The case for five to be 'the greatest number of all time'
Five may be simple and easy, but this number has a rich and complex personality
*This is the third episode in our series, The Greatest Numbers of All Time. Listen to more episodes.
Responding to rival bids from three and zero, number five is bursting to showcase why it's not only deserving of a spot in the IDEAS series, The Greatest Numbers of All Time, but of the coveted top rank above all other participants.
Whereas zero could lay claim to encompassing all of infinity, and three gained support from its association with spellcraft, rhetoric, and triangles, five's bid eschews the grandiose or the magical. Instead, it appears to be pursuing a "middle way" strategy. The compromise candidate. The centrist.
"I feel like five is just quite so special," said Tomoko Kate Kitagawa, co-author of The Secret Lives of Numbers, speaking in favour of the number's candidacy.
"Five started to become my favourite since my younger days at school in Japan."
Kitagawa points to the meanings given to five in such cultural traditions as feng shui and other branches of Chinese numerology, where it represents a central position, unlike its fellow single-digit integers.
The number one usually implies the northern direction and the winter season, Kitagawa explained. So does the number six. Two and seven connote the south, along with summer and red. Three and eight face east, and they are associated with spring. Four and nine are both west and the season of autumn.
This leaves five without a direction or season of its own, or even a pair among the other small whole numbers.
Alone, it has come to imply centrality and the so-called "Changing of Seasons," a time that may take place between summer and fall, or occupy the space at the end of each other season, according to different sub-cultures.
Circle of Fifths
In the worlds of art and science, the number five similarly finds itself taking a central spot.
Two of the most significant examples are its roles in music and statistics.
"It's like a question mark," says trumpeter and composer Rebecca Hennessy, describing what musicians most often call "the five" — a chord built off the fifth note of a major scale.
"Anything can happen on a five chord," Hennessy added.
This chord, along with the related interval known as the "perfect fifth," is absolutely central to classical and pop music theory. Leaving aside its role in the so-called Circle of Fifths (a relationship connecting all the different musical key signatures), the five chord reaches our ears at a frequency exactly midway between the 'tonic' notes.
In reality, this perfection can be subject to subtle distortions relating to the physics of musical instruments, but in theory, the chord labelled with the letter 'E', for example, contains all the frequencies in a 3:2 ratio with those in the chord labelled 'A' five notes below it.
Just as the letter E is the fifth letter in the alphabet, it is also the fifth note in the A major scale.
Pentagon power
While the number three's geometrical form, the triangle, can claim more widespread use in the construction industry, five's geometrical form is vastly more interesting.
The pentagon is the simplest shape that cannot tile a plane (such as a floor) without leaving gaps.
However, by attempting to tile pentagons, a person will likely build the model for a three-dimensional dodecahedron, the 12-sided regular object whose faces are all pentagons.
The challenge of achieving a pattern of five-fold symmetry in two dimensions eluded Western mathematicians and designers until 1972, despite appearing many centuries earlier in Islamic tiles.
Mathematician John Baez delves into this fascinating but complex topic here.
The infamous 'five per cent'
Five's reputation as a trustworthy centrist, albeit one with hidden complexities, may face challenges arising from its role in statistics, where the notorious "five per cent" line defines a border between results considered "statistically significant" and those considered "null."
"I remember when I learned, like — oh yes, five!" recalled Sherry Hou, a PhD student in epidemiology at McGill University. "Above five, it's not good. Below five, it's good!"
Hou notes that this convention has begun to change in many branches of science, but not before causing serious problems of publication bias and unhealthy incentives in research.
In 2016, the American Statistical Association issued a specific caution, for the first time in its history, against the misuse of so-called "p values," of which the five per cent standard was the most common type.
Five can hope that this scandal will fail to taint its reputation in the general public awareness simply due to the complexity of the problem and the fact that most of this controversy is expressed in statistical jargon terms few outsiders understand.
Nevertheless, given the number's centrality in the conventions of reporting scientific results across many disciplines for more than a century, it will surely become a weakness that other numbers will seek to exploit as they compete for the title of "Greatest Number of All Time."
Guests in this episode:
Clayton Fisher is an audiologist at Treat Hearing Health Care in Ottawa.
Sherry Hou is a PhD candidate in the department of epidemiology at McGill University.
Jonathan Hoff is a data scientist and senior consultant for Splunk.
Tomoko Kate Kitagawa is director of the Space Education Office for the Japan Aerospace Exploration Agency (JAXA), and co-author of The Secret Lives of Numbers.
Arthur Benjamin is a professor of mathematics at Harvey Mudd College, and a public speaker. His lecture series on The Joy of Mathematics is available through The Great Courses. His books include The Magic of Math.
Nathan Taback is a professor in the department of statistical sciences at the University of Toronto.
Stavroula Kousta is the chief editor of Nature Human Behaviour, an online journal of the social and natural sciences.
Andrew Gelman is professor of political science and statistics at Columbia University in New York.
A special thank you to Sesame Street for giving numbers personality, especially for their 1978 song, Gimme Five.
*This episode was produced by Tom Howell.
