2.1.5 - Everyday Large Numbers for a Modern World

2.1.5

Everyday Large Numbers for a Modern World

" In the world of large numbers size is relative, all numbers shrink before the human imagination " -- Sbiis Saibian

INTRODUCTION

In newpapers around the world we constantly hear about huge figures, often in the millions or billions , being reported as annual production or profit rates. The reason for this is that we now live in a world economy. Global trade has lead to everyday familiarity with numbers which would otherwise be quite remote from our day to day existence. In fact, numbers of this size are often highly underestimated do to their constant use.

In addition, mathematicians, scientists and above all amateurs have named numbers much much larger than these numbers in ways one can scarely imagine prior to being exposed to lengthy definitions and discussions of occasionally dicey algorithms. In comparison numbers in the low -illion range can seem pretty pitiful. We forget just how far these numbers stretch beyond the immediacy of our day to day existence.

So as a way of introduction to very large numbers, I want to remind my readers of the sizes of these numbers, as it makes the size of the truly large numbers all that more impressive. Let's Begin ...

ONE

If there is one natural number that will never seem big it's one. Quite simply, it's unique in being the very smallest natural number, and also in being the only one not larger than any other given natural. One is so simple that we barely even think of it. When we look at 1 apple, we only see the apple, the oneness of the collection seems irrelevant, its just an apple. It is difficult for me to imagine primitive humans coming up with the concept of oneness, without first coming up with the idea of a collection. In order to imagine a collection we are first taught to imagine a number of objects grouped together. The collection is that vague conceptual notion that all the objects are somehow bound together as one set. but we have no use for binding a single object to itself, it is a set on its own.

Perhaps the first number humans perceived was 2, and then in retrospect came to recognize loneliness as oneness.

In any case one is still important and unique. After all who doesn't want to be in 1st place. As an ordinal 1 is a useful way to label a beginning of any sequential process. In this sense the purpose for one in a discussion of large numbers is in having a non-ambiguous point of departure. There is no more fundamental place to start then 1, the very first natural number. But if 1 is but the first step in the journey to infinity, then we must at some point proceed to greater values, for example ...

TEN

"10" may seem small, but from a successive stand-point it's a considerable sum. To see what I mean think about sequels to movies. It's fairly common that a popular action / fantasy movie may have 1 or 2 sequels ( that is a series of 2 or 3 movies ). A few rare exceptions might be Scary Move 4, and the 6 star wars movies. But Imagine a 10 movie series ?! Video Games are alittle more libral when it comes to sequel counts. Even though the pattern of 3 is still prevalent, it has not been unheard of for series to have 4 or more games. The original Megaman series will soon have 9 official sequential games, and Final Fantasy has reached as far as 12. Still the occurance of numbers smaller than 10 is much more common.

we are comfortable with grouping things into pairs and triplets. Notice that when viewing 2,3, or even 4 objects it is unneccessary to count. Our brains instantly precieve oneness, twoness, threeness, and to some extent fourness. But when presented with 5 or more objects are brains begin to faulter. 5 is almost always precieved as 2&3, 6 as 3&3 , 7 as 3&4 , and lastly 8 as 4&4 . After this we can never be totally certain from a momentary glance how many objects are there.

It is common for people to have 1,2 or 3 kids, but 10 would be exceptional. What about twins, and triplets ? Quadruplets, Quintuplets, sextuplets, septuplets , octuplets, ... by the time we reach ten ( decuplets i suppose, though I've never heard the term used ) we have a birth that borders on the miraculous.

When considering these facts 10 begins to seem big once again. It's very difficult to have any sense of 10 objects. Our mind usually will bounce back and forth, but is not able to take in the wholeness of the collection.

It is often cited that the reason we use a decimal system, is because man was formed with 10 fingers. It seems to me that there must have been a time at the dawn of man at which 10 must have seemed a sensably large figure. When man could only precieve numbers up to 4, he could always say to himself "there is a number even larger which is the count of all fingers on both hands". Impressive in comparison.

But today we are so used to 10 we scarely give it thought. Most people probably have more dollars in their wallet. So let's move on ...

ONE HUNDRED

100

"100" , or One Hundred. This is a popular "large number" for young grade schoolers. I think almost everyone would agree that there was a point early in their lives at which a hundred was a very large number.

When we begin to learn to count past 10, we are usually taught by groups of 10's. For example, some kids only know how to count up to 40. Instinctively children will anticipate the continued pattern with "fourty-one" , "fourty-two" , "fourty-three" , all the way to "fourty-nine", but without knowledge of the next "group of ten" will resort to "fourty-ten" , "fourty-eleven" etc. in order to continue. Even when children don't know all the groups of ten however, they will often already know of a hundred from hear say, and already understand it as "ten groups of ten".

