In 2004, Brady Haran published the infamous Astounding: $1+2+3+4+5+\cdots =-\frac{1}{12}$ video in which Dr Tony Padilla demonstrates how sum of all natural numbers equals minus a twelfth. The video prompted a flurry of objections from viewers rejecting the result. But riddle me this: Would you agree the following equation is true: $$1+\frac{1}{2}+\frac{1}{4}+\cdots =2$$

Obviously the equation does not hold. How could it? The thing on the left side of the equal sign is an infinite series. The thing on the right side is a real number. Those are completely different objects therefore they cannot be equal. Anyone saying the two are equal might just as well say $\mathbb{N}=🐘$ (and while I grant you that elephants are quite large, they’re not sets of natural numbers).

And yet, people usually agree the infinite sum equals 2. Why is that? They use ‘mathematical trickery’ and redefine the meaning of the equal sign. Indeed, $1+\frac{1}{2}+\frac{1}{4}+\cdots$ does not equal 2. Rather, given an infinite series $(a_n)$ where $a_n=2^{-n}$, we can define an infinite series $(S_n)$ where $S_n=\sum _0^n{a}_n$ and only now we get: $$\underset{n\to \mathrm{\infty }}{lim}{S}_n=\underset{n\to \mathrm{\infty }}{lim}2-\frac{1}{2^n}=2$$

But that’s a different equation than the one I’ve enquired about.

Historical perspective

Let’s consider some other mathematical statements:

• There exists a number such that adding it to 5 results in 4.¹
• Square root of two cannot be represented as a ratio of two natural numbers.²
• There exists a number such that squaring it produces -1.³
• There are more real numbers than natural numbers.⁴
• Two parallel lines can intersect.⁵

All of those claims were at one point considered an ineffable twaddle; now they are intricately woven into the tapestry of modern mathematics and engineering. We shan’t be too harsh on our ancestors. We only need to think back to grammar school to understand their sentiments. One day we’re taught squaring produces a non-negative quantity; another we’re taught about imaginary numbers. Many a student has labelled complex numbers as nonsense. Yet, accepting them opens a wondrous universe of possibilities.⁶

In the video no one actually claims sum of all integers doesn’t diverge. But what if we assigned a number to it anyway? What if we allow for the equal signs to mean something different just like we allow square root of -1 to exist? Just as accepting non-Euclidean geometry opens up a whole new marvellous world to explore, what discoveries can we make if we use summation method which does say all integers sum to minus a twelfth?

School imparts people with the wrong idea of maths as a linear progression where all new discoveries mustn’t contradict what has been established already. But that’s not the case. It’s all made up and someone can come, make up other rules and see where those lead.

Rigour

Not everyone has an issue with assigning $-\frac{1}{12}$ to the infinite series $1+2+3+\cdots$. Another criticism of the video is its lack of rigour. That I don’t disagree with as much. Indeed, the video could make it more explicit non-standard summation rules were used.

On the other hand, Numberphile (YouTube channel where the Astounding video has been published) is not a channel with academic lectures. Brady is a popular science communicator and his videos are not replacement for maths classes. For people interested in more rigour, Dr Padilla has published a supplementary article with a formal derivation.

Over the years Brady published other videos on the subject (many linked from his blog post which tries to calm the situation). It can be argued the video did decent job at popularising mathematics. I would certainly say, the amount of animosity still hurled at the video is unreasonable.

Generalities

I leave you with the following poem by Cyprian Kamil Norwid. It’s something my physics teacher made us learn by heart in high school. What does it have to do with physics? What does it have to do with Numberphile’s video? I leave it to your interpretation.