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: \[\lim_{n\to\infty} S_n = \lim_{n\to\infty} 2 - \frac{1}{2^n} = 2\]
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.
Generalities
Cyprian Kamil Norwid
When like a butterfly the Artist’s mind
In spring of life inhales the air,
It can but say:
‘The Earth is round — it is a sphere.’
But when autumnal shivers
Shake the trees and kill the flowers,
It must elaborate:
‘Though somewhat flattened at the poles.’
Amid the varied charms
Of Eloquence and Rhyme
One persists above the rest:
A proper word each thing to name!
Translation © All Peotry
1 Geometry had a big influence over early mathematics and negative quantities were considered absurd. Al-Khwarizmi, the guy whose book is the etymology of the word ‘algebra’, rejected negative solutions to quadratic equations. He also rejected negative coefficients in quadratics instead considering three separate equation ‘species’: \(ax^2 + bx = c\), \(ax^2+c=bx\) and \(ax^2=bx+c\). (See al-Khwarizmi, The Algebra of Mohammed ben Musa, edited and translated by Friedrich Rosen, Oriental Translation Fund, London, 1881). ↩
2 Legend has it that Greeks could be drown at sea for revealing the existence of irrational numbers. Perhaps a fitting end to someone who spread knowledge of the ‘inconceivable’! In reality probably just a metaphor but illustrated how Pythagoreans disregarded irrational numbers. (See William Thomson and Gustav Junge, The commentary of Pappus of Book X of Euclid’s Elements, Harvard University Press, 1930). ↩
3 We had to wait until 16th century until Gerolamo Cardano used what we now call complex numbers. As he laid out various rules for solving polynomial equations, square roots of negative numbers helped him find solutions to the problems. Even still, he called square roots of negative numbers ‘as refined as they are useless.’ (See Girolamo Cardano, The Great Art or The Rules of Algebra, edited and translated by T. Richard Witmer, The MIT Press, 1968). ↩
4 Proof that there are more real numbers than natural numbers is particularly late addition to mathematics. It was presented in 19th century by Georg Cantor who introduced concept of set cardinality. A work for which he received sharp criticism from his contemporaries. With time, some even suggested that contemplating the infinite drew Cantor mad, though there are no evidence his theories were source of his mental struggles. (See Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, 1979). ↩
5 The parallel postulate is the fifth postulate in Euclid’s Elements. It stands out for not being self-evident and throughout the centuries consensus among mathematicians was that the postulate was true but simply needed to be proven. It took over two millennia before independence of this postulate was discovered. (See Florence P. Lewis, History of the Parallel Postulate, The American Mathematical Monthly, Vol. 27, No. 1, 1920). ↩
6 Complex numbers found multiple uses in physics and engineering. But openness to them created opportunities for other benefits as well. For example, William Hamilton introduced quaternions which have three distinct numbers which all square to -1. They proved useful in computer graphics as representation of rotation. (See John Vince, Quaternions for Computer Graphics, Springer, 2021). ↩