is 2 to the 49th. power really equal more than all the grains of sand in the world?

Question

someone once told me....

Answer 1

2^49 is 5.63 x 10^14.

I'm sure there is more grains of sand on one beach.

Answer 2

i dunno.. lets do an "order of magnitude calculation"
the mass of the earth is about 6X10^24 kilograms,
which is pretty heavy.
lets suppose that the mass of a grain of sand is 1 miligram

if the earth was made out of grains of sand there would be
6X10^30 of them.

2^49 is bigger to 8^16, which is smaller than 10^16... so... 2^49 and 10^16 are about the same bigness.

so, uh, if the world were made of sand, there would be a lot more sand than 2^49 grains.

of course, our guess that the world is made of sand is a huge overestimate. lets zoom in a little more.

let just suppose that the outer 1 km or so of the earth was made of sand. about how much sand would that be?

the earth is about 6000 km in radius, a sphere with a r= 6000 km radius will have a volume (3/4)pi*r^3= 508.68 x 10^9

a shell which is 1 km thick and with a radius of 6000 km will have a volume (3/4)*pi*(r^3-(r-1)^3)
so the % of the total mass in the shell will be

volume shell / total volume earth
( r^3 - (r-1)^3 ) / r^3

which simplifies to

1 - ( (r-1)/r )^3
1 - ( 5999/6000)^3
= 0.0005

(if we wanted to make the shell a meter instead of a kilometer... dividing by a thousand is probably a good guestimate)

so the total mass of the sand on the outside of the earth is
0.00017 * 6X10^24 = 10X^21 kg

if the mass of a grain of sand is about a miligram, the earth in our new estimation will have about 10 x 27 grains.

so, if we broaden our estimation a little, and say that the sand is anywhere from 1 meter to a kilometer thick, then there should be somewhere between 10X27 to 10X24 grains,
and if we say that the average grain of sand has a mass somewhere between 1 mg and 1 gram, then there will be somewhere between 10X27 to 10X21 grains...

which is still a hundred thousand times more sand than
the 10^ 16 that we estimated 2^49 to be.

what else could be throwing our estimation off?
1) our guess that sand has the same average density as the earth:

density of earth: 6X10^30 miligrams /5x 10^29 mm^3
which is about = 10 mg/ mm^3
which is about how much we've estimated our sand to weigh (we guessed that each grain weighed between a miligram and a gram, and it's probably a good guess to say that a grain of sand is about a cubic meter in volume).

2) there isn't enough sand to cover the earth in a shell of sand 1 meter thick. (a shell that was about 1 cm or smaller would give us a number closer to 2^49)...

I'm not sure if this is a problem... deserts aparently cover about 1/5 of the earth's land surface, and since land covers 1/4 of the globe, that means that deserts cover 1/20th of the earth's surface. covering the earth with a layer of sand 1 meter deep would require the deserts to have sand that was 20 meters deep. and i think that that's probably a pretty viable approximation. also, there's probably a lot of sand that isn't in deserts, and also under water.

so, my order of magnitude calculation says that there are way more grains of sand on the earth than 2^49... probably a million or a billion times more.

i guess, if you glazed over when i started doing math, what the moral of the story is that even though 2^49 is a really really really big number, the world is also very big, it has a huge surface, and sand is very small even though there is lots of it. when all these factors combine, 2^49 is still too small. maybe 2^65 or more would be closer to the right number.

Answer 3

It's pretty easy to see if this is true or not. A grain of sand is around 0.1 mm across. That means there are 10^12 grains in a cubic meter. Your number, 2^49 = 6 x 10^14. Dividing the grains of sand per cubic meter into the number of grains gives the cubic meters of sand.

(6 x 10^14 grains) / (10^12 grains/cubic meter) = 600 cubic meters.

That's the amount of sand on a beach 600 m long x 10 m wide x 10 cm deep.

That's a pretty small beach. Even if my estimate of grain size is off by a bunch, the result stands that there are way more than 2^49 grains of sand on all the beaches.

If you start pumping up the exponent, it doesn't take long to exceed any value that you pick. At 2^64 you need 20,000 of those small beaches. That's probably about right for all the beaches on Earth. At 2^110 the volume of sand is equal to the volume of the earth. I belive around 2^220, the mass of sand equals the mass of the entire universe.

Answer 4

How many grains of sand would you say are in a cubic inch? Let's say 20*20*20 = 800, make it 1000 for round numbers.

How big is the earth? r = 4000 miles = 4000*5280*12 inches. Volume of the earth is pi* this squared, is about 2.01E17 inches^3, so about 2E20 grains of sand would fill a volume as large as the earth.

2^10 is about 1000, 2^50 is about 1E15, 2^49 = 5E14.

Dividing this, 2E20/5E14 = 4E7, or 40,000,000 Is 1/40,000,000 of the earth sand? I would think that is pretty reasonable.

I found this:

"The calculation is detailed here:
http://www.hawaii.edu/suremath/jsand.html

That number is 7.5 x 10^18 or 7.5 billion billion."

So I think they are about equal.

Answer 5

2^49=5.629*10^14. I am not sure if this is less or more than all the grains of sand on Earth. There is no published data and no way to verify.

Answer 6

I've never counted all of the grains of sand on earth, so I really can't say. My initial reaction would be no, there's more sand than this.

Answer 7

whoever told you this is not wrong. however this must have given you this answer out of sheer intelligence.

as such there is no one on this planet who can count the number of grains of sand in the world.

however, since grains cannot be counted, he's just given you an imaginary number.

Answer 8

honestly, i would have to say no, but i wouldnt count this out, considering that there are more possible games of chess than there are atoms in the universe.

Answer 9

NO...
who told you that??