Carl Sagan on atoms, the googol, and the googolplexShow Video Details ↓ [music] …Carl Sagan: If you wish to make an apple pie from scratch, you must first invent the universe. Thank you very much. Suppose I cut a piece out of this apple pie [music] … … … Crumbly, but good. And now suppose we cut this piece in half, more or less. And then cut this piece in half and keep going. How many cuts before we get down to an individual atom? The answer is about 90 successive cuts. Of course, this knife isn't sharp enough, the pie is too crumbly and an atom is too small to see in any case. But there is a way to do it. It was here, at Cambridge University in England that the nature of the atom was first understood, in part by shooting pieces of atoms at atoms and seeing how they bounce off. A typical atom is surrounded by a kind of cloud of electrons. The electrons are electrically charged as the name suggests and they determine the chemical properties of the atom. For example, the glitter of gold or the transparency of the solid that's made from the atoms silicon and oxygen. But deep inside the atom, hidden far beneath the outer electron cloud, is the nucleus composed chiefly of protons and neutrons. Atoms are very small. A hundred million of them, end to end, would be about so big and the nucleus is a hundred thousand times smaller still. Nevertheless, most of the mass in an atom is in the nucleus. The electrons are by comparison just bits of moving fluff. Atoms are mainly empty space. Matter is composed chiefly of nothing. [silence] When we consider cutting this apple pie, but down beyond a single atom, we confront an infinity of the very small. And when we look up at the night sky, we confront an infinity of the very large. These infinities are among the most awesome of human ideas. They represent an unending regress which goes on, not just very far, but forever. Have you ever stood between 2 parallel mirrors in a barber shop, say, and seen a very large number of you? Or, you could use 2 flat mirrors and a candle flame. You would see a large number of images each the reflection of another image. You can't really see an infinity of images because the mirrors are not perfectly flat and aligned and there's a candle or candle flame at least in the way and light doesn't travel infinitely fast. When we talk of real infinities, we're talking about a quantity larger than any number, no matter what number you have in mind, infinity is larger. [silence] There's a nice way to write large numbers. You can write the number 1000 as 10 to the power 3, meaning a 1 followed by 3 zeros, or a million is written as 10 to the power 6, meaning a 1 followed by 6 zeros. There's no largest number; if anybody gives you a candidate largest number you can always add the number 1 to it. But there certainly are very big numbers. The American mathematician Edward Casner once asked his young nephew to invent a name for an extremely large number: 10 to the power of 100, which I can't write out all the zeros on this page because there isn't room on the page. The boy called it a googol. If you think of googol as large, consider a googolplex. It's 10 to the power of a googol, that is a 1 followed, not by a hundred zeros, but by a googol zeros. Now, by comparison with these enormous numbers, the total number of atoms in that apple pie is only about 10 to the 26, tiny compared to a googol and of course much much less than a googolplex. The total number of elementary particles, protons, neutrons, and electrons in the accessible universe is of the order of 10 to the 80th, a 1 followed by 80 zeros. Still much much less than a googol and vastly less than a googolplex. And yet, these numbers, the googol and the googolplex do not approach, they come nowhere near, infinity. In fact, a googolplex is precisely as far from infinity as is the number 1. We started to write out a googolplex but it wasn't easy. It's a very big number. [music] … … … … … Writing out a googolplex is a spectacularly futile exercise. A piece of paper large enough to contain all the zeros in a googolplex couldn't be stuffed into the known universe. [music] Fortunately, there's a much simpler and more concise way to write a googolplex. [silence] [noise] Like this. And, infinity can be represented like this. |