シャドーイング練習: Every Giant Number Explained - 動画で英語スピーキングを学ぶ
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194 文
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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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このレッスンの語彙とスピーキングのポイント
このスピーキングレッスンは、動画「Every Giant Number Explained」を教材にしています。 繰り返し出てくる語は次のとおりです：universe, arrow, Google, atom, colour。 この動画には、シャドーイング用の文が194文、単語が1954語あります。 音声の長さは12:13です。 話す速さは1分あたり約160語で、日常会話に近い自然なペースです。 英語の頻出3,000語に含まれる単語は81%だけなので、語彙は難しめです。
この動画の重要語彙
動画に出てくる覚えておきたい単語15語を、発音と意味つきで紹介します。
| 単語 | 発音 | 意味 |
|---|---|---|
| arrow 名詞 | /ˈæɹəʊ/ | 矢 |
| atom 名詞 | /ˈætəm/ | 原子, アトム |
| digit 名詞 | /ˈdɪd͡ʒɪt/ | 数字 |
| infinity 名詞 | /ɪnˈfɪnɪti/ | 無限 |
| formula 名詞 | /ˈfɔː.mjʊ.lə/ | 公式, 定式 |
| ordinary 形容詞 | /ˈɔɹdɪnɛɹi/ | 通常な, 普通な |
| dimension 名詞 | /daɪˈmɛn.ʃən/ | 面積, 寸法 |
| cube 名詞 | /kjub/ | 立方体, 正六面体 |
| connect 動詞 | /kəˈnɛkt/ | 繋げる |
| symbol 名詞 | /ˈsɪmbəl/ | 記号, 符号 |
| triple 形容詞 | /ˈtɹɪp.əl/ | 三重 |
| estimate 名詞 | /ˈɛs.tɪ.mət/ | 推定 |
| enormous 形容詞 | /ɪˈnɔː.məs/ | 巨大な |
| mathematics 名詞 | /mæθ(.ə)ˈmæt.ɪks/ | 数学 |
| ladder 名詞 | /ˈlæd.ɚ/ | 梯子 |
動画に出てくる句動詞
| 単語 | 発音 | 意味 |
|---|---|---|
| write down 動詞 | 書き留める, 記す | |
| come out 動詞 | /ˌkʌm ˈaʊt/ | 出る |
| point out 動詞 | 指す, 指し示す |
繰り返し練習したい文
動画に出てくる短く完結した文です。日常会話でそのまま使えます。
- You'd think it would always be that way.
- It feels like we've gone very far.
- We haven't gone far at all.
この動画の文法
話し手がよく使っている文型を、動画の実際の表現とともに紹介します。
| 文型 | 動画での表現 |
|---|---|
| 受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる | was invented · been found · can be computed |
| 現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している | have had · hasn't been found · we've gone |
注意したい発音
話し手はisn't, you'd, can'tなど、短縮形や弱形を23回使っています。聞こえたとおりの短い形で発音しましょう。
- 「th」の音: mathematics /mæθ(.ə)ˈmæt.ɪks/, sixth /sɪksθ/, thirty /ˈθɝti/, ninth /naɪnθ/, theorem /ˈθiərəm/
- 「sh」と「zh」の音: dimension /daɪˈmɛn.ʃən/, sheet /ʃit/, combinations /kɑmbɪˈneɪʃənz/, logician /ləˈd͡ʒɪʃən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- 長い単語（アクセントの位置に注意）: infinity /ɪnˈfɪnɪti/, ordinary /ˈɔɹdɪnɛɹi/, infinitely /ˈɪnfɪnɪtli/, mathematics /mæθ(.ə)ˈmæt.ɪks/, practically /ˈpɹæk.tɪ.k(ə.)li/
日本語話者が苦手な音：
- /r/ と /l/ の区別 — /l/ は舌先を歯茎につけ、/r/ はどこにもつけない: roughly /ˈɹʌf.li/, triple /ˈtɹɪp.əl/, carl /kɑɹl̩/, practically /ˈpɹæk.tɪ.k(ə.)li/
- /v/ — /b/ にならないように、上の歯を下唇に当てる: server /ˈsɝvɚ/, divide /dɪˈvaɪd/, whichever /wɪt͡ʃˈɛvɚ/, invent /ɪnˈvɛnt/, observable /əbˈzɜː.və.bəl/
- /f/ — 「フ」ではなく、上の歯と下唇で出す: infinity /ɪnˈfɪnɪti/, formula /ˈfɔː.mjʊ.lə/, flip /flɪp/, infinitely /ˈɪnfɪnɪtli/, roughly /ˈɹʌf.li/
この動画での練習方法
- まず声を出さずに動画を最後まで聞き、知らない単語をメモします。
- まず0.75倍速で一文ずつシャドーイングし、慣れてきたら通常の速度に戻します。
- 自分の声を録音して元の音声と比べます。arrow, atom, digitなどの単語に特に注意しましょう。
シャドーイングとは？英語上達に効果的な理由
シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。