Explain this exponent
March 23, 2017 6:24 AM Subscribe
1600 to the power of 16. How can I effectively explain just how large this number is to a non-math audience in common language? My calculator says this number is 1.8446744e+51. That is kind of meaningless to me (e+51?) Is this number a sextillion? Can we put this number in context to other large numbers?
I was about to give the same answer as koolkat.
The number of atoms in the entire planet earth can be estimated to be around 1e50. So your number is the number of atoms in 18 earths.
I tried to work it out in terms of the number of pints of water in all of the earth's oceans. In the end I had to work on the basis of every single star in the universe having a couple of million earths orbiting it. In that scenario, I think there ought to be somewhere around that number of pints of water in total.
It's a pretty big number.
posted by pipeski at 6:40 AM on March 23, 2017 [1 favorite]
The number of atoms in the entire planet earth can be estimated to be around 1e50. So your number is the number of atoms in 18 earths.
I tried to work it out in terms of the number of pints of water in all of the earth's oceans. In the end I had to work on the basis of every single star in the universe having a couple of million earths orbiting it. In that scenario, I think there ought to be somewhere around that number of pints of water in total.
It's a pretty big number.
posted by pipeski at 6:40 AM on March 23, 2017 [1 favorite]
1051 is a 1 with 51 zeros after it: 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. A sexdecillion. Your number is 1.8446744 sexdecillion.
posted by zsazsa at 6:45 AM on March 23, 2017 [2 favorites]
posted by zsazsa at 6:45 AM on March 23, 2017 [2 favorites]
WolframAlpha says it's roughly 35 times bigger than the number of chess positions, but people probably don't have a feel for just how mindbogglingly big that number is either.
1 e+51 is one sexdecillion on the short scale. According to this source, the volume of the milky way galaxy is about a sexdecillion cubic miles. Your number is one and four-fifths that big.
posted by radwolf76 at 6:46 AM on March 23, 2017
1 e+51 is one sexdecillion on the short scale. According to this source, the volume of the milky way galaxy is about a sexdecillion cubic miles. Your number is one and four-fifths that big.
posted by radwolf76 at 6:46 AM on March 23, 2017
That is kind of meaningless to me (e+51?)
For folks who don't already know, scientific notation is a format for expressing really large, or really small numbers. They're typically written in the form m x 10n. A main number m, multiplied by ten to the power of n.
3 x 102 = 3 x 100 = 300
3 x 10-2 = 3 x 0.01 = 0.03
You just move the decimal point to the right n spaces if the power of ten is positive, left if it's negative.
E-notation is how calculators and computers that can't use superscript represent scientific notation.
3 e+2 = 3 x 100 = 300
3 e-2 = 3 x 0.01 = 0.03
posted by zamboni at 6:48 AM on March 23, 2017 [9 favorites]
For folks who don't already know, scientific notation is a format for expressing really large, or really small numbers. They're typically written in the form m x 10n. A main number m, multiplied by ten to the power of n.
3 x 102 = 3 x 100 = 300
3 x 10-2 = 3 x 0.01 = 0.03
You just move the decimal point to the right n spaces if the power of ten is positive, left if it's negative.
E-notation is how calculators and computers that can't use superscript represent scientific notation.
3 e+2 = 3 x 100 = 300
3 e-2 = 3 x 0.01 = 0.03
posted by zamboni at 6:48 AM on March 23, 2017 [9 favorites]
You could show them the classic Powers of Ten movie, which goes from a scale of 10^24 meters, or the size of the observable universe, to quarks in a proton of a carbon atom at 10^−16 meter. That's a relative change of "only" 10^40.
posted by Mr.Know-it-some at 6:51 AM on March 23, 2017 [3 favorites]
posted by Mr.Know-it-some at 6:51 AM on March 23, 2017 [3 favorites]
A sextillion is 10^21 (aka 1e+21) so your number is 1.8 billion sextillion sextillion (i.e. 1.8 x 1e+9 x 1e+21 x 1e+21 = 1.8e+51).