When we think of 100 in terms of counting, it is definitely a sizable number. Even if one counts quickly (assuming that you have to fully pronounce each number name) it will still take you between 45 to 60 seconds to count to a hundred. It takes more than "a moments effort" to count to a hundred. The fact that counting to a hundred causes one to become aware of the passage of time is one reason to consider it a early example of a "large number".

One Hundred is a very human kind of large number. Even in a prehistoric age, a man could concievably have impressed himself by describing an exact number, much larger than he would have words to describe. If their were as many people as fingers on both hands, the total number of fingers would be an impressive one hundred. If this same person was to see a 100 trees, or a 100 herding animals, there would be an immediate visual impact of "much more than I can precieve at once".

In fact when considering a 100 of anything we find that the visual impact is "satifyingly large" even fleetingly impressive, especially when we are comparing it to smaller numbers like 10.

Ultimately I think the appeal of 100 draws from it's ability to give us a sense of largeness, while still being humanly attainable even in an immediate and very real sense. A 100 is impressive while it lasts, but it certainly doesn't strike fear into us. I would describe it as "comfortably large".

Once we begin to get used to 100, once it becomes too common or safe, the human desire for still larger numbers arrises. As we will see many very large numbers draw their inspiration from the impressiveness of a 100, but this will come much later. Let us consider the next step up first ...

ONE THOUSAND

1000

If a hundred was big, how about Ten Hundreds, which is commonly called One thousand (1000). To a kid this is pretty big as numbers go, in that counting to a 1000 is a bit of a chore. Where as counting to a hundred might be considered an easy but mildly rewarding excersize, counting to a 1000 is more of a drudgery. First off counting to a thousand is likely to take at least a good 15 minutes. It won't be over in the blink of an eye, and more importantly it will take enough time that you might begin to wonder why your wasting it counting in the first place. There are plenty of better ways to spend 15 minutes after all. The point is that 1000 is not a number which can be easily dismissed.

A thousand of anything looks impressive even when the individual objects are relatively small. Consider a thousand pennies. But inside a cup they could fill it at least half way. And most peoples penny collections probably amount to no more than a few thousand pennies.

A hundred dollars is a good amount of money, but people will feel that this can disappear quickly. On the other hand, a thousand dollars is about a months pay on minimum wage.

How about food ? It would take at least 7 bags of Doritos to have about a 1000 chips. A Thousand apples would fill up even a large basket easily.

How about usage. It is reported that the average person uses about 120 gallons of water per day. That means a 1000 gallons would last the average person at least 8 days, or just over a week.

Now imagine a running fascet. Even if you ran your sink at full force it would take over 8 hours for a thousand gallons to flow through the pipes !

Let go back to the ancients. A thousand was a pretty good way up the number scale for them. The most common roman numerials go up to a thousand ( I V X L C D M ). After this one had to resort to less familiar constructions. For example there was a time when a bar was written over the number to signify x1000. So for example to specify 5000 you would write a V with a bar over it. But the standard symbols only went up to a thousand.

In today's world a thousand is a pretty common occurance though. After all a Ten-dollar bill is still worth a thousand pennies. We routinely budget our money in the thousands. A thousand is large, but it is also fairly common place. And although it presents some challenge it is by no means unattainable either by measure or count.

Now let's start to consider some truly large numbers ...

ONE MILLION

1,000,000

The first time I learned about a million wasn't from school. In fact I remember that our math text book in 2nd grade made no such mention of it. The highest number you could express was 999,999. I felt good knowing that the next grouping was a million, and I remember feeling somewhat resentful to the school system for keeping numbers like a million a secret. Perhaps this was the first time I felt that there was a secret world of large numbers beyond those in the class room just waiting to be discovered.

First off, try to think of a million as a thousand thousands. A thousand is pretty big, but that is only a single thousand, now imagine not ten not a hundred, but a thousand thousands ! A journey of a thousand miles may begin with one step, but even if you travelled a thousand miles you still would have only scratched the surface of a million mile walk !! In fact could anyone ever walk a million miles ? At a good pace a person can walk a mile in about half an hour. Let's say for argument sake someone wanted to walk a thousand miles. If we make allowances for sleep ( 8 hours a day ), and our traveller walks for 16 hours a day it will still take them at least a month to walk that distance ! Even if a person was able to keep up such a pace for their entire lives, they would not succeed in walking a million miles.

