Números y más números. A través de diecisiete entretenidos ensayos, Asimov nos pasea por el universo de los números desde casi todos los puntos de vista concebibles. Así nos habla del cero, del infinito, de los números binarios, de los imaginarios, de la cuadratura del círculo. Calcula el tamaño del universo en protones, las islas de la Tierra según su superficie, así como la escala según su altitud de las montañas.

Part I:

Chapter 1: Nothing Counts
Numbers, ancient numbers, the invention of zero. A typical dig at numerology, which of course deserves it. Apparently, when we talk about the score in a game we're really saying "What's the twenty?". How strange.

Chapter 2: One, Ten, Buckle My Shoe
Binary number system, and number systems using various bases. Conversion of base 2 to base 8 by grouping. It is easy to prove this must always be correct and does not require induction. The method is of course generalizable to other powers, e.g., base 3 and 27., base 2 and 4. Conversion between bases generally is more or less covered.

Chapter 3: Exclamation Point!
(1 + 1/n)^n approaches e from below as n approaches infinity. The derivative of this function is (1/x+1)^x*(ln(1/x+1)-1/((1/x+1)*x)) which is big and confusing (obtained from derivative-calculator.net, I couldn't have done it myself). It's not clear that this is positive for n > 0, just by looking, due to the negative term. So, I haven't shown that (1 + 1/n)^n is monotonically increasing, although Asimov seems quite sure.

Another way to express e is 1/0! + 1/1! + 1/2! + 1/3! + ... This is a nice expression, because every term you use increases the precision of your value. 1/n! > 1/(n + 1)! + 1/(n + 2)! + .... Asimov asserts this without proof.

Part II: Numbers and Mathematics
6. A Piece of Pi
Perimeter and diameter go together better than circumference and diameter. For one thing, circumference is Latin and the other two are Greek. Also, perimeter starts with pi, and when the diameter is one ms the perimeter is pi ms. There is a cute joke: if the law declaring pi to be 3 were passed, wouldn't all the wheels in the state obediently become hexagons? 22/7 is a fine rational approximation for pi, a bit too large by .04%. Archimedes used the "method of exhaustion" with 96 sided polygons and bounded pi from above by 22/7 and from below by 223/71. If you average the two and use that value for pi, the error is only 0.0082%, so precision increased by about 100 times. In the sixteenth century somebody arrived at the value 355/113 which is greater by only about 0.000008%. If you used that to calculate the circumference of the earth assuming it was a perfect sphere you would be off by 11 feet. 335/113 is 3.1415[92]. In the 16th century Francois Vieta uses a similar but more algebraic approach than Archimedes. Leibniz derived a series: pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11.... According to Asimov, this series does not converge very rapidly. You can look at a bunch of graphs online and see that among the various approximations it is one that is distinguished by oscillating wildly about the true value. With the naive approach, getting a value correct to 10 decimal places requires about five billion terms according to Wikipedia. Many people tried to calculate pi to various precisions, up to 72 decimal places by 1717, hoping to find a repeating term, hence proof that pi was a rational number. pi was proved irrational in 1761. For all known sizes, even the known size of the universe in Asimov's day, 35 decimal places is quite enough to leave one with a ridiculously small error when calculating lengths. Various people continued to calculate the value of pi more or less by hand into the late 1800s. In the 1940s that job was taken on by the ENIAC which got the value of pi to 2035 places and by 1955 10,000 or so.

Chapter 7: Tools of the Trade
Straightedge and compass constructions. By using compass and straightedge you can arrive at any rational number geometrically, apparently. It's not obvious to me how you would even trisect a line, but you can find convincing examples on Wikipedia. Clever, yet indirect. It might seem that it would be impossible to arrive at lines of irrational length using straightedge and compass construction but this, on reflection, is obviously not true. Think of square root of 2, which is irrational and which is also just the diagonal of a unit square. The Greeks believed that any conceivable number could be represented by some such construction, although it might take hundreds of years to discover, e.g., Gauss's division of the circle into 17 equal parts. The method could be generalized to various numbers, although certainly not all, but it also constituted a method for constructing n-sided polygons for the various n. Straightedge and compass numbers are a subset of the algebraic numbers (you can't take roots higher than 2 with a straightedge and compass). In 1844, it was proved that non-algebraic numbers exist. A likely but not certain candidate was e and in 1873 it was demonstrated that e was certainly not algebraic and it was called transcendental. In 1882 it was proved that pi was transcendental.

