The Penguin Dictionary of Curious and Interesting Numbers: Revised Edition

by David G. Wells

This dictionary of numbers, arranged in order of magnitude, exposes the fascinating facts about certain numbers and number sequences.

Genres: Mathematics, Nonfiction, Reference, Science, Number, Popular Science

256 pages, Paperback

First published January 1, 1968

About the author

David Wells is a writer on mathematics and puzzles.

Librarian Note: There is more than one author in the Goodreads database with this name.

Reviews

[Patrick · Rating 4 out of 5]

This book is wonderful for skimming. And something about its direct, mathematical style is very funny to me. I sometimes wonder if mathematics might not be the best training for writing terse prose. Here's a pretty much randomly chosen entry:
33,705 The first exception to the rule that 945+630n is an odd abundant number. It is deficient.

[Charles · Rating 5 out of 5]

While the content is dated by the ongoing research in mathematics as well as what can now be done with modern computers, it is still a fascinating book to examine. Wells writes with a high degree of clarity and understanding, explaining quite clearly why the selected numbers are so interesting.
The book is organized in the form of a dictionary, starting with -1 and i then moving in the general direction of ascending order. Most of the numbers in the dictionary are integers. The largest number that appears is Graham’s number, 3↑↑↑3, a number so large that a new notation had to be invented to represent it.
If you are someone that loves numbers and all of the strange and continually fascinating properties that they exhibit, then this is a book that is timeless even though dated.

This review also appears on Amazon

[Jerry · Rating 5 out of 5]

Mathematicians do not always state the obvious.

I hadn’t planned on reading this book all the way through. I brought it downstairs thinking it might be a useful book for browsing randomly next to the armchair. But it’s so interesting I’m already fifty pages in. This is a bathroom reader for fans of the movie π. Each entry is a short description of what makes that number interesting.

Pi itself comes in on page 48, “the most famous and most remarkable of all numbers”. In seven pages—one of the longest entries in the book—he describes how the Greeks attempted to determine more and more accurate representations of π, as well as the Chinese and then Ludolph van Cuelen.

Many numbers feature something to do with the Fibonacci sequence. In the number five, with ten pages it is probably by far the longest entry, six of the pages are taken up by the Fibonacci sequence.

It is one of the curious coincidences that occur in the history of mathematics that a problem about rabbits should generate a sequence of numbers of such interest and fascination. Rabbits, needless to say, do not feature again in its history.

The book is filled with tricks that will be difficult to remember unless you remember what number they’re associated with. The casting out of nines is probably an easier one, since it appears under the number 9.

Arithmetical sums may be checked by the process called ‘casting out nines’. This came to Europe from the Arabs, but was probably an Indian invention. Leonardo of Pisa described it in his Liber Abaci. Each number in a sum is replaced by the sum of its digits. If the original sum is correct, so will the same sum be when performed with the sums-of-digits only.

Some of the tricks make Darren Aranofsky’s movie look tame. To test if a number is divisible by 11, start from either end, and add and subtract the numbers alternately. That is, 121, 1-2+1 = 0, or 2781, 2-7+8-1 = -4. If the result is divisible by 11 (which includes zero, as in 121), the original number is also divisible by 11. So, 121 is divisible by 11 and the somewhat randomly chosen 2781 is not. The reason this works is “because 11 = 10 + 1”.

Among the reasons that the numbers 12 and 13 are interesting is that they can be reversed to get the reverse of their squares. That is, 12 squared is 144; 21 squared is 441.

Part of the reason there are so many interesting numbers is that mathematicians have come up with so many reasons for them to be interesting. “Abundant” numbers also come in under the number 12; it is the first abundant number, which is to say, the first number that “is less than the sum of its factors excluding itself.”

All numbers, in fact, are interesting up to 39:

39
This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting.
It is therefore also the first number to be simultaneously interesting and uninteresting.

It is the third of I suspect nine integers that are equal to the sum of their digits added to the product of their digits. Which is, admittedly, uninteresting.

While 42 has an entry, he does not mention that 42 is the meaning of life, though the calculation probably was spurious. He does include many predictions that a number is prime and explains when those predictions were falsified. The entry for 42 entry also describes Catalan numbers, because 42 is the fifth Catalan number.

There are a couple of numbers that repeat powers when divided into 1, a feature that would have had Aronofsky’s protagonist drilling into his head in the first ten minutes of the movie. For example, 1/97 repeats the powers of three, and appears to do so for a long time, in a very strange way.