Another way to think about this number is that it's roughly the odds against flipping a fair coin 171 times and having every single one come up heads.
posted by firechicago at 7:09 AM on March 23, 2017 [5 favorites]
Another way to think about this number is that it's roughly the odds against flipping a fair coin 171 times and having every single one come up heads.
posted by firechicago at 7:09 AM on March 23, 2017 [5 favorites]
When trying to get people to understand the bigness of numbers, you can can get a good effect by jumping around and combining different visualisations. Here's one for your number:
Imagine everyone in the world. That's 7 billion people, or 7 x 10^9.
Now let's arrange for every single person alive to shake hands with everyone else, one after the other. One handshake at a time. So, first, I'm going to shake hands with everyone on earth, one at a time. That's take me a while. Then you are. Then Frank is.
Now we're up to 4.9 times 10^19.
Oh, and one more twist. Let's make sure we can't shake hands too fast. Somewhere far away, there's a mountain, a hundred miles high and a hundred miles wide. Once every thousand years a little bird comes to this rock and pecks it. And when the mountain is worn away? That's one handshake. Then we start again with a new mountain. And when that's worn away? One handshake.
How long does that take? All of this handshaking, and waiting around for countless millennia to shake hands? According to this estimate re: the bird question, it'll take (4.2 x 10^27) * (4.9 x 10^19) = 2.058e+47 years.
So we'll just do the whole thing over, when we're finished. Ten thousand times.
And that's how big your number is.
posted by piato at 7:26 AM on March 23, 2017 [26 favorites]
Imagine everyone in the world. That's 7 billion people, or 7 x 10^9.
Now let's arrange for every single person alive to shake hands with everyone else, one after the other. One handshake at a time. So, first, I'm going to shake hands with everyone on earth, one at a time. That's take me a while. Then you are. Then Frank is.
Now we're up to 4.9 times 10^19.
Oh, and one more twist. Let's make sure we can't shake hands too fast. Somewhere far away, there's a mountain, a hundred miles high and a hundred miles wide. Once every thousand years a little bird comes to this rock and pecks it. And when the mountain is worn away? That's one handshake. Then we start again with a new mountain. And when that's worn away? One handshake.
How long does that take? All of this handshaking, and waiting around for countless millennia to shake hands? According to this estimate re: the bird question, it'll take (4.2 x 10^27) * (4.9 x 10^19) = 2.058e+47 years.
So we'll just do the whole thing over, when we're finished. Ten thousand times.
And that's how big your number is.
posted by piato at 7:26 AM on March 23, 2017 [26 favorites]
Just start them down the inductive understanding path; show how large 16^2 is, then multiply that by 16 for the next power, showing the difference in magnitude, then one more time... then tell them this numeric explosion happens 13 more times.
The number is incomprehensibly big, but that gives a sense of how incomprehensible and why.
Why are you talking about this in the first place? I'd guess that the effect that you are talking about, whatever that is, is more impressive and important than the number itself.
posted by amtho at 7:28 AM on March 23, 2017
The number is incomprehensibly big, but that gives a sense of how incomprehensible and why.
Why are you talking about this in the first place? I'd guess that the effect that you are talking about, whatever that is, is more impressive and important than the number itself.
posted by amtho at 7:28 AM on March 23, 2017
Something like this?
https://www.youtube.com/watch?v=T69cguFzZ_w
You could say :
"This is a litre of water.
An oil barrel holds about 158 litres of water.
A train tank car holds about 720 barrels of water.
The great lakes hold 2.3×10^16 Litres of water, or about 200 Billion train cars.
The worlds oceans hold about 700,000 Great Lakes worth of water, or 1.3e21 litres. 1,300,000,000,000,000,000,000 Litres.
To have 1.8e51 Liters of water, you'd need 1.3e30 earth's worth of oceans. You'd need more earth's than there are litres of water in each ocean, a billion times more.
posted by Capn at 7:30 AM on March 23, 2017 [1 favorite]
https://www.youtube.com/watch?v=T69cguFzZ_w
You could say :
"This is a litre of water.
An oil barrel holds about 158 litres of water.
A train tank car holds about 720 barrels of water.
The great lakes hold 2.3×10^16 Litres of water, or about 200 Billion train cars.
