Category: Math
One Two Three Infinity Facts and Speculations of Science
All right. One, Two, Three Infinity: Facts and Speculations of Science, George Gamow. All right, Mr. George Gamow.
All right. Playing with numbers, space, time, and Einstein, micro cosmos, macro cosmos, and then a whole bunch of illustrations. Whole bunch of them. Big numbers. No, let’s see what he’s talking about. My eyes are kind of funny.
How high can you count? A gazillion. Boom. Just did it. Count as high as possible. Infinity. All right. “There’s a story about two Hungarian aristocrats who decided to play a game in which one who calls the largest number wins. ‘Well,’ said one of them, ‘You name your number first.’ After a few minutes of hard mental work, the second aristocrat finally named the largest number he could think of. ‘Three,’ he said. Now it was the turn of the first one to do the thinking. But after a quarter of an hour, he finally gave up. ‘You’ve won,’ he agreed.
“Of course, these two Hungarian aristocrats did not represent a very high degree of intelligence, and this story is probably just a malicious slander. But such a conversation might actually have taken place if the two men had been not Hungarians, not Hottentots. We have indeed on the authority of African explorers that many Hottentot tribes did not have in their vocabulary the name for numbers larger than three. Ask a native down there how many sons he has or how many enemies he has slain. And if the number is more than three, he will answer, ‘Many.’
“Thus in the Hottentot country in the art of counting fierce words would have been beaten by an American child of kindergarten age who could boast the ability to count up to 10. Nowadays, we’re quite accustomed to the idea that we can write as big a number as we please, whether it is to represent war expenditures, incense or stellar distances in inches by simply setting down a significant number of zeros on the right side of some figure. You can put in zeros until your hand gets tired. And before you know it, you will have a number larger than the following number of atoms in the universe, which identically is a lot.”
That, and that’s what I meant. Nowadays, what’s the largest number you can think of? Infinity. All right, going on and on and on. But this is interesting. “This arithmetic wasn’t known in ancient times. In fact, it was invented less than 2000 years ago by some unknown Indian mathematicians before his great discovery. And this was a great discovery. Although we usually do not realize it, numbers were written by using a specific symbol for each of what we now call decimal units and repeating this symbol as many times as there were units. For example, the number 8,073 was written in ancient Japanese.”
How? What are you even… Okay. Makes sense. So you have, look, you have eight of these, seven of these, three of these. You have eight of these, seven of these, three of these, two of these. Whereas, a clerk in Caesar’s office would have represented it in this form. So remember, this is what Egyptians and this is the Romans. So obviously, Roman numerals, we still use Roman numerals. Think of the Super Bowl.
Well, and think about it. Okay. What he’s saying right here, the only way that you would have been able to write one million is to take the thousand and write that down. Or what is it? A thousand M’s in succession, and then that would have been a million, because a thousand times a thousand is a million. Right?
Psammites or Sand Reckoner, Archimedes says, “There are some who think that the number of sand grains is infinite in multitude. And I mean, by saying not only that which exists about [inaudible 00:06:59], and the rest is Sicily, but all the grains of sand which may be found in all the regions of the earth, whether inhabited or uninhabited. Again, there are some who, without regarding the number as infinite, yet think that no number can be named which is great enough to exceed that, which would designate the number of the earth’s grains of sand.
“And it is clear that those who hold this view, if they imagine a mass made up of sand and other respects as large as the mass of the earth, including in it all the seas and all the hollows of the earth filled up in the height of the highest mountain, would be still more certain that no number could be expressed which would be greater than that needed to represent the grains of sand that’s accumulated. But I will try to show that of the numbers named by me, some exceed not only the numbers of grains of sand, which would make a mass equal in size of the earth, fill in a way to describe, but even equal to the mass of the universe.”
All right. So the way that he did this is kind of the way we do it today. He begins with the largest number that existed in ancient Greek arithmetic, a myriad or a 10,000. Then he introduced a new number, a myriad myriad, a hundred million, which he called an octate or a unit of a second class octate, octates, or 10 to the 16th is called a unit of the third class, octate, octate, octate. A unit of the fourth class of writing of large numbers may seem too trivial a matter to which to devote several pages of a book, but in the time of Archimedes, the finding of a way to write big numbers was a great discovery and important step in the forward to science and mathematics.”
I’m very interested. I am very interested. That is super interesting. And it keeps going on and on and one, playing with the big numbers. I’m glad I found this. I never would’ve thought about that. But if you think about where we came from, that’s true. We talked about writing in a different book this morning. We’re talking about numbers now. Man, just think a thousand years from now, all the things that they’re doing, they’re going to look back at us and be like, “Man, those were ancient people.” You know what I’m saying?
They only used 10% of their brain? Right? Because back here, it seems like they’re only using maybe 5%, maybe 1%, right? They didn’t know how to write. They didn’t know how to read. They didn’t know how to create. They didn’t know how to do large numbers. So let’s say 3% of their brain. So we’re studying off of people like 3% of their brain.
Imagine some of the top smartest people nowadays use 15, maybe 10%. Right? So in 2000 years, hopefully it doesn’t take that long, and people are using 80% of their brain, they’re going to look back at us and just think of us as the Neanderthals. Does that make sense?
