Shadowing Practice: Every Giant Number Explained - Learn English Speaking with Video
בחר מצב תרגול
כל מצב פותח חדר תרגול במסך מלא.
194 משפטים
אם המשפטים קצרים או ארוכים מדי, לחץ על ערוך כדי להתאים אותם.
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If you wrote one digit on every atom in the observable universe,
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you'd run out of atoms before you finished writing one of the numbers in this video.
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And that number isn't even the biggest one here.
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But let's start small.
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A million, a billion, a trillion.
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They sound like neighbours, but each one is a thousand times bigger than the last, and the easiest way to feel that is in seconds.
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A million seconds is a little over 11 days.
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A billion seconds is almost 32 years.
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And a trillion seconds is about 32 ,000 years.
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That's six times longer than humans have had writing.
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32 ,000 years ago, the world was in an ice age and people were hunting mammoths.
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So counting with words gets awkward fast.
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A million is a one followed by six zeros.
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Short version, ten to the sixth.
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A billion is ten to the ninth.
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A trillion, ten to the twelfth.
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Every three zeros get a new word.
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Quadrillion, quintillion, all the way up to a decillion, ten to the thirty -third.
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The words run out soon.
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The numbers don't. In 1920,
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the American mathematician Edward Kasner asked his nine -year -old nephew what to call a one followed by a hundred zeros.
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The boy, whose name was Milton, said, A Google.
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A Google is ten to the hundredth.
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It's easy to write down.
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It's one line.
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But what can you count with it?
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Atoms In the entire observable universe, in all the stars, planets and gas clouds, there are about 10 to the 80th.
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That's less than a Google.
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Not a little less, a hundred quintillion times less.
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Imagine every atom in the universe turned out to be a whole universe full of atoms.
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Now you'd have more than a Google.
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But in our real universe, there just aren't that many things.
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There are that many possibilities, though.
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In 1950, Claude Shannon estimated how many different chess games there could be.
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He got about 10 to the 120th, more than a Google.
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Combinations grow much faster than stuff.
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By the way, one company's name is a misspelling of Google, and its servers still live at the domain 1E100,
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which is how programmers write a Google.
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Milton came up with the next word too.
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A Googleplex, he said, is a one followed by writing zeros until you get tired.
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Casno pointed out that different people get tired at different times, so he made it precise.
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A Googleplex is a one followed by a Google zeros, ten to the power of a Google.
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And this is where the paper runs out.
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To write a Googleplex, you need to write a Google zeros.
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Even if you put one zero on every atom in the universe, you wouldn't have enough atoms there are only 10 to the 80th of them,
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and you need 10 to the 100th zeros.
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If you printed a million zeros per book, you'd need 10 to the 94th books.
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That's more books than there are atoms in the universe.
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Carl Sagan made this point in Cosmos back in 1980.
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There isn't room for a Googleplex in the known universe.
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But numbers like this do show up in physics.
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If the universe is infinite, then somewhere out there is an exact copy of Earth, and an exact copy of you watching this video.
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The physicist Max Tegmark estimated how far away, roughly 10 to the 10 to the 28th miles.
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And at that size, it doesn't matter whether you say miles, feet or inches, the difference gets lost somewhere in the last digits of the exponent.
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And a Googleplex is bigger still, much bigger.
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The Googleplex was invented as a joke, but once an even bigger number was needed for real.
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Prime numbers divide only by themselves and 1, 2, 3, 5, 7, 11.
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The further you go, the rarer they get.
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Gauss came up with a formula that roughly tells you how many primes there are below any number.
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People checked it for billions, for trillions, for numbers with 19 zeros.
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And every time, the formula came out a little too high.
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Every single time.
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You'd think it would always be that way.
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But in 1914, John Littlewood proved it isn't.
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Somewhere far out, the formula starts missing in the other direction, and it flips back and forth infinitely many times.
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Where exactly, he didn't say.
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In 1933, his student Stanley Skews found a limit.
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The first flip has to happen before 10 to the 10 to the 10 to the 34th.
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Three stories of exponents.
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G. H.
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Hardy called it the largest number that had ever served any definite purpose in mathematics.
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Today we know the first flip happens much earlier, somewhere around 10 to the 316th.