The worlds oceans hold about 700,000 Great Lakes worth of water, or 1.3e21 litres. 1,300,000,000,000,000,000,000 Litres.
To have 1.8e51 Liters of water, you'd need 1.3e30 earth's worth of oceans. You'd need more earth's than there are litres of water in each ocean, a billion times more.
posted by Capn at 7:30 AM on March 23, 2017 [1 favorite]
Buy a Powerball lottery ticket every second.
Every time you win the jackpot, remove one teaspoonful of rock from Mt. Everest.
Every time you remove the last bit of rock from Mt. Everest, drain one teaspoonful of water from the ocean, and then replace all the rock in Everest and start over again.
Every time you finish draining the ocean, fill it back up and start over again.
After you've drained the ocean about 80 times, 1600^16 seconds will finally have passed.
Wolfram Alpha link
posted by CustooFintel at 7:48 AM on March 23, 2017 [4 favorites]
Every time you win the jackpot, remove one teaspoonful of rock from Mt. Everest.
Every time you remove the last bit of rock from Mt. Everest, drain one teaspoonful of water from the ocean, and then replace all the rock in Everest and start over again.
Every time you finish draining the ocean, fill it back up and start over again.
After you've drained the ocean about 80 times, 1600^16 seconds will finally have passed.
Wolfram Alpha link
posted by CustooFintel at 7:48 AM on March 23, 2017 [4 favorites]
There are (per random google result) about 1x10^23 stars in the universe. If each of those stars contained its own universe with the same number of stars, you'd still need to multiply by 180,000 to get your number of stars.
posted by papayaninja at 7:55 AM on March 23, 2017
posted by papayaninja at 7:55 AM on March 23, 2017
There is about ~10^19 atoms in a single grain of sand.
As a non math person this doesn't make any sense to me, since the number in the question (1600^16) doesn't sound that far off from 10^19, yet you say it's more than the number of atoms in the entire planet. How can both be true?
posted by AFABulous at 8:06 AM on March 23, 2017
As a non math person this doesn't make any sense to me, since the number in the question (1600^16) doesn't sound that far off from 10^19, yet you say it's more than the number of atoms in the entire planet. How can both be true?
posted by AFABulous at 8:06 AM on March 23, 2017
It's because "sounding far off" isn't a reliable guide to how exponents work. The number of atoms in a grain of sand is roughly 10^9. The other number, 1600^16, works out to 1.8 X 10^51. Let's round down and say it's just 10^51. So, the second number is 10 quintillion (10,000,000,000,000,000,000) times bigger than the first.
posted by Ipsifendus at 8:14 AM on March 23, 2017 [1 favorite]
posted by Ipsifendus at 8:14 AM on March 23, 2017 [1 favorite]
Perhaps you could ask your audience to think about money. Have them try imagine 1.84e+51 pennies. If you distributed this amount of money evenly to the 7 billion people on earth, and each person could spend a billion dollars a second, how long would it take to spend all the money?
1.84e+51/ 100 pennies per $ / 7e+9 people / 1e+9 $ per second = 2.6 e+30 seconds
Still too long to picture!
2.6 e+30 seconds / 60 seconds per minute / 60 minutes per hour/ 24 hours per day / 365 days per year = 8.4 e+22 years
Yikes! The big bang 'only' happened 13.8 billion (13.8 e+9) years ago. So if 1.84 e+51 pennies were spent by everyone on earth, at a rate of a billion dollars a second it would still take more than a billion times the age of the universe to spend it all (about 6 e+12 times the age of the Universe).
Ugh. That's still pretty tough to picture. Good luck.
posted by Sauter Vaguely at 8:21 AM on March 23, 2017
1.84e+51/ 100 pennies per $ / 7e+9 people / 1e+9 $ per second = 2.6 e+30 seconds
Still too long to picture!
2.6 e+30 seconds / 60 seconds per minute / 60 minutes per hour/ 24 hours per day / 365 days per year = 8.4 e+22 years
Yikes! The big bang 'only' happened 13.8 billion (13.8 e+9) years ago. So if 1.84 e+51 pennies were spent by everyone on earth, at a rate of a billion dollars a second it would still take more than a billion times the age of the universe to spend it all (about 6 e+12 times the age of the Universe).
Ugh. That's still pretty tough to picture. Good luck.
posted by Sauter Vaguely at 8:21 AM on March 23, 2017
As a non math person this doesn't make any sense to me, since the number in the question (1600^16) doesn't sound that far off from 10^19, yet you say it's more than the number of atoms in the entire planet. How can both be true?
This is a bit like saying that the difference between 3 and 7 is only 4, so how can the difference between 3 x 100 and 7 x 100 be 400?
When it comes to exponents, changes to the base are compounded. So the difference between 1600^16 and 10^16 isn't a factor of 160, it's a factor of 160^16 (or about 1.8 x 10^35).
posted by firechicago at 8:25 AM on March 23, 2017
This is a bit like saying that the difference between 3 and 7 is only 4, so how can the difference between 3 x 100 and 7 x 100 be 400?
When it comes to exponents, changes to the base are compounded. So the difference between 1600^16 and 10^16 isn't a factor of 160, it's a factor of 160^16 (or about 1.8 x 10^35).
posted by firechicago at 8:25 AM on March 23, 2017
AFABulous, 10^19 means that you multiply 10 by itself 19 times. In practical terms, that means you add a zero each time: 10^2 is 10*10= 100; 10^3 is 100*10=1000, etc. After doing it 19 times, you have a 1 with 19 zeros after it.
When you multiple 1600 by itself, by contrast, each time you are adding at least 3 more digits to the number each time: 1600^2 is 1600*1600=2,560,000; 1600^3 is 2,560,000*1600=4,096,000,000; etc. After doing it 16 times, you've got a 1 plus 51 digits after it.
See how 1600^3 already is a 4 with (basically) 9 zeros after it, while 10^3 is just 1000 (1 with 3 zeros)? That difference literally multiplies with each round of multiplication.
As for the original question - I find that talking about atoms is hard to picture. I like going big: There are about 10^24 stars in the universe. You know the theory of multiple universes? If every star represented a parallel universe with its own 10^24 number of stars, you'd still have 1000 times fewer stars than 1600^16 stars.
posted by alligatorpear at 8:43 AM on March 23, 2017
When you multiple 1600 by itself, by contrast, each time you are adding at least 3 more digits to the number each time: 1600^2 is 1600*1600=2,560,000; 1600^3 is 2,560,000*1600=4,096,000,000; etc. After doing it 16 times, you've got a 1 plus 51 digits after it.
See how 1600^3 already is a 4 with (basically) 9 zeros after it, while 10^3 is just 1000 (1 with 3 zeros)? That difference literally multiplies with each round of multiplication.
As for the original question - I find that talking about atoms is hard to picture. I like going big: There are about 10^24 stars in the universe. You know the theory of multiple universes? If every star represented a parallel universe with its own 10^24 number of stars, you'd still have 1000 times fewer stars than 1600^16 stars.
posted by alligatorpear at 8:43 AM on March 23, 2017
I realize I came up with the exact same thing as papayaninja. D'oh!
posted by alligatorpear at 8:50 AM on March 23, 2017
posted by alligatorpear at 8:50 AM on March 23, 2017
This video, with it's cubes made up of cubes made up of cubes made up of suns also does a good job explaining big numbers
posted by Capn at 9:30 AM on March 23, 2017
posted by Capn at 9:30 AM on March 23, 2017
I believe y'all! The question was how to describe numbers to a non-math person. I am a non-math person, so I am pointing out that your explanations, while correct, aren't necessarily easy to grasp.
posted by AFABulous at 9:49 AM on March 23, 2017 [1 favorite]
posted by AFABulous at 9:49 AM on March 23, 2017 [1 favorite]
I believe y'all! The question was how to describe numbers to a non-math person. I am a non-math person, so I am pointing out that your explanations, while correct, aren't necessarily easy to grasp.
That is why I put the grain of sand bit in there. People know that atoms are small, just not how small. so if you think that there are 10^19 atoms in a single grain of sand then knowing that there are 10^50 atoms in the entire earth really gives a sense of scale. I think for the stars version people really should keep to only the stars able to be seen with a naked eye, otherwise people will think that the stars that they can see number 10^20, when in reality it is more like 10^5 (just a guess) visible stars.
posted by koolkat at 9:59 AM on March 23, 2017
That is why I put the grain of sand bit in there. People know that atoms are small, just not how small. so if you think that there are 10^19 atoms in a single grain of sand then knowing that there are 10^50 atoms in the entire earth really gives a sense of scale. I think for the stars version people really should keep to only the stars able to be seen with a naked eye, otherwise people will think that the stars that they can see number 10^20, when in reality it is more like 10^5 (just a guess) visible stars.
posted by koolkat at 9:59 AM on March 23, 2017
This number is so large that comparing it to quantities of things is hard without doing some of the "doubling up" that others have suggested (take the number of stars in the universe, now give each star its own universe, etc). Probability will give a more direct approach here. Someone else already translated this into coin flips, but here's a different way of doing this that might be more immediate for your students:
Wander around the center of a large city, approach a random stranger, and ask them to guess your birthday. (Month and day only, not the year.) Do this 20 times, walking for a few minutes between approaching each stranger so nobody overhears your previous conversation. Now imagine that every single one of these 20 strangers guesses your birthday correctly.
The odds of this happening purely by chance are about 1 in 365.25^20 ≈ 1.8x10^51, almost exactly the number you're trying to explain.
posted by freelanceastro at 12:23 PM on March 23, 2017 [2 favorites]
Wander around the center of a large city, approach a random stranger, and ask them to guess your birthday. (Month and day only, not the year.) Do this 20 times, walking for a few minutes between approaching each stranger so nobody overhears your previous conversation. Now imagine that every single one of these 20 strangers guesses your birthday correctly.
The odds of this happening purely by chance are about 1 in 365.25^20 ≈ 1.8x10^51, almost exactly the number you're trying to explain.
posted by freelanceastro at 12:23 PM on March 23, 2017 [2 favorites]
I've always enjoyed using time-and-probability-based analogies for numbers this big. Like:
Pack a deck of cards. Once a year, shuffle and deal yourself a 5-card poker hand. Each time you get a royal flush, buy yourself a PowerBall ticket. If that ticket wins the jackpot, place one grain of sand into the Grand Canyon. When you’ve filled up the canyon with sand, remove one ounce of rock from Mt. Everest. Now empty the canyon and start all over again. When Mt. Everest has been completely removed, you've reached 160016 seconds.
posted by Mayor West at 1:10 PM on March 23, 2017 [7 favorites]
Pack a deck of cards. Once a year, shuffle and deal yourself a 5-card poker hand. Each time you get a royal flush, buy yourself a PowerBall ticket. If that ticket wins the jackpot, place one grain of sand into the Grand Canyon. When you’ve filled up the canyon with sand, remove one ounce of rock from Mt. Everest. Now empty the canyon and start all over again. When Mt. Everest has been completely removed, you've reached 160016 seconds.
posted by Mayor West at 1:10 PM on March 23, 2017 [7 favorites]
To win the (UK) lottery, you have to pick six correct numbers from between 1 and 59 inclusive, your odds of winning are about 1 in 45 million. Your number would be like winning a lottery where you had to pick six correct numbers between 1 and a billion.
posted by lucidium at 6:22 PM on March 23, 2017
posted by lucidium at 6:22 PM on March 23, 2017
I am sort of a math person and I would suggest staying away from odds. If there's one thing that people understand less than large numbers, it's probability.
Piato's suggestion is great, it deals with concrete things like years, people on earth, the size of a pebble vs a large mountain.
posted by FirstMateKate at 8:58 AM on March 24, 2017 [1 favorite]
Piato's suggestion is great, it deals with concrete things like years, people on earth, the size of a pebble vs a large mountain.
posted by FirstMateKate at 8:58 AM on March 24, 2017 [1 favorite]
This thread is closed to new comments.
http://education.jlab.org/qa/mathatom_05.html
There is about ~10^19 atoms in a single grain of sand.
https://scienceyourfacein.wordpress.com/2014/01/12/craigs-questions-sand-and-atoms/
posted by koolkat at 6:35 AM on March 23, 2017 [1 favorite]