Part III: Numbers and Measurement
9: Forget It!
A tirade about how stupid it is to have a measurement system which does not match the base of the number system. The English finally went to decimal coinage and the metric system; but the US still sticks w/ the English system of measurement, which is a dumb thing to do. Makes some general remarks about simplifying English grammar, which are pretty stupid. Yet the publishers numbered the parts with Roman numerals anyway.

Part IV: Numbers and the Calendar
TODO

Part V: Numbers and Biology
TODO

Part VI: Numbers and Astronomy
TODO

Part VII: Numbers and the Earth
This is not quite a collection of miscellaneous geographical statistics, but nearly so.

15. Water, Water, Everywhere
The sizes of lakes and oceans.

16. Up and Down the Earth
Ranking mountains and trenches by how high and how deep using different metrics. This is the best of the three.

17. The Isles of Earth
Points out that the division of continents is kind of arbitrary. Europe and Asia run together and both are joined to Africa, really. One nice etymological point: island is derived from the Anglo-Saxon while isle is from the Latin, terra in sala -> insula. "island" gained an "s" late in its life to make it more like "Isle".

Each time I see a numeral
A prime, a two, a zero
I can’t resist a fine-eyed urge
To count and say
Thank Asimov, his “Numbers”
Have properties
Deducive, it’s their way.
Thank Asimov, his “Numbers,”
They rise up in
The most delightful ways.
Those little (pi)s
So helpful and revealing
When in equations
Send you answers
That you needed.
Thank heaven that their power
Brings favors to us all
No matter where
No matter who
Without them
What would we ever do?
Thank heaven
Thank heaven
Thank heaven for all numbers.
[Inspired by lyrics written by Alan Jay Lerner and Frederick Loewe and sung by Maurice Chevalier—but do not blame these gentlemen for the lines above.]

Numbers have a lot of personalities in this collection of essays: abundant, deficient, perfect, complex, Arabic, Roman, base 2 or 3 or anything, imaginary, real, rational, irrational, integral, natural, transcendental, negative, positive, prime, one or two or zero. Hey, how about that zero? Isaac Asimov teaches us not to underestimate it. Its magnitude may be zero but don’t be fooled—its impact has been huge.

I think my favorite bit had to do with the units for measuring volumes (of, e.g., beer!). Here are some that folks used in England: drams, firkins, kilderkins, barrels, hogsheads, puncheons, butts, pottles, bushels, pecks, strikes, cooms, quarters, chaldrons, weys, and lasts. If you feel those Dickensian names promise too simple a system of measurement, take note that some of these units change depending on
(1) What you are measuring (a firkin of ale does not equal a firkin of beer).
(2) Where you are measuring (a chaldron in London differs from a chaldron somewhere else).

Shouldn’t we Americans look to embrace this elegant system rather than that awfully inconvenient metric mess? ( )

I came across this book at my school library by chance, I probably wouldn't have picked it up if it didn't have "Asimov" printed in large type down the spine. The book is a collection of nonfiction essays written by Isaac Asimov that deal in some way with numbers.

I was expecting a book about maths when I picked it up, and that's what it mostly was (written in a way that would be approachable by anyone who has an interest), but nearing the end they essays seemed to be included just because they had a bunch of numbers in them (but I can't blame the title for lying to me).

One of the essays included about the metric system (when it was written, most of the English world had not adopted it, by the time the book was published only the United States were left) threw me off because it sounded like one big attack against all other units of measure rather than an explanation of why the Metric system is better. I have no sympathy for any other systems, but this essay felt out of place from the others because of the style. ( )

I suspect many folks would find this collection dry and a bit boring (made for nerds,) but I've always enjoyed Asimov's essays and have found his other collections insightful and fun to read. This collection was no exception. This specific collection featured essays on mathematics (more specifically numbers or arithmetic.) The book takes us from the discovery of concepts like "zero" to using numbers to scale, size, and categorize almost anything you can think of. By reading these you not only gain a better understanding of the history of math, insights into geography, biology, etc., but also the simple processes that can be used in our own every day efforts to understand and deal with the world around you. Along the way Asimov also includes extra embedded pieces that help illuminate a piece history or place that is connected to an essay. Of course there is also the Asimov charm and his corny humor which takes the edge off the technical thrust of this work. I've have also used some of the presentation ideas of Asmov in the book in teaching my 6-12 grade students math. ( )