Numbers too dangerous to imagine

 kw: book reviews, nonfiction, mathematics, number theory

For us ordinary folk, and for most scientists, the numbers on the balance are the limits of practical need. On the right: Avogadro's number, the quantity of protons in one gram of hydrogen. On the left, the long number with lots of leading zeroes is the weight of a proton in grams. Note the little "1/" above it: that makes the quantities balance. I left neutrons out of the discussion because the mass of a neutron is 0.14% greater than that of a proton.

For our more day-to-day experience, we've become accustomed to hearing "trillion" (a million millions) in news about government spending (but I remember a time that the entire Federal budget was much less than one trillion dollars), but few of us ever handle amounts larger than a few hundred thousand dollars, such as writing the check (heavily underwritten by our bank loan) for a house. And we may be familiar with numbers like milligrams or even micrograms (the weight of an average sand grain). That's about it.

Antonio Padilla would like to expand our horizons. A lot. A whole universe/multiverse of a lot. He begins his book Fantastic Numbers and Where to Find Them: A Cosmic Quest From Zero to Infinity by promising to describe a number so compendious that, if it were possible to memorize it in total, the mass of the information would collapse our brain into a black hole. Yes, information has mass. One "bit" of information stored in a computer memory chip has a tiny mass, much less than that of a proton. So does the same information held in our memory. However, the number of neurons available to hold memories (most are used for various thinking tasks), may be a few billion, although the "bits" of memory may be in synapses, at the rate of a few thousand per neuron. So perhaps we have the equivalent of several trillion bits of memory. Some think it is more like a quadrillion (1,000 trillion). Regardless, the mass of our memories is less than the mass of one proton. How many digits must be in a number that "weighs" 67,331,765,482,045,288,639,033,395 kilograms? (I got this number from the Schwarzschild Radius Calculator, using 10 cm for the radius of the "brain".)

Time for a sidebar about scientific notation. We don't readily comprehend numbers written out in full when they have more than 4-6 digits. Zeroing out most of the digits above, we can write that number as 6.733x10²⁵ kg. The exponent 25 tells us how many digits are after the decimal point. Similarly, the number on the left of the scale above is easier to comprehend as 1.672623x10^-24, the mass in grams of a proton. The exponent -24 tells us how many digits, including the leading "1", follow a decimal point far to the left. Its companion on the right is 6.022x10²³.

The "black-hole-in-a-brain" number has so many digits that the exponent has more digits than there are protons in the observable universe. It is called Graham's Number, and is so far the largest number used in a mathematical proof.

Before he gets to Graham's Number, though, Dr. Padilla takes us through some "easy" steps. His first chapter deals with the number 1.000 000 000 000 000 858. There are 15 zeroes in there. Scientific notation doesn't help here! This is the amount of time dilation experienced by Usain Bolt at his peak running speed of just under 30 mph (13.4 m/s). In sub-chapters, we read of various phenomena besides relative velocity that affect the flow of time, such as changing gravitational force on mountain tops or in ocean deeps. Chapters for Googol and Googolplex follow. A Googol is a one followed by 100 zeroes, or 1.0x10¹⁰⁰. A Googolplex has a Googol of zeroes: 1.0x10^Googol or (I have to use a graphic):

Think about that repeated exponent for a moment. It gets one somewhat ready to think about Graham's Number. Using a somewhat different notation, there are 64 levels…or perhaps 64 groups of levels of increasing size. I wasn't too clear, but, I don't want to be swallowed up by a black hole anyway. Graham's Number represents the extreme limit of the number of steps that may be needed to solve a particular problem. Needless to say, no physical computer system will even be able to count the steps, let alone perform them. Perhaps God's computer could do so, but I can't imagine Him being willing to devote the resources to perform the task.

The second section of the book starts with Zero, "a most beautiful number", and goes on to a few special numbers that are very, very close, but not quite there. One is "the most embarrassing number", 1x10^-120 (it has 119 zeroes before the "1"). That is the ratio between the amount of "vacuum energy" we can measure using the Casimir Effect, and a calculation of what the energy ought to be, using quantum theory.

It's the biggest blunder in physics! If quantum theory is true, right down to the smallest scales, no universe larger than a golf ball could exist without immediately collapsing back into a "big crunch". Clearly, we are missing something. Even if we limit the reach of quantum theory to the scale of a proton, no universe larger than the size of Manhattan is possible. The author and others think string theory will rescue the situation. I doubt it; not anytime soon. The last I learned, the number of possible variations of "string theory" is about 10⁵⁰⁰. That's a Googol to the fifth power (Googol⁵). It isn't even possible to make a list of string theory candidates so they can be compared.

So what could come next? Infinity of course! The famous sideways-eight (∞). One chapter is plenty, even though, as we come to find out, there is more than one kind of infinity. Rather than try to explain the difference between Georg Cantor's Aleph-null and Aleph-one infinity—and we don't know if there isn't a "real" Aleph-one, or more, in between!—I'll leave it to you to read this enjoyable, idea-packed book.

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Now I must touch on less pleasant matters. I wish Dr. Padilla had employed a copy editor; if he had one, that person needs a new day job.

1. While discussing time dilation (Usain Bolt's running speed), he dwells on a solar sail that might reach 1/5th the speed of light. Then he goes on to say that time dilation would mean a bacterial stowaway (or an onboard clock) would experience time slowed such that the 20-year journey to Alpha Centauri (4.37 light-years away) would seem to take less than nine years. He slipped a decimal somewhere. The time dilation factor is the square root of [1 - v²/c²], which I calculate to be √[1-0.04] = 0.9798. Now, 4.37x5 = 21.85 years (classical), which time dilation reduces, for shipboard bacteria and clocks, to 21.4 years. To reduce the time to nine years requires about 90% the speed of light, not 20%.
2. Two places I find the phrase, "The precise value of..." something, and a number with two or three digits of precision. The number "precise" should have been left out, or even it should have been stated that "The value of [whatever] is about..." The numbers in question are known to a precision of nine digits. This I attribute to the author.
3. This is a little less consequential: "100 billion neurons in the human brain." An up to date number is 86 billion, plus or minus a couple of billion. Also, 80% of our neurons are in the cerebellum, where they regulate our bodily functions and communicate with the 16 billion (more or less) neurons of our cortex, which is where thinking happens, and where all memories seem to be stored. Our vaunted intelligence primarily resides in those 16 billion.
4. This is a biggie, but it comes down to a simple copy/paste error. On page 190 we find "A quick reference guide to all the particles you'll encounter in this chapter", along with a chart of the Standard Model of particle physics. However, the Lepton portion of the chart is a copy of the Quark portion. The Lepton portion should show the electron, the muon and the tau, and their associated neutrinos. Big oops! I can't believe all the folks who wrote glowing blurbs praising the book (for its back cover) didn't see that. I suspect none of them read the book in full.
5. In the chapter on 10^-120 I find the phrase "external to spacetime", about an "external mechanism". As I understand cosmology, there is nothing external to spacetime. This is nonsense.
6. Finally, a simple typo. At the bottom of a table on page 282, "…infinities of site [Aleph-1]" should read "…infinities of size [Aleph-1]".

This is what I noticed without "looking for trouble." I have edited for others. In "editor mode" I might have found 2-3x as many items needing attention. Now, for those who've read this part, please don't hold it too much against the author. The book is full of wonderful ideas. I still mightily enjoyed the book.