0.010309 278350 515463 917525 773195…

If you look closely, 1, 3, 9, 27, are obvious. But 83 is in fact 81 (3^4) overlapped with 243 (3^5). The 2 of 243 adds to the 1 of 81, to give 83. Likewise, the 7 of 729 (3^6) adds to the 43 of 243 to give 50.

1/98 does the same with powers of two, 1/96 the powers of four, and so on.

I have a lot more sympathy with people who believe God has left secret messages in strange numbers after reading this book.

Some of the entries include interesting history that you’d never be able to find without already knowing it. For example:

641
Euler found the first counter-example to Fermat’s conjecture that 2^2^n + 1 is always prime, when he discovered in 1742 that 2^2^5 +1 is divisible by 641.

The number 1,089 includes a potential magic trick. “If a 3-digit number is reversed and the result subtracted, and that answer added to its reversal, the answer is always 1089: 623-326=297 and 297+792 = 1089.

This book came out in 1986. On page 204, very near the end, he talks about factorizing large numbers; at this point, the numbers are so large that he doesn’t reproduce them, and uses, in brackets, the number of digits. The CRAY at this time could factor 100-digit numbers “in about 40 seconds”. This is important because “in 1975, Whitfield Diffie and Martin Hellman invented the trapdoor function, and shortly afterwards, Rivest, Shamir and Adleman showed how to make it a practical proposition.”

He’s almost certainly talking about the RSA algorithm used in public key cryptography.

From that point on, modern computers get a lot more mention, though always big ones. Three pages later, on page 207, he mentions a personal computer for the first (and only) time, the Apple II.

This is an amazing book for browsing through. There is a lot of mathematical history literally hidden among its numeric entries, and a lot of the superficially, at least, strange behavior of numbers.

A Python script for calculating Pi, based on the method on page 55:

#!/usr/local/bin/python3
# calculate pi using David Wells, p. 55

from sys import argv, exit
from decimal import Decimal, getcontext
from argparse import ArgumentParser

parser = ArgumentParser(description="calculate pi")
parser.add_argument('precision', type=int, nargs='?', default=5)
parser.add_argument('--verbose', action='store_true')
arguments = parser.parse_args()

getcontext().prec = arguments.precision

d1,d2,d4,d6 = Decimal(1), Decimal(2), Decimal(4), Decimal(6)

a=d1
x=d1
b=d1/d2.sqrt()
c=d1/d4
previousPi = d1

while True:
	y=a
	a=(a+b)/d2
	b=(b*y).sqrt()
	c=c-x*(a-y)**d2
	x=d2*x
	pi = (a+b)**d2/(d4*c)

	#leave loop if pi has stopped changing
	if pi == previousPi:
		break
	previousPi = pi

	if arguments.verbose:
		print(pi)

print(pi)

[Paulo · Rating 2 out of 5]

It's exactly what it says: a dictionary. It's not appropiate for sequential reading from start to end. Sometimes it's hard to follow, especially since it provides too much repetitive and trivial data about the numbers (like which numbers are palindromic or equal to basic operations).

So this is reference book, increasingly ordered, but at times it seems a little boring when it chains data without explanation or context. A good book to consult when you are looking for some numerical curiosity (keeping in mind its year of publication and therefore that it's outdated, of course), but not a particularly enjoyable read.

[Harry · Rating 3 out of 5]

Overall I enjoyed this book, and there was a lot of interesting information in here. There were some errors, one of the more obvious being the claim that "if any prime between 7 and 210 is subtracted from 210, the result is another prime", which has counterexamples 23, 41, 67, 89. However, the ideas described are well explained and a large range of properties is covered, so as the reader will generally check the results they find interesting anyway the occasional mistake is forgivable. A worthwhile read for anyone who is interested in numbers.

[David · Rating 4 out of 5]

I have to admit, you need to have a bit of an interest in numbers and maths to read this book. I found it from a review whilst looking into another book. It is exactly what it says on the cover; a dictionary of numbers. It starts with -1 and i and works its way up in value pointing out interesting and noteworthy numbers on the way. Obviously, some entries are more interesting than others and many involve some hefty maths but I did find the book enjoyable although it would be pretty difficult to find any particular entry again unless you already know the evalue.

One thing that becomes apparent fairly early on in the book; superstars like pi and e really put themselves about a bit, something like carbon does in chemistry - I have heard carbon described as the slut of the periodic table. 51 gets an entry as being the first number with nothing interesting to say about it apart from its uninterestingness.

The last quarter of the book flashed past really quickly because by that stage, the numbers are about 20 digits or more in length and so occupy plenty of space. A lot of the entries from here on document ever larger Mersenne primes and here it does show it's age somewhat as it doesn't make it it as far as the age of distributed computing and the GIMPS project.