If a person were able to walk a thousand miles every 31 days, and they lived for 75 years ( average life expectancy ) they would only have managed to walk about 883,000 miles. In order to make it they would need to live to at least 85, but I doubt an 85 year old would be able to make a 30 minute mile, let alone a 2 year old. The bottom line is that a million miles is just out of reach of human ability.

So perhaps a man can't walk a million miles, how about counting to a million instead ? How hard could that be ? It seems deceptively easy. Let's assume the person has to speak out loud, and say the names completely. It's true that it is fairly easy to breeze through 1,2,3,4,5,6,7,8,9,10 but it is not so easy to get through 287,541 287,542 287,543 etc. let's say the person devotes 8 hours to counting each day. It would still take at least 103 days of counting, or more than 3 months ! On average the person would be tallying an extra 10,000 every day. Such a tideous routine is scarely imaginable, and comparable to torture. A more realistic plan to reach a million, would be to tally up about 2700 daily, this task is more managable and practical. At this rate it will only take 371 days, or about a 6 days past a year. If one wants to really take a long but practical approach one can tally 900 and do the count in 3 years. Counting to a million is a task of comparable magnitude to reading the bible ( Note: many bible plans involve a commitment of 1 to 3 years ).

Speaking of bibles, there is probably around a million letters used to write out the entire scriptures.

First off there are 66 books of the bible which on can contain anywhere from 20 to 50 chapters. It turns out that there are a total of 1189 chapters in the bible ( I have verified this myself actually, the number might seem lower than expected, but that is because a number of bible books contain 16 chapters or less, and 5 books containing only a single chapter. In addition the minor prophets rarely contain more than 10 chapters.). Later I started to check the number of verses within each chapter, and using a small sampling (the first 50 chapters of genesis ) I found the average to be roughly 25 verses per chapter. Multiplying this by the number of chapters leads to an approximation of about 30,000 verses contained in the bible. This estimate has proven to be fairly accurate, I have the figure 31,103 from one source as the exact value with a book by book break down.

We can use a verse to come up with a rough estimate of words and letters in the bible. Take for example this commonly cited verse ...

1 Timothy 6:10 "For the love of money is a root of all sorts of injurious things, and by reaching out for this love some have been led astray from the faith and have stabbed themselves all over with many pains."

there are 38 words and 152 letters. Assuming this was a typical verse that would mean the bible should contain around 1,181,000 words and 4,727,000 letters.

These numbers aren't far from fact. One source holds that there are in fact 773,692 words and 3,556,480 letters in the bible. So we can honestly say that the bible is in the order of the millions. No wonder reading through it is such a monumentous task.

One last illustration before we move on. Let's think about the faucet again. So you abandoned your home and moved out of state and left the faucet running at full force out in the middle of nowhere. It would take 347 days for a million gallons of water to travel through the pipe. In otherwords you would need to run a faucet full force for almost a year straight in order to waste a million gallons ! That is a LOT of water !

So yes a million is pretty big, it is massive and not easily subdued. The population of states are measured in the millions ! Yet it is not completely beyond human comprehension. Afterall people have read the entire bible before which means that over a million letters flashed before their eyes. It is possible to see a million things in a life time. And undoubtably there have been people who have successfully counted to a million in one manner or another, though I'm sure not without some trial. The point is the million exists on a tenuous border line between human attainability and utterly beyond reach. If a million doesn't hit the WOW button just yet try the next step ...

ONE BILLION

1,000,000,000

If a million was massive, a billion is super massive. A billion is a thousand millions ! How can one even grasp that at once, when we needed to think of a thousand thousands just to grapple with the million?

There is about 3 billion miles between the sun and pluto, which says alot about the crazy scales of our universe. Interestingly a human will live for around 3 billion seconds ( Note: 3 billion seconds is about 95 years , so most people don't hit the 3 billion mark, however plenty of people live at least a billion seconds since thats only about 32 years ). There are about 6.6 billion people living in the world.

So how about counting to a billion ? If counting to a million could take a year, then we know that a person couldn't count to a billion at that rate, because it would take a 1000 years ! Even a diligent counter tallying about 8 million a year ( which could be achieved provided the person counts for 16 hours a day with only 8 hours for sleep ) it would still take 125 years. The world record for longest documented lifespan was 120 years ! Even under these hellish conditions a person would fail to count to a billion. Even if it were possible, the person would have to average a tally every 3 seconds ! The person would barely have any time to sleep, eat , drink , use the toilet, let alone the more meaningful activities like talking to someone else. A persons sanity can only take so much. Also there would be rather longish names that would be difficult to cram into 3 seconds. For example try this one ...

eight hundred fifty three million two hundred sixty five thousand three hundred twenty nine

( 853,265,329 )

A count to a billion stretches human life to its absolute limits. It is doubtