A couple of maths-related questions..

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No. There's no last number. There are infinitely many counting-numbers (positive integers). Likewise of course there are infinitely-many integers, infinitely many rational numbers, and infinitely-many real numbers.

Cantor showed that the infinity of real numbers is greater than the infinity of integers.



No, a human is made of a lot more than a billion atoms.

A 6 foot (~2 meter) human weighs about a billion times as much as a 2 millimeter ant.

Michael829

No. You wouldn't need a number 1000 digits (in base 10) for all of the atoms in the known Universe.

A number 85 digits long would suffice. It is a bit hard to believe but that's the consensus (actually the latest wiki says between 10^78 and 10^82).

If one person is made up of a billion atoms that's one time ten to the ninth. A thousand persons would be to 12th power, the whole seven billion strong human race would be 10 to the 19th.

Off the top of my head I don't know the mass of the earth. But lessee an adult human is the equivalent of a ball three feet in diameter. The Earth is a ball 40 million feet in diameter. So the earth is 10e7 times as wide as an adult human, or (10e7) cubed the volume of a human. Or 10E10 the volume of a human. The earth is denser than a human, but lets say they are the same density (the same weight per unit size). So that would mean that the earth has something like ten followed by ten zeros as many atoms as a human.

So when you go up from the size of one human to the size of the planet, yes the number goes up hugely, but the number of zeroes you need to tack on to the number to write it does NOT go up as much as you might think.

And so it goes up the cosmic scale.

Actually MIcheal is right. Not only is an adult human several hundred million times the size of an ant, but an adult human has about 50 trillion cells. So we must have more than a billion atoms if we have 50 thousand times a billion cells.

Huge numbers just give me a headache. They confuse me. Sometimes I wonder if the world exaggerates.

Like the other day I was watching a TV programme about cats, and they said something like they save 4000 abandoned cats a year in just England. 4000 is a massive number. There are only 365 days per year, so they must save...um...how many a day? (I can't work it out).

Then during fire training at work once, the tutor said that 1000 care homes catch fire a year, in the UK. I didn't even know fires in care homes were that common. It puts me off working in the care home where I work. Even 100 care homes a year experiencing a fire would throw me, so 1000 just gives me a headache.

But with the card shuffling thing, there is no way a pack of cards has several billion different orders. I think that 'fact' is just made up.

I assure you that it's a lot more than a few billion.

It's a trillion trillion trillion trillion trillion, multiplied by ten million. ...times 8.

That means a trillion multiplied by itself all those times. ...and then multiplied by 80 million.

Each trillion is 1000 billions

(I'm using the U.S. system, in which a billion is a thousand million, and a trillion is a million million)

That number can be written 8E67

One way to verify that it's true: Ask teachers of math or science. You'll find that you pretty much always get that same answer: 8E67

But let me give you a number bigger than that. You know those scan-arrays, on posters, packages, and so on? An array consisting of a pattern of black squares? You scan it, and that takes you to a website. You know what I mean. I don't know what they're called.

Anyway, it's a square array, 35 millimeters to a side, divided into little square regions each of which is about 1 millimeter to a side.

That means that the total number of little squares in the array is 35X35 =1225.

Each of those little squares can be black or white. Well, each possible configuration of black and white little squares could send you to a different website. How many possible configurations of black squares and white squares are possible in that array of little squares?

2 to the power of 1225. How much is that?:

About 5.78E368

You could round that to about 6E368

That's more than 8E67 to the 5th power. That is, 8E67 multiplied by itself 5 times.

In other words, it's a lot larger number than the already huge number of ways a deck of cards can be shuffled.

Michael Ossipoff

I wrote out an explanation of this yesterday, but cloud flare obliterated my explanation. I gave up posting here for the day out of sheer disgust with cloud flare. (I think the flare part should be changed to something else that begins with an 'f'.)

I may try again later.

I emphasize that the 80 million trillion trillion trillion trillion trillion orders in which a deck of cards can shuffle is incomparably more than the typically-estimated billion trillion or so grains of sands on the worlds beaches.

