≡ m × (1/n), if m is zero then anything on the RHS must be zero, because of that inescapable nature of nothingness - thus, 0 ÷ 0 = 0 × (1/0) = 0, with all other possible alternatives having been rendered impossible.
> you're back to adding one of those limits into itself an other-limit number of times.
> some other definition of multiplication that's impossible to express as self-summation.
Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
> If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.
I mean... it is in fact the case that tₙ never actually reaches zero; otherwise, if 1/n = 0 for some n, then by multiplying both sides by n we obtain 1 = 0.
What's meant by tₙ ⟶ 0 is that any neighborhood centered about 0 of any radius (call the radius "epsilon") always contains at least one point from the sequence {tₙ}.
To hammer the point that sₙ · tₙ ⟶ 1 home, and since you are fond of using a computer to perform arithmetic (note: not prove mathematical statements), here's what computers have to say about the limit of n · (1/n): https://www.wolframalpha.com/input?i=limit+as+n-%3Einfinity+...
> You just need to accept that zero is just as much of a mathematical special case as infinity - unsurprisingly, since it's the inverse of infinity and vice versa.
> the inverse of infinity
You again throw around words like "inverse" whose meaning you don't understand. Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1? Or an additive inverse that yields the additive identity, in which case +∞ + 0 = 0? Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
> if m is zero then the line being drawn never stops being vertical
Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.
> Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
You add it to itself 3 times, then shift the decimal point and repeat with 1, then shift the decimal point and repeat with 4, and so on with each digit of π. 1 × π = 1 + 1 + 1 + 0.1 + 0.01 + 0.01 + 0.01 + 0.01 + 0.001 + 0.0001 + 0.0001 + 0.0001 + 0.0001 + 0.0001 and so on forever.
> To hammer the point that sₙ · tₙ ⟶ 1 home
That point doesn't need hammered. sₙ · tₙ ⟶ 1 can absolutely be true when you haven't yet reached zero. That doesn't mean it's true in the event that you do indeed manage to reach zero. It indeed can't be true in the event that you do indeed reach zero, because n × 0 = 0 for all values of n.
> Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1?
You obviously already know that's what I meant, since that's exactly what I described further down - including how ∞ × 0 ≠ 1 because the multiplicative identity breaks down when one of the factors is zero, specifically because having zero of something will always produce zero no matter what that something is.
> Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
If you're seriously calling multiplication-as-summation pseudomathematics, then you're in no position to assess whether or not I "don't understand" the meanings of words.
I've been nothing but civil toward you, and you've been nothing but condescending toward me. That normally wouldn't be a problem (condescension is par for the course on the Internet), but if you're going to be condescending, the least you can do is not be blatantly wrong in the process.
> Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
That's exactly what I'm presenting here (since apparently you believe adding numbers together is a spook). You don't even need a concept of numbers to see plain as day that any multiplication wherein one of the factors is zero will always be zero.
> Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.
You know what? Maybe I will. And I'm willing to bet you'll find some other pedantic reason to be a condescending prick when that happens.
Last word's yours if you want it. I have better things to do than argue with people engaging in bad faith.
There's no need for me to continue engaging you with formal mathematical arguments when you reply with the mathematical equivalent of climate change denialism or vaccine conspiracy theory and uneducated statements that are "not even wrong" [0], so instead I will just refer you to expert opinions on the topic; though at this point I doubt that your level of mathematical literacy is sufficient to understand any of this subject matter.
The real numbers can be extended to include two special elements ∞ and -∞, but this extension does not constitute a field, and the range of expressions in which these symbols make sense is very strictly and narrowly defined (see Rudin's PMA, Definition 1.23):
The extended real number system is most commonly used when dealing with limits of sequences, where you may also see such symbols appear: In no other contexts do the symbols ∞ and -∞ make any sense. They only make sense according to the definitions given.It's usually the case that when you see people discussing infinity that they are actually talking about sequences of numbers that are unbounded above (or below). The expression "sₙ ⟶ +∞" is meant to denote such a sequence, and the definitions that extend the real number line (as in Definition 1.23 above) are used to do some higher-level algebra on limits of sums and products of sequences (e.g. the limit of sₙ + tₙ as n becomes "very large" for two sequences {sₙ}, {tₙ}) to shortcut around the lower-level formalisms of epsilons and neighborhoods of limit points in some metric space, which is how the limits of sequences are rigorously defined.
In no case do the symbols ∞ and -∞ refer to actual numbers. They are used in expressions that refer to properties of certain sequences once you look far enough down the sequence, past its first, second, hundredth, umpteenth, "Nth" terms, and so on.
Thus when you see people informally and loosely use expressions such as "infinity times zero" they're not actually multiplying two numbers together, but rather talking about the behavior of the product of two sequences as you evaluate terms further down both sequences; one of which is unbounded, while the other can be brought arbitrarily close to (but not necessarily equal to) zero. You will notice that no conclusions can be drawn regarding the behavior of such a product in general, whether referencing the definitions comprising the extended real number system or the lower-level definitions in terms of epsilons and neighborhoods of limit points.
So much confusion today comes down to people confidently using words, symbols, and signs they don't understand the definitions nor meanings of. Sometimes I wonder if this is the real esoteric meaning of the ancient Tower of Babel mythos.