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But the exact spot still hasn't been found.
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And that's the lesson.
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A pattern can hold for every number we can reach and still be false.
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To go further, we need a new way of writing.
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and it's built like a ladder.
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Adding three to itself three times is multiplication.
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Multiplying three by itself three times is a power, and if you raise three to a power and then raise three to that result,
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you get a tower of powers.
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In 1976, Donald Newth suggested arrows for this.
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One arrow is an ordinary power.
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Three arrow three is 27.
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Two arrows make a tower.
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3 double arrow 3 is 3 to the 3 to the 3.
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That's 3 to the 27th.
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7 ,625 ,000 ,000 ,000 ,000 and change.
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Not scary yet, but add one more story to the tower.
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3 to the power of that number.
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The result has more than 3 .5 trillion digits.
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Now 3 arrows.
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3 triple arrow.
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3 is a tower of 3's more than 7 .5 trillion stories tall.
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Not that many digits.
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that many stories.
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For comparison, a Googleplex fits in a tower of just four tens and skews his number in a tower of five.
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Four stories of tens and the number already has more digits than there are atoms in the universe.
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Each new arrow repeats the previous one as many times as the last step produced,
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and these are the arrows you need to describe the next number.
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Take a cube and connect every corner to every other corner.
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Colour each line red or blue.
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Now look for four corners that lie in one plane with all six lines between them the same colour.
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With an ordinary cube, you can colour the lines so that no such four exist.
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But what about a cube in more dimensions?
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From which dimension on is there always such a group of four, no matter how you colour?
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In 1971, Ronald Graham and Bruce Rothschild proved that such a dimension exists.
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And in 1977, Martin Gardner wrote about a number
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Graham had found as an upper bound for the answer.
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Three years later, it was in the Guinness Book of World Records as the largest number ever used in a serious proof.
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Here's how it's built.
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Take three, four arrows, three.
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That's step one, and it's already monstrously bigger than anything so far.
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In step two, there are as many arrows between the threes as step one produced.
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In step 3, as many as step 2.
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And so on, 64 times.
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Physicists say a region the size of your head can hold only a limited amount of information.
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Cram in more, and that region becomes a black hole.
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The digits of Graham's number won't fit in your head.
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They won't fit in the whole universe.
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And yet we know how it ends.
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Its last digits are 2464195387.
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And the actual answer to the problem?
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Today we know it's at least 13.
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The answer might be a small number.
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Only the bound was enormous.
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Graham's number came from a problem.
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The next one comes from a game you can play on a sheet of paper.
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You draw trees.
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Dots connected by lines, with no loops.
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Each dot can be one of several colours.
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The first tree can have at most one dot.
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The second, at most two.
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The third, at most three.
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And so on.
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The main rule?
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No earlier tree may fit inside a later one with the same colours and the same branching.
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As soon as you can't draw a new tree, the game is over.
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In 1960, Joseph Kruskal proved a theorem that means this game always ends, no matter how many colours you use.
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With one colour, the game lasts one move.
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With two colours, three moves.
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And for three colours, the logician Harvey Friedman called the length of the longest possible game tree of three.
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It's finite.
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The game really does end.
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But next to tree of three, Graham's number is practically zero.
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Friedman showed that if you try to prove this game ends using modest arithmetic,
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the proof itself would be longer than anything you could write down in the universe.
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Three colours and simple rules.
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And the result is a number that makes all of Knuth's arrows look like toys.
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Every number so far can be computed.
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Slowly, longer than the life of the universe, but by the rules.
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And then there are numbers that can't be computed at all.
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Take the simplest computer, a Turing machine.
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It's an endless tape of cells and a head that reads a symbol, writes a new one, and moves, following a table of rules.
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Back in 1936, Turing proved that there's no way to tell in advance, for every program, whether it will ever stop or run forever.
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In 1962, Tibor Rado came up with a game called The Busy Beaver.
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Take all machines with a given number of rules and run them on a blank tape.
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Among the ones that stop, find the one that runs the longest.
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With one rule, one step.
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With two, six.