An Introduction To Probability And Statistics
No more car videos or car books for just a second. I mean, they’re great. They’re interesting, but no more, at least for two books and then we’ll go back to finishing it up, four more car books. It’s going to be a whole series, the car books.
So we got an Introduction to Probability and Statistics.
We’ve got probability, random variables in there, probability distributions moments in generating functions, multiple random variables, some special distributions limit theorems, samples.
A whole bunch of stuff. But however, let’s look at what is probability? I’ll give you a quick little introduction on there and then random variables in our probability distributions. So we’re going to do a little bit on page 1 in a little bit on page 40. So don’t let me forget. So page UNO.
Okay. The theory of probability had its origin in gambling and games of chance. This is part of the reason why I was like probability. You guys might not know, but if you’re a degenerate, like I am like the gamble or like the stock market then probabilities is really important for you. [inaudible 00:01:48] in gambling and games of chance, it owes much of the curiosity of gamblers who pestered their friends in the mathematical world with all sorts of questions. Unfortunately, this association with gambling contributed to a very slow and sporadic growth of probability theory as a mathematical discipline. The mathematicians of the day took little or no interest in the development of any theory, but looked only at the combinational reasoning involved in each problem. The first attempt at some mathematical rigor is credited to Laplace.
In his monumental work, Theorie analytique des probabilities, so he is Latin or French. I would say that’s more Latin, in 1812. Laplace gave the classical definition of the probability of an event that can occur only in a finite number of ways as a proportion of the number of favorable outcomes to the total number of all possible outcomes, provided that all the outcomes are equally likely. According to this definition, computation of the probability events was reduced to combinatorial counting problems. Even in those today, this definition was found inadequate in addition to being circular and restrictive, it did not answer the question of what probability is. It only gave a practical method of computing, the probabilities of some simple events, all right.
But he did pretty good for 1812, right? And to think it’s even saying that games of chance and gambling was the origin of probability. The first person that came up with thinking of probability was in 1812, right? So gambling and gaming games of chance has been around for a very long time in trying to find out the probability. So, like I said, I’m going to go more into this, but I want to get into 42. And I don’t want these videos to be too long. Right.
Probability for kind of what he was talking about. What is the chances of something happening? If everything is the same, right? So in cards you can count the probability. If you know how many sets, how many cards are. There’s 52 cards is 13 per color or symbol, right? So when you have that finite amount of numbers you can count and you can do mathematical equations. You can figure out the probability of something happening next, right? With the stock market, there’s a million different variables, but in all actuality, there’s only like five or six things that somebody can do. It can go up, it can stay the same, or it can go down right as three things. However, it can go up slow, fast. That’s two. It can stay the same violently or very calm. So either volatile or not volatile, or it can go down very fast or very slow. So off of three things. You have actual six things right now. There’s volume, there’s variables, there’s times of the year. There’s all these different variables. But in all essence, there’s only six things that this can do. It can stay the same. It can go up, it can go down, it can stay the same peacefully. It can stay the same violently. It can go up fast. It can go up slow. It can go down. Slope can go down fast, right? So it’s probable, it’s going to do something now with all the different variables and everything you can calculate using calculations. And then you can come up with formulas and everything to better hypothesize which direction or what it’s going to do. But in the same, you have six different possibilities for it to do, right? I didn’t know all those different variables. Now you have your probabilities now. Now, 40.
Random variables and their probability distributions. Right? And in chapter one, we dealt essentially with random experiments that can be described by, okay, no random variables where does exist. We defined a random variable. And in study two, three, we study the notion of probability distribution of a random vehicle variable. No, just what is a random, okay. Actually in chapter one, we were concerned with such functions without defining term random variable. Here we study the notion of random variable and examine random variables’. In chapter one, we study probabilities of a set function P defined on a sample space. Rome.
I can’t remember what those signals saying. I’ll just show you. You can figure it out. Those two, since P is a set function, it is not easy to handle. We cannot perform arithmetic or algebraic operations on sets more note, moreover in practice one frequently observed some function of elementary events. When a coin is tossed repeatedly, which replication resulted in heads is not of much interest. Rather, one is interested in the number of heads and consequently, the number of tails that appear in, say, N tossings of the coin. It is therefore desirable to introduce a point function on the sample space when we can then use knowledge or calculus or real analysis to study properties of P .
So then we get into a whole bunch of, yeah, baby. Excited to learn a whole bunch of gibberish, right? I mean, it’s not, sorry. Shanaya said it’s gibberish. It might be more advanced for my feeble mind that I have over here because this is a lot of numbers. I don’t know what they mean. This is the introduction. What’s the second part. What I’m saying, this is level one. What’s level five. Look like a whole bunch of gibberish I’ll never understand.
Yes, I will try to decipher as much as the first. What is probability and why is it important? And I will try to decipher as much as the little, the equations as possible. Don’t make very many promises, but I will do my best to make it nice and simple for you to understand in the essay. All right, done with that one. Almost done with this box, man, almost done with this box. This box was massive.