There was a story about someone who'd done a favor for a king, and the king, as a reward, offered to grant any reasonable request. The man asked for one grain of wheat for the first square on a chessboard, two grains for the 2nd square, 4 grains for the 3rd square, etc., doubling the added amount for each next square.

A chessboard has 64 squares.

At first it was possible to put one grain on the 1st square, 2 on the next one, 4 on the next one, but after a while it was necessary to use a cup or bowl, and then a pot, and then a bushel basket, and then a wagon, and soon a whole caravan of wagons (obviously by this time they weren't counting individual grains, but were only doubling volumes).

By the time they got to the last square, the king owed the man so much grain that the man became the owner of the kingdom.

It's mathematically plausible.

I think the grain as wheat, but I'm not absolutely sure.

But let's be extra-conservative, and assume that each grain, of whatever grain it is, takes up only a cubic millimeter, when the grain is packed together.

The number of grains that are given for the last square on the chessboard is 2 the the 64th power.

That's about 1.84E19

1.84 times 10 the the 19th power.

That's 18 million trillion.

That's about a thousand times less than that typical estimate of the number of grains of sand on all the world's beaches. But it's still a lot. It would fill a cube that's 2.64 kilometers to a side. That's 164 miles to a side.

But what if the grain were spread out one foot thick on the ground? Then it would cover a square piece of land about 119 miles to a side. But, in an actual grain-field, there isn't nearly as much grain per square foot, as there'd be if it were in a one-foot-thick layer. And so the size of that piece of land, as a grain-field, would have to be a lot more than 119 miles to a side. A lot more. It would probably cover the entire kingdom, and then some.

...and that's if each of these grains occupies only a cubic millimeter, when the grain is in a pile. A grain of wheat is bigger than that.

...and that's just counting the last square.


Michael829

I guess some folks are just more at home with numbers than others.

In a country with as big a population as Britain (60 million humans) it wouldn't phase me at all to learn that they rescue ONLY 4000 stray cats a year. That's 4000 spread out over a land area of 90 thousand square miles (the size of Britain). So its one cat a year for every 25 square miles. Or its one cat a year for every 15 thousand people in the nation. Neither seems like a lot at all. Both sound about right. Even a bit low.

And if it 'gives you a head ache' to figure out "how many per day 4000 cats a year is" just do this: the 365 days in a year is basically 400 (rounded to the nearest hundred). And then think "400 goes into 4000 ten times, so its like ten or eleven cats a day". Close enough to get the picture, and easy as pie.

I do admit that I am a BIT surprised that a thousand nursing homes catch fire every year in the UK.

But there are probably a lot more assisted living homes in industrial countries than either of us realizes. So in proportion to the total actual number of assisted living and nursing homes its probably nothing to be alarmed about.

But if you're willing to settle for a few billion possible orders, then you can get by with only 14 cards.

14 cards can be arranged in 87 billion orders.

Say you start a stack of cards on the table by putting down only one card.

The first card of course can only go in one place in the order, because it's by itself.

The 2nd card has two places it can go. It can go above or below the 1st card.

The 3rd card has 3 places it can go. It can go at top, bottom or middle.

The 4th card has 4 places it can go.Above any of the previous 3 cards, or at the bottom.

The 5th card has 5 places it can go. Above any of the previous 4 cards, or at the bottom.

Each Nth card has N places it can go.

Do that all the way to the 52nd card, multiplying together all of those numbers of places where the card can go.

...because if there are 2 ways to arrange 2 cards, and 3 places to put the 3rd card, then that makes 2X3 possible arrangements.

So, you multiply together all of the numbers from 1 to 52.

That's called 52 factorial.

A scientific calculator can give factorials. Just enter 52, and then press the key, or key-sequence, that will do the factorial operation on the number in the display. You'll get 8E67.

...for 52 cards.

But, if you're willing to settle for 87 billion, then you could just multiply the numbers from 1 to 14, or (the same thing) from 14 to 1. You'll get about 87 billion.

Michael829

The number of combinations for each number (its factorial) increases very quickly. If you have 3 items, there are 6 possible ways you can arrange them. If you have 4 items, there are 24 possibilities. For 5 it's 120, for 6 it's 720, and for 7 it's 5040. Given how fast it grows, it's not so hard to believe that the number of possibilities for 52 is so high.