It was an enjoyable read, though and it had to end somewhere. It was never going to reach infinity and nor should it and Graham's number is a reasonable milestone.

[Ronnie Childs · Rating 4 out of 5]

This book is a dictionary--it says so right on the front cover. Unlike with other dictionaries, I ended up reading it all the way through, cover to cover, and quite enjoyed it. Its first entry is -1 and 𝘪, tricky devils they can be, and it truly did illuminate some of their more slippery qualities over the next four pages, making it one of the longer entries. Almost as long is the next entry, 0, which has a fascinating history itself. So the first two have me hooked, and although it slips occasionally into tedium, the overall effect of reading it is a pleasant and relaxing, with a little edification thrown in. Some of the math is over my head or outside of my interest, but it quickly comes back down to more accommodating grounds. As a reference book, it has been unfortunately rendered not so useful in this Wikipedia age. Clearly it's not for everyone; a satisfied reader will be one with a pretty fair splash of geek in their makeup. But it's got a good beat and you can dance to it.

[Matt · Rating 4 out of 5]

An excellent taster to see if you're really into mathematics, but be aware there are many mistakes.

I combed it entirely (both editions), checking every fact (I'm meticulous like that), and found over a hundred.

But he does a good job for all that.

A few more details on the citations would have been admirable (often he just gives the name of the journal and the year, making it difficult to track down the precise source of a nugget). And some of the sources are obscure magazines that have never made it to the internet, so, as they say, good luck with that.

He's good at what he does, but he lacks attention to details. So, get it for your teenage relatives, you know, those ones who are always buried in science books and toys and things.

[Kevin McDonald · Rating 4 out of 5]

Pretty cool that someone made this. It’s exactly what you think it would be. A glossary of numbers. Obviously this is only going to appeal to people interested in math, with a lot of it being historical or somewhat esoteric.

It is cool, however, that you can basically pick any number and look up some things about it. IE, so and so is the third perfect cube and the lowest that is a product of the sum of perfect squares. More “famous” numbers have a lot more detail, sometimes pages of historical context. I did like that, since a lot of these are not anything that would have any purpose being taught, and were new to me. Certainly not casual reading though, and more for lookup from the bookshelf.

[Chris · Rating 1 out of 5]

really boring.

[John Mann · Rating 5 out of 5]

As a mathematics graduate I remember this book with great fondness - well written, concise and informative. Thoroughly recommended!

[Arianne X · Rating 5 out of 5]

The Mathematical and The Mystical

Number as the basis of nature is the overarching and guiding concept of this book just as it was for the Pythagoreans. That is, I cannot help but think, at least implicitly, that the author compiled a compendium of curious and interesting numbers because he presumes that number is the basis of nature. The idea of the Pythagoreans, and implicit in this book, is that the physical world works on numerical and mathematical principles. However, the Pythagoreans combined mathematics and mysticism, what we pejoratively call numerology was a legitimate part of mathematics. Up until at least Newton, mathematics and mysticism were meshed together but even Newton, thought of as the first of the moderns, was referred to as the last magician by John Maynard Keynes since Newton spent more time and energy on uncovering the bible code than he did in developing the calculus. This tradition starts with the Pythagorean who were very much a religious cult. Mathematics was as much mystical as analytical. In the modern era, we of course separate anything having to do with the science of mathematics from anything approaching the mystical, or do we?

The modern paradigm is that mathematics can explain anything that nature throws at us, but is this necessarily true? Is the modern paradigm just an updated myth? We believe that all of nature must be reducible to mathematics. Why must this be? Can this really be the case given the unprovable foundational assumptions of arithmetic and geometry? Is this just not the overarching and guiding assumption of the era in which we live and flatter ourselves in calling modern? The idea of number as the basis of nature still informs modern science. We look for mathematical patterns in nature because we start out with the idea no matter what the question about nature is, the answer will be, must be, a mathematical answer. We tend measure how advanced a given field of inquiry is and how solid our knowledge of a given field of study is by how mathematical it is. But still, mathematics as a final ontology is rather austere. The explanations provided by mathematics build the rudimentary scaffolding of explanation and understanding of the human condition. The more mathematically certain we are the more austere is our explanation. We dwell in the world poetically (Holderlin), not mathematically.

There is still with us the implicit presumption that reality must reduce to mathematics. Desiring a mathematical explanation of existence is very Pythagorean. Any physicist doing work on all encompassing unified theory is looking back to his or her Pythagorean roots. Mathematics is a set of rules that we can accept or not accept. I chose to accept them. Rather like playing chess, if one wants to play, one must understand that the game is constitutive of rules and accept the rules as such to play. This is a presumption that has worked well in securing human knowledge, but still, mathematics is an interesting human curiosity.

[Koen Crolla · Rating 3 out of 5]

This is one of those concepts that sound good until you stop to think about them. It's very hard to come up with number trivia that more than one person thinks is interesting, particularly if you're writing for a lay audience.

And since Wells *is* writing for a lay audience, it's not surprising that most of the trivia in the book is of the kind that can be turned into a party trick for non-mathematicians: base-10 trivia rather than actual number trivia. Maybe it's because I'm a programmer rather than a 6-year-old, but I don't see the appeal of yet another palindromic number, or of "happy numbers", or of numbers that are equal to some basic operation on their individual digits in base-10 (8208 = 8^4 + 2^4 + 0^4 + 8^4). I suppose it's better than English-language trivia (111,777 is "the least integer not nameable in fewer than nineteen syllables", which is a description of eighteen syllables), but that doesn't justify the fact that maybe 90% of the trivia in the book is of this sort.

It's also hard to figure out how the book is supposed to be read. On the one hand you have a lot of entries referring back to previous ones, which suggests it can be read like a normal book, from start to finish; on the other, you have things like the σ function, which he mentions in the entries for eighteen different numbers before finally defining it (relatively poorly) in the nineteenth, which suggests that it's one of those books you leave in your bathroom and open at random, ready to consult the index, whenever you've had Mexican food.

Then there's the fact that some of his information is either contradictory or so unclearly put that it might as well be, that he keeps misspelling Paul Erdős' name as "Erdös", that he occasionally slips into surprisingly outdated jargon (perhaps in the mistaken belief that non-mathematicians will find that easier to follow than the modern equivalent, which tends to have a higher information density), and that a lot of the trivia is genuinely trivial (1000 = 10^3 in any base!). This revised edition came out in 1994, so obviously some of it is now outdated, too (he lists the largest confirmed Mersenne prime as being M_216,091, with M_756,839 and M_859,433 listed as maybes; nowadays the latter two have been confirmed, and fourteen larger ones have been discovered, the biggest being M_43,112,609).

All in all, not a very enjoyable book.
This David Wells is the same David Wells who wrote You Are a Mathematician, which I read a few months ago. That one was a lot better.

[Greg Ross · Rating 5 out of 5]

This book does what a good popular math book ought to -- it conveys the author's love of the subject without seeming to compromise or condescend to the general reader. Essentially it's one long list of interesting numbers, with a brief but lucid description of each, illustrating by example the depth and fascination of mathematics and written with Wells' characteristic intelligence and clarity. Penguin deserves credit for publishing such a book -- it may not have broad appeal, but for the right kind of reader it's like discovering a trove of diamonds. (Wells has published three other books in the "curious and interesting" series, collecting curiosities in mathematics, geometry and puzzles. All are outstanding.)

[Tim]

I inherited the 1988 edition of this dictionary from my grandfather. It starts with -1 and i, passes through 0 and a number of small fractions and irrational numbers before reaching 1. The numbers climb to a googol (1 followed by 100 zeros) on page 205 with a reference to a googolplex. On page 208 it lists the largest prime as (2^216091)-1 with only 65050 decimal digits!

Instead of a discussion about infinity, the dictionary ends with "Graham's Number". Otherwise I would have rated it more highly.

[Aaron · Rating 3 out of 5]

Enjoyable read, although it was more of a scan. I can imagine reading every page, but I just don't have that kind of time in this lifetime. I could see doing it had I been a math major, and it is possible that there still is a math major lurking within me. Lots of enjoyable descriptions in this book.

[Tiffany · Rating 2 out of 5]

It was okay, but not quite what I was expecting. There were some interesting tidbits, but since the book is laid out like a reference book, the facts seem boring. They're more interesting when in books about math, not a reference book of numbers.

[Maurizio Codogno · Rating 4 out of 5]

La mia recensione: http://xmau.com/notiziole/archives/00...

[Natalie · Rating 4 out of 5]

The stuff that is relevant to you personally is very interesting but sometimes it is hard to follow and there is a lot of information that is not relevant to you.

[Harry Lawrence · Rating 4 out of 5]

A good thing to own if you have an interest in mathematics, some quite interesting stuff.

[Chris · Rating 2 out of 5]

boring but not bad to have lying around

[OMKAR]

I want to read it now

[Nikola Š · Rating 3 out of 5]

What a uniquely pointless book! I love it!

[Johan Koolwaaij · Rating 4 out of 5]

One of these books that provides many triggers for new thoughts.

[Abhiram Devesh · Rating 5 out of 5]

Amazing book... True to its title.