Ultimately, the value considered correct and accurate is the one in the middle of 1 and 0 (the arithmetic mean).
It occurred to me the same logic applies to powers of i…the imaginary unit equal to the square root of -1.
i=i
i^2=-1
i^3=-i
i^4=1
i^5=i
i^6=-1
i^7=-i
i^8=1
.
.
.
The mean…average…of 1 and -1 is zero. I assume…but am not certain…that the average between i and -i is also zero (imaginary numbers are not “ordered”…that is you can’t say one imaginary number is “bigger” or “smaller” than another.)
That’s not quite what he says, and there’s a bit of a leap in logic at the end of your OP. But one step at a time.
What he is saying is that there’s a way of defining the sum of an infinite series where 1/2 makes sense in this case.
And note the way that is phrased: ‘defining the sum of an infinite series’. Turns out how you define that is important, as tends to be the case with pretty much all things in mathematics.
In the conventional way we tend to define infinite series, 1+1-1+1-1+1-1+… does not converge - it does not have a proper value with our conventional definition. But there’s a different way of summing (he mentioned Grandi’s series) where you could assign a value to an infinite series that would diverge with our conventional definition. And in this case, that is 1/2.
It doesn’t mean the “true” value is 1/2. It just provides a different way of looking at the series that provides a somewhat meaningful value when it does not have a conventional convergent value.
So, now onto powers of sqrt(-1). What you are doing is very different. There’s no longer a sum. Instead, you are examining i^n.
If you examined i - i + i + i + -i, then something similar would pop out. Or even 1-i-1+i+1-i-1+i+1+… But that’s not what you are doing.
ETA: a lot of stuff at the end there is a generalization of Grandi’s series - check Ramanujan summation or Cesaro summation, which is what he was basically describing. Lots of good stuff on assigning values to divergent series. But not really applicable to i^inf in that way.
Under the standard axioms most typically used for such things, neither 1+1-1+1… nor i^infinity is defined.
Other axioms can be chosen, under which either or both of those expressions is defined. What value they have then depends on the axioms. One can come up with axioms under which they have any value at all. But what you really want is a set of axioms which defines those nonstandard expressions, but which is also consistent with the standard axioms, and gives the same value for all of the expressions which were defined under the standard axioms.
I would have said that, under the standard definition most typically used for such things, 1-1+1-1+… is defined to be the limit, as N approaches infinity, of \sum^N_{k=0} (-1)^k; but this limit does not exist. (But, as noted, there are other ways to define it so that it does have an existing value.)
As others have said Grandi’s convention only applies to the sum of an infinite series numbers not the value of the the “last” value.
The sum from 1 to n of i^n for increasing values of n is:
i, (i-1)/2, -1/3, 0/4, i/5, (i-1)/6 …
which tends to 0 as k tends to infinity which under Grandi’s convention would defind the sum of i^n as n increases from 1 to infinity as 0.
A similar convention of the “last” value could well be average of all the values of greater than N as N approaches infinity which would indeed produce a possible solution to i^infinity as 0. I do not know if any mathematician (before you) has proposed this convention.
To consider a similar problem without complex numbers this convention would mean the value of sin(infinity) could be considered to equal 0. So if i^infinity = 0 so does sin(infinity)
The powers of i all lie in the circle of radius 1 centered at the origin in the complex plane. For example i^{1/2}=1/\sqrt{2}+(1/\sqrt{2})i. Although of course -1/\sqrt{2}-(1/\sqrt{2})i is another one.
Yes, of course, but what is the phase of infinity? If “i^infinity” is to have any meaning at all, it would have equal claim to any point on the unit circle. It doesn’t seem unreasonable to suppose that that would make the actual value the average of all points on the unit circle, though I don’t know if there’s a consistent axiom that would make that rigorous.
This might be a hijack so mods let me know and I will start a new thread…
What I do not get from the video is why a mathematician is ok putting parenthesis wherever they want. Why not say you (general “you”) cannot do that? Start at the front and take it as written.
For standard addition with a finite number of terms, there’s no reason you cannot put parentheses wherever you want. Caveat: treat minus sign as indicating addition of negative numbers, though.
So, for 1-1+1-1 = (1-1) + (1-1) = 1 + (-1+1) - 1 = 0. All the same. And we’d rather that always be true when we have a finite number of terms.
But with an infinite number of terms, what does that even mean anymore?
That’s one reason even the definition of the sum of an infinite sequence of numbers needs to be handled very carefully. Define it haphazardly and suddenly we get a bunch of nonsense. Worse, we get nonsense that doesn’t give us any interesting insights. Mathematicians study interesting nonsense all the time, after all.
Precisely - that is why he did NOT actually say the sum was 1/2.
If you get different answers by looking at it two different ways, there’s definitely something hinky going on.
If you add an infinite set of terms together, we have a definition that calls it “convergent” if, basically and somewhat oversimplifying, we can come up with an answer most people would agree to.
Example, 1/2 + 1/4 + 1/8 + 1/16 + … = 1. And that would make sense to a lot of people. It wouldn’t even matter if you put parentheses everywhere. It still comes out the same however you group terms together.
If it is not “convergent”, we consider it “divergent” or that it diverges. Or (oversimplifying a smidge again) there’s not a single, good value we can assign to it.
Often times, that’s because it blows up. For example, 1 + 10 + 100 + 1000 + … just blows up. It’s divergent. We can’t assign a conventional number that works (note: infinity is not a conventional number). Other times, it’s because there’s no way to assign a value that makes sense in every situation.
That is the case for 1-1+1-1+1-1+… It is a divergent series. There’s no one, good value to assign to it that makes conventional sense.
But it is the case that at least for some divergent series, we can pull some nonsense that makes us think about summation in interesting ways. And that’s what happens here. In some sense, we can assign a value of 1/2 to it by using a method that gives us interesting insights into the process of summation.
It doesn’t mean it is really 1/2 but that there’s an interesting way of looking at it that results in 1/2.