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With three, twenty -one one, with four a hundred
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and seven with five forty seven million one hundred seventy six thousand eight hundred seventy steps
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that was only proven in july 2024
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and it was done mostly by amateurs working together online with six rules in 2025 someone found a machine
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that runs longer than two triple arrow five this function grows faster than any function
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that can be computed sooner or later it overtakes the arrows, graham and tree.
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And for a machine with 215 rules, the answer can no longer be proven with ordinary mathematics.
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That was shown in October 2026.
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There's one last step.
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In 2007, at MIT, two philosophers held a duel.
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Who can name the bigger number?
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Agustin Rayo won like this.
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Take every number that can be described in the language of logic with no more than a Google symbols.
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Rayo's number is the smallest number bigger than all of them.
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A million seconds, the atoms in the universe, a Google, a Googleplex, Skews,
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the Arrows, Graham, Tree of Three, the Beaver, Rayo.
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It feels like we've gone very far.
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But here's the strange part.
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Whichever of these numbers you pick, there are infinitely many numbers bigger than it, and only finitely many smaller.
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So almost all whole numbers are bigger than Rayo's number.
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We haven't gone far at all.
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We're standing at the very start of the number line.
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So is the biggest number infinity?
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No. Infinity isn't the last number in line.
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It answers a different question.
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Not which number, but how many numbers there are in total.
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And that question has a ladder too.
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Georg Cantor showed that infinities come in different sizes.
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There are infinitely many whole numbers, but there are even more points on a line.
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And above that infinity, there's another one, and so on, forever.
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And any number in this video, even Rayo's number, is no better than one when you put it next to infinity.
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Enormous, but finite.
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We started with the idea that the universe doesn't have enough atoms to write down one number.
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Now it's clear the problem isn't the universe.
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Numbers simply grow faster than any place you could write them down.
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אוצר מילים והערות הגייה לשיעור הזה
שיעור הדיבור הזה מבוסס על הסרטון “Every Giant Number Explained”. המילים שחוזרות הכי הרבה: universe, arrow, Google, atom, colour. בסרטון הזה יש 194 משפטים ו-1954 מילים לתרגול Shadowing. הדיבור נמשך 12:13. הדובר מדבר בקצב טבעי של כ-160 מילים בדקה, קרוב לשיחה יומיומית. רק 81% מהמילים נמצאות בין 3,000 המילים הנפוצות ביותר באנגלית, כך שאוצר המילים מאתגר.
אוצר מילים מרכזי בסרטון הזה
15 מילים מהסרטון שכדאי ללמוד, עם ההגייה והמשמעות שלהן:
| מילה | הגייה | משמעות |
|---|---|---|
| arrow שם עצם | /ˈæɹəʊ/ | חֵץ |
| atom שם עצם | /ˈætəm/ | אָטוֹם |
| digit שם עצם | /ˈdɪd͡ʒɪt/ | סִפְרָה |
| infinity שם עצם | /ɪnˈfɪnɪti/ | אֵינְסוֹף |
| formula שם עצם | /ˈfɔː.mjʊ.lə/ | נוסחה \ נֻסְחָה |
| ordinary שם תואר | /ˈɔɹdɪnɛɹi/ | רָגִיל |
| dimension שם עצם | /daɪˈmɛn.ʃən/ | מֵמַד |
| cube שם עצם | /kjub/ | קובייה, קֻבִּיָּה |
| connect פועל | /kəˈnɛkt/ | חיבר |
| symbol שם עצם | /ˈsɪmbəl/ | סֵמֶל |
| mathematics שם עצם | /mæθ(.ə)ˈmæt.ɪks/ | מָתֵמָטִיקָה |
| ladder שם עצם | /ˈlæd.ɚ/ | סולם, סֻלָּם |
| lesson שם עצם | /ˈlɛs.ən/ | שיעור, שִׁעוּר |
| cloud שם עצם | /ˈklaʊ̯d/ | עָנָן, עָב |
| server שם עצם | /ˈsɝvɚ/ | שָׁרָת |
Phrasal verbs שתשמעו
| מילה | הגייה | משמעות |
|---|---|---|
| write down פועל | רָשַׁם | |
| come out פועל | /ˌkʌm ˈaʊt/ | יָצָא |
משפטים שכדאי לחזור עליהם
משפטים קצרים ושלמים מהסרטון שאפשר להשתמש בהם בשיחה יומיומית:
- You'd think it would always be that way.