To reiterate what the above three are saying you can do the experiment yourself.

Take three cards out of pack and put them on table and count how many ways they can be lined up.
You will get 6 possible ways.

Add a fourth cards and the number of possibilities goes up to 24.

Add a fifth and the number of ways you can line them up goes up to 120.

And so on. It gets astronomic very fast. LONG before you get to the full pack of 52.

Let me expand a little on my suggestion, in my previous post, to multiply the numbers from 1 to 14, to show that the result will be 87 billion It's easier than it might sound. It will only take a few minutes. (...considering how small most of the numbers from 1 to 14 are)

Suppose that a scientific calculator isn't available. Say you have an ordinary calculator. Sure, it's display doesn't have the 11 digits that are needed to display 87 billion. No problem:

Just multiply the numbers from 1 to 10. That's a good round number at which to temporarily stop.

Now divide by 1,000,000 Of course that's 1 followed by six zeros.

Now continue the multiplications. You've already multiplied the numbers from 1 to 10.

So now you multiply what's in the display by the numbers from 11 to 14.

You'll get about 87,000

Because you've divided by a million, the fact that the display says 87,000 means that the product of multiplying all those numbers is really 87 billion.

So try it yourself, with an ordinary calculator, if a scientific calculator isn't available.

Michael829

I sort of get it now, thanks for the explanations. I never thought math could be so interesting.

I do believe now that a pack of 52 cards can be shuffled 87 billion times, after reading the cool posts in this thread. But I still don't believe that a pack of cards can be shuffled more times than every grain of sand on every beach in the entire world, because there's more than 87 billion grains of sand on this Earth.

OK 87 billion is a painfully huge number, but the world is HUGER than I can imagine, so 87 billion is too small to cover every grain of sand on this enormous planet of our's. Yes I know Earth is small compared to most the other planets in our solar system, but it still feels enormous to us. It's a bit like saying there are more orders to shuffle a pack of cards than there are blades of grass on the whole planet. No way, right?

Also I still don't understand how the numbers can go on forever. People have told me the infinity theory but that didn't explain much. I mean, supposing you had a list of every person in the world, how many orders can you list that?
Probably a trillion trillion trillion trillion trillion times.

14 cards can be shuffled in 87 billion orders. Multiply the numbers from 1 to 14. (but when you've multiplied from 1 to 10, divide by 1,000,000 before continuing the multiplications from 11 to 14.) The whole product will be 1,000,000 times the number in the display. 87,000 is in the display when you're done, so the product = 87 billion.

52 cards can be shuffled in 80 million trillion trillion trillion trillion trillion orders.

That's so much bigger than the number of grains of sand on all the world's beaches, that the comparison isn't even close. The number of grains of sand on all the world's beaches is only about a billion trillion. (That's my estimate, and I haven't seen one much larger).



The number of ways to shuffle 52 cards is much greater than the number of blades of grass on the Earth.

With only about a billion trillion brains of sand on the beaches, how much greater than that can the number of blades of grass be?

80 million trillion trillion trillion trillion trillion is so many time bigger than a billion trillion, that there's no chance that the number of blades of grass on the Earth could be even close to the number of orders for 52 cards.



A lot more than that.

80 million trillion trillion trillion trillion trillion is the number of orders for 52 cards.

So the number of order for a list of all the people in the world, will be hugely larger than that.

As others have pointed out, for any number that can be named, you can add 1 to that number. So that means that there's no number that's the largest, because you can add 1 to any number.

Michael829

Yeah, the concepts of infinity and eternity boggle my mind. But nor do I understand how space or time could have an end.

In our lives, time doesn't have an end.

At a person's end of lives (or at the end of this life if there's no reincarnation), there's a well-deserved peaceful timeless sleep.

But of course you never experience a time when you don't experience.

Only your survivors experience the time after your complete shutdown.

So a person, even when they die, never reaches oblivion. They just go into sleep, but never experience a time when they don't experience.

And, even at the beginning of the arrival of that sleep, the person has no idea that there ever was, or could have been such things as worldly life, body, identity, time, or events--or trouble, misfortune, need, want or lack.

Michael829