Comments by "MusicalRaichu" (@MusicalRaichu) on "Mathologer"
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@adomaster123 That's totally correct, but you're making assumptions: you're assuming conventional arithmetic. Assigning a value to a divergent series is not conventional, it's something new and arbitrary. Look, even within the constraints of conventional arithmetic, 1+2+3+... is not a positive integer either! It's plain meaningless.
If for some reason we want to give 1+2+3+... a meaning, we can do it any way we like. The only constraints on the definition we invent is that when the definition applies to previously defined series, it still yields the same result as before. For example, with our new definition of infinite series, 0.5+0.25+0.125+... must still be assigned the value 1.
One way is to give 1+2+3+... a meaning is to say that it's infinite. So long as you explain what infinite means and how it relates to the rest of arithmetic, it's a perfectly fine approach. If you like, you can even distinguish positive and negative infinity and that's OK too so long as it's consistent with everything else (like with complex numbers, I don't think we can distinguish positive and negative infinity.)
Ramanujan's approach is different. He sought to define non-convergent series in a way that they have a finite value. We're free to do that. So long as our definition of the value of a series still gives the same result for convergent series, then it's free to give any value at all for divergent series, even if you end up with a negative number for some series.
The whole point of this video is that 1+2+3+... does not "equal" anything in conventional mathematics, neither positive nor negative. Normal arithmetic breaks down because the series diverges. The video describes an approach that someone could take if they want to somehow "assign" a finite value to divergent series.
What I'm trying to say is that when you're thinking using conventional arithmetic, positive numbers add to give positive results. But the assumption is that you're adding a finite number of terms. Even when you extend your arithmetic to deal with convergent infinite series, sure, positive result. But in that extension, you've lost something: rationals can suddenly add to give an irrational! Believe me, I was horrified when I realized that. Can you imagine the shock a teenage boy suffered when he realized that the definition of infinite series violated the closure property of rationals? It's more than the astonishment I felt as an adult when I heard someone invented a way to add positives and get a negative.
When you're inventing completely new mathematics, all bets are off. The old properties of numbers don't have to hold in new situations, so long as your new definitions still maintain familiar properties in old situations.
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@adomaster123 What I'm trying to say is that you're right using conventional definitions of addition and convergence. However, if you extend the definition so that things that were undefined before now have some kind of meaning, then (as the video states) you can lose some properties.
If you extend the definition of addition to include summing an infinite number of terms (still conventional maths), then some closure properties of addition can break, e.g., adding an infinite number of rational numbers can give you an irrational number.
In the same way, when addition is extended to include adding non-convergent series (like 1, -1, 1, -1, ...), the definition you use to assign a value to such a series might break some existing properties, e.g., adding integers can give you a non-integer.
When you come up with some sort of definition that assigns a meaning to a divergent series like 1+2+3+..., then so long as your new definition works the same for existing maths, you can do what you like for situations that were previously undefined. So so long as your new definition makes sure convergent positive series add up to a positive number, you're fine, but you're not limited to yielding a positive result for a divergent series.
Historically maths has always gone forward by defining what was undefined before. That's how negative numbers, rational numbers, etc. came about. Subtracting 10 from 4 makes no sense ... until you give it sense. Likewise, taking the square root of -2 is meaningless ... until you give it a meaning. Likewise 1+2+3+... is nonsense ... until someone with too much time on their hands gets hold of it. In each case, the result is counter-intuitive, but so long as it's consistent, people can work with it.
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@ozaman-buzaman9300 Your confusing numbers with representations of numbers.
Consider these numbers expressed as the number of o's and the way we represent them using conventional writing:
oooo 4
oooooooo 8
oooooooooooooooo 16
oooooooooooooooooooooooooooooooo 32
That's called a base 10 representation. I could just as easily write them in another base, e.g., base 8
oooo 4
oooooooo 10
oooooooooooooooo 20
oooooooooooooooooooooooooooooooo 40
or base 9
oooo 4
oooooooo 8
oooooooooooooooo 17
oooooooooooooooooooooooooooooooo 35
In mathematics, there is nothing special about base 10 representation. A number exists independently of how you write it. The "oooo..." notation is closer to the real meaning of the number than any representation involving digits.
If you want to posit a largest allowable integer, that's fine, but there is no reason why it would have to end in 9 or anything else because the use of 9 as the last number before you click over to 0 is an artifact of the choice of base. If people used base 8, the biggest number would be 7 and you would be asking about 7 instead of 9.
If you would like to create an arithmetic where the set of allowable numbers is {0, 1, ..., 9}, that's fine to do, except that that system of arithmetic would not have a means of representing the size of the set. Even if you said it was {0, 1, ... 10}, then again the size of the set is 11 which is not a member of the set.
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Definitely there is an association between 9.9 repeater and 1, but I think the problem is that you are calling that association "equals".
In reality, there is no such thing as 9.99... because it involves an infinite number of operations. Your explanation involved adding an infinite number of fractions and also doing an infinite number of multiplications, which is not realizable. In mathematics, what we are doing is arbitrarily defining infinite series to make sense (when they converge).
What the relationship is between 9.9 repeater and 1 is that 1 is the limiting value of the decimal as the number of decimal places gets arbitrarily large. What you really should say is that we can make 9.9 repeater as close to 1 as we like by including more digits in the sum of fractions 9/10, 99/100, ... Saying that the inifinte sum of fractions actually EQUALS 1 is a mere convenience that, under certain assumptions that we make in mathematics, forms a reasonably useful and consistent body of theory.
That body of theory gives results that agree with the real world (for example, Achilles overtakes the tortoise), but we need to be careful because it also produces results that do not (for example, dividing a sphere along fractal boundaries and reforming the pieces into two spheres).
For this reason, I think the use of the term "equal" in your title needs to be taken with a grain of salt ...
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