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# Numbers Too Large (and Small) to Count

At one point in the children’s novel “The Phantom Tollbooth,” its protagonist, Milo, sets out to reach infinity. When he abandons the quest as hopeless, he is advised “Infinity is a poor place.”

“Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity,” by Antonio Padilla, holds a different view. As the author shows, infinity can be terrifying, but it is filled with an endless amount of numbers.

This is not a book just about numbers. It is about the relationship between numbers and physics, and how the world works at both the largest and the smallest scales. The numbers Padilla examines are fantastic in two senses. They are so extreme as to challenge belief and they are so extravagant as to seem fancy. And they define how the universe works.

Padilla goes to extremes. The first third of the book deals with big numbers, while the second third examines small numbers. The third section, Infinity, wraps everything up, including a discussion on the theory of everything.

He presents a slew of curious numbers. He introduces readers to the googol (that is how it is spelled) and the googolplex. A googol is a one followed by 100 zeros. A googolplex is a googol to the googolth power. It is not the largest number he discusses. He introduces Graham’s number, a number so large it is impossible to write it out before the universe ends – even if you write it barely under light speed.

At the other end of the scale, he looks at zero and the numbers close to it. He presents the history of zero. (It turns out many philosophers and scientists really hated it.) He looks at several numbers barely greater than zero and shows why these seemingly insignificant values determine our destiny.

He also takes readers on a trip through the extreme ends of modern physics. This includes chapters that venture to the edge of the universe, and chapters examining the smallest particles possible. He also shows how much physics has changed in the last fifty years.

There is math involved. Readers with technical training will be better able to follow Padilla’s discussions than those who abandoned mathematics at their first opportunity in high school. Even engineers may have difficulty following everything. Yet “Fantasic Numbers” is worth reading even if you do not grasp everything perfectly. It is an entertaining and informative book, with a high “wow” factor.

*“Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity,” by Antonio Padilla, Farrar, Straus and Giroux, 2022, 341 pages, $30.00 (Hardcover), $14.99 (ebook)*

*This review was written by Mark Lardas, who writes at Ricochet as Seawriter. Mark Lardas, an engineer, freelance writer, historian, and model-maker, lives in League City, TX. His website is **marklardas.com*.

This sounds like it would fill in many holes in terms of my understanding of math. I hear or see the word “googolplex” quite a lot, yet never knew what it was until I read yr fine review.

I have been listening to that young man who has Youtubes up regarding Theory of Relativity, how gravity doesn’t affect anything and more. He often has to delve into mathematical explorations, as only through math can it be proven whether Einstein’s Theory of relativity is right or not.

He discusses how not all “infinities” are equal, as was a topic of discussion among math experts when they were investigating whether there are more numbers between -1 and zero, or 1 and infinity.

One of his videos: https://www.youtube.com/watch?v=XRr1kaXKBsU

You can subscribe to him here once you are on youtube:

Veritasium

12.6M subscribers

Tony appears on Numberphile occasionally. I’ve been enjoying that channel for years. Also Derek’s channel that you mentioned.

I am partial to the number 5. Worn by the great Royals HOFer George Brett. And my favorite line on the multiplication table. In the grand scheme of infinitely small and infinately large, 5 is comfortably in the middle, nice and safe and accessible and understandable. No complex abstractions. Easy math. Heck, you can count to it on one hand. It’s a nice number. The kind of number you want your daughter to bring home. Not one of those skeevy non-repeating infinite decimals like pi or the square root of two. Or god forbid, that stupid imaginary number.

You want to know a secret? I’m still not perfectly reconciled to the notion of infinity, and I’d wager I spend more time there than most.

Infinity is a strange concept. It’s not actually a number.

CarolJoy’s comment has me thinking:

This doesn’t seem to make sense to me. What does it mean for “infinities” to be “equal.” Infinity is not a number, so I don’t see how it can be “equal” to anything.

The second part doesn’t make sense to me either. There are an infinite number of numbers between -1 and zero, and an infinite number of numbers “between” 1 and infinity, though in the latter case, it doesn’t really make sense to use the word “between.”

I can give a couple of examples.

Are there “more” positive whole numbers than positive even numbers? Well, there are an “infinite” number of both. You might think that there are twice as many positive whole numbers than positive even numbers, because for every positive number that you might pick greater than 1 — call it x — there are at least twice as many positive whole numbers less than x than there are positive even numbers less than x. On the other hand, there is a one-to-one correspondence between the positive whole numbers and the positive even numbers (defined by the formula y=2x, where x is a positive whole number).

So what does it mean for there to be “more,” when we’re talking about “infinity”? What does it mean for one “infinity” to be bigger (or smaller) than another?

Maybe there’s a way in which this makes sense. It doesn’t make sense to me, at the moment.

To look at the second issue raised by CarolJoy, consider whether there “are more numbers between -1 and zero, or 1 and infinity.”

I can easily construct a function that creates a one-to-one correspondence between the numbers in the interval (-1,0) and the numbers in the interval (1, infinity). Take y=-1/x. When x is -1, y is 1. As x approaches zero (from the negative side), y approaches infinity. Pick a number, any number for y — say 1 billion. There is one, and only one, corresponding x — in this case, -1/1,000,000,000.

By this approach, there are the “same number” of numbers between -1 and 0 and “between” 1 and infinity.

On the other hand, consider the function y=-(abs(sin x)). (“abs” indicating the absolute value function.) The sine function varies between -1 and 1, so the negative absolute value of the sine varies between 0 and -1. This creates a correspondence between the intervals (1,infinity) and [-1,0], though in this case the correspondence includes the endpoints of the range (-1,0), an in this case, it is not a one-to-one correspondence. For this function, there are an infinite number of x values that correspond to each y value, because the sine function repeats (

i.e.the sine wave).This ends up making no sense, to me, because I don’t think that it’s conceptually sound to consider whether different “infinities” are equal or not.

Infinity is just a weird idea.

Maybe this video can help. It’s the concept of countable, or as James from the video prefers, listable.

Thanks, but I know about the idea of countable and uncountable. I did a year of grad school in math, many years ago.

Any countable set of real numbers turns out to have measure zero. This is weird, because the rational numbers are countable. (Rational numbers are those that can be expressed as a fraction, with a whole number as the numerator and denominator.) This seems strange because the rational numbers are “dense.” There are a couple of alternative, equivalent definitions of “dense” — one way to think about it is that a set of real numbers is “dense” if there are an infinite number of them in any interval. So, for example, there are an infinite number of rational numbers between 0.00001 and 0.000001. This is true, no matter how small the interval might be. (An “interval” must have non-zero length.)

It turns out that there are

uncountablesets of real numbers that also have measure zero. The example that I recall is the middle-third Cantor set. The video guy mentions Cantor at the end of the video.Even something as apparently simple as the number line turns out to have strange properties.