- It feels like we've gone very far.
- We haven't gone far at all.
דקדוק בסרטון הזה
המבנים שהדובר משתמש בהם הכי הרבה, עם המילים המדויקות מהסרטון:
| מבנה | בסרטון |
|---|---|
| סביל (Passive voice) be + past participle — הדגש הוא על מה שקורה, לא על מי שעושה את הפעולה | was invented · been found · can be computed |
| הווה מושלם (Present perfect) have/has + past participle — פעולה שקרתה בעבר ועדיין יש לה חשיבות או השפעה עכשיו | have had · hasn't been found · we've gone |
הגייה ששווה לשים לב אליה
הדובר משתמש ב-23 קיצורים וצורות מקוצרות, כמו isn't, you'd, can't. אמרו אותם בצורה הקצרה, כפי שאתם שומעים אותם.
- הצלילים של “th”: mathematics /mæθ(.ə)ˈmæt.ɪks/, sixth /sɪksθ/, thirty /ˈθɝti/, ninth /naɪnθ/, theorem /ˈθiərəm/
- הצלילים של “sh” ו-“zh”: dimension /daɪˈmɛn.ʃən/, sheet /ʃit/, combinations /kɑmbɪˈneɪʃənz/, logician /ləˈd͡ʒɪʃən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- מילים ארוכות — שימו לב להטעמה: infinity /ɪnˈfɪnɪti/, ordinary /ˈɔɹdɪnɛɹi/, infinitely /ˈɪnfɪnɪtli/, mathematics /mæθ(.ə)ˈmæt.ɪks/, practically /ˈpɹæk.tɪ.k(ə.)li/
צלילים שדוברי עברית מתקשים בהם:
- /w/ — שפתיים מעוגלות, לא /v/: awkward /ˈɔk.wɚd/, whichever /wɪt͡ʃˈɛvɚ/, twelfth /twɛlfθ/ [tʰw̥ɛɫfθ]
- /æ/ — פתוח יותר מה-“אֶ” העברית, לא כמו “a” ב-“cup”: arrow /ˈæɹəʊ/, atom /ˈætəm/, graham /ɡɹæm/, mathematics /mæθ(.ə)ˈmæt.ɪks/, ladder /ˈlæd.ɚ/
- /r/ — הלשון לא נוגעת בשום מקום, לא ה-“ר” הגרונית: arrow /ˈæɹəʊ/, graham /ɡɹæm/, ordinary /ˈɔɹdɪnɛɹi/, roughly /ˈɹʌf.li/, triple /ˈtɹɪp.əl/
איך לתרגל עם הסרטון הזה
- הקשיבו לכל הסרטון פעם אחת בלי לדבר, ורשמו את המילים שאתם לא מכירים.
- התחילו במהירות 0.75×, תרגלו Shadowing משפט אחר משפט, וחזרו למהירות רגילה כשזה מרגיש קל.
- הקליטו את עצמכם והשוו למקור, תוך תשומת לב למילים כמו arrow, atom, digit.
מהי טכניקת השאדואינג?
שאדואינג היא טכניקת למידת שפה מבוססת מדע שפותחה במקור להכשרת מתורגמנים מקצועיים ופופולרית על ידי הפוליגלוט ד"ר אלכסנדר ארגואלס. השיטה פשוטה אך עוצמתית: אתה מקשיב לאודיו באנגלית ילידית וחוזר עליו בקול רם מיד — כמו צל העוקב אחרי הדובר עם עיכוב של 1–2 שניות בלבד. בניגוד להאזנה פסיבית או תרגילי דקדוק, שאדואינג מכריח את המוח והשרירים בפה לעבד ולשחזר דפוסי דיבור אמיתיים בו זמנית. מחקרים מראים שזה משפר משמעותית את דיוק ההגייה, האינטונציה, הקצב, הדיבור המחובר, הבנת השמיעה ושטף הדיבור — מה שהופך אותו לאחת השיטות היעילות ביותר להכנה לIELTS Speaking ולתקשורת באנגלית בעולם האמיתי.