A Numberphile video clip posted earlier this month claims that the sum of all the positive integers is -1/12. Is it true?
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A Numberphile video posted earlier this month claims that the sum of all the positive integers is -1/12.
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I"m usually a tín đồ of the Numberphile crew, who vày a great job making mathematics exciting and accessible, but this đoạn clip disappointed me. There is a meaningful way lớn associate the number -1/12 lớn the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series. Furthermore, the way it is presented contributes to lớn a misconception I often come across as a math educator that mathematicians are arbitrarily changing the rules for no apparent reason, and students have no hope of knowing what is và isn"t allowed in a given situation. In a post about this video, physicist Dr. Skyskull says, "a depressingly large portion of the population automatically assumes that mathematics is some nonintuitive, bizarre wizardry that only the super-intelligent can possibly fathom. Showing such a crazy result without qualification only reinforces that view, and in my opinion does a disservice lớn mathematics."
Addition is a binary operation. You put in two numbers, và you get out one number. But you can extend it to more numbers. If you have, for example, three numbers you want to showroom together, you can địa chỉ cửa hàng any two of them first & then showroom the third one khổng lồ the resulting sum. We can keep doing this for any finite number of addends (and the laws of arithmetic say that we will get the same answer no matter what order we địa chỉ them in), but when we try to showroom an infinite number of terms together, we have to lớn make a choice about what addition means. The most common way to deal with infinite addition is by using the concept of a limit.
Roughly speaking, we say that the sum of an infinite series is a number L if, as we địa chỉ cửa hàng more và more terms, we get closer and closer khổng lồ the number L. If L is finite, we gọi the series convergent. One example of a convergent series is 1/2+1/4+1/8+1/16…. This series converges khổng lồ the number 1. It"s pretty easy to lớn see why: after the first term, we"re halfway to 1. After the second term, we"re half of the remaining distance khổng lồ 1, và so on.
Zeno"s paradox says that we"ll never actually get lớn 1, but from a limit point of view, we can get as close as we want. That is the definition of "sum" that mathematicians usually mean when they talk about infinite series, và it basically agrees with our intuitive definition of the words "sum" and "equal."
But not every series is convergent in this sense (we điện thoại tư vấn non-convergent series divergent). Some, lượt thích 1-1+1-1…, might bounce around between different values as we keep adding more terms, & some, lượt thích 1+2+3+4... Might get arbitrarily large. It"s pretty clear, then, that using the limit definition of convergence for a series, the sum 1+2+3… does not converge. If I said, "I think the limit of this series is some finite number L," I could easily figure out how many terms to địa chỉ to get as far above the number L as I wanted.
There are meaningful ways to lớn associate the number -1/12 khổng lồ the series 1+2+3…, but I prefer not to gọi -1/12 the "sum" of the positive integers. One way khổng lồ tackle the problem is with the idea of analytic continuation in complex analysis.
Let"s say you have a function f(z) that is defined somewhere in the complex plane. We"ll gọi the tên miền where the function is defined U. You might figure out a way khổng lồ construct another function F(z) that is defined in a larger region such that f(z)=F(z) whenever z is in U. So the new function F(z) agrees with the original function f(z) everywhere f(z) is defined, & it"s defined at some points outside the tên miền of f(z). The function F(z) is called the analytic continuation of f(z). ("The" is the appropriate article to lớn use because the analytic continuation of a function is unique.)
Analytic continuation is useful because complex functions are often defined as infinite series involving the variable z. However, most infinite series only converge for some values of z, and it would be nice if we could get functions lớn be defined in more places. The analytic continuation of a function can define values for a function outside of the area where its infinite series definition converges. We can say 1+2+3...=-1/12 by retrofitting the analytic continuation of a function to its original infinite series definition, a move that should come with a Lucille Bluth-style wink.
The function in question is the Riemann zeta function, which is famous for its deep connections to lớn questions about the distribution of prime numbers. When the real part of s is greater than 1, the Riemann zeta function ζ(s) is defined lớn be Σ∞n=1n-s. (We usually use the letter z for the variable in a complex function. In this case, we use s in deference khổng lồ Riemann, who defined the zeta function in an 1859 paper
Analytic continuation is not the only way khổng lồ associate the number -1/12 to lớn the series 1+2+3.... For a very good, in-depth explanation of a way that doesn"t require complex analysis—complete with homework exercises—check out Terry Tao"s post on the subject.
The Numberphile đoạn clip bothered me because they had the opportunity to lớn talk about what it means lớn assign a value khổng lồ an infinite series and explain different ways of doing this. If you already know a little bit about the subject, you can watch the đoạn clip and a longer related đoạn clip about the topic và catch tidbits of what"s really going on. But the video"s "wow" factor comes from the fact that it makes no sense for a bunch of positive numbers to lớn sum up to a negative number if the audience assumes that "sum" means what they think it means.
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If the Numberphiles were more explicit about alternate ways of associating numbers to series, they could have done more than just make people think mathematicians are always changing the rules. At the kết thúc of the video, producer Brady Haran asks physicist Tony Padilla whether, if you kept adding integers forever on your calculator and hit the "equal" button at the end, you"d get -1/12. Padilla cheekily says, "You have to go to lớn infinity, Brady!" But the answer should have been "No!" Here, I think they missed an opportunity lớn clarify that they are using an alternate way of assigning a value khổng lồ an infinite series that would have made the video clip much less misleading.
Other people have written good stuff about the math in this video. After an overly credulous Slate blog post about it, Phil Plait wrote a much more levelheaded explanation of the different ways khổng lồ assign a value khổng lồ a series. If you"d lượt thích to work through the details of the "proof" on your own, John Baez has you covered. Blake Stacey and Dr. Skyskull write about how substituting the number -1/12 for the sum of the positive integers can be useful in physics. Richard Elwes posts an infinite series "health & safety warning" involving my old favorite, the harmonic series. I think that the proliferation of discussion about what this infinite series means is good, even though I wish more of that discussion could have been in the video, which has more than a million views on YouTube so far!
The views expressed are those of the author(s) & are not necessarily those of Scientific American.
ABOUT THE AUTHOR(S)
Evelyn Lamb is a freelance math & science writer based in Salt Lake City, Utah.Follow Evelyn Lamb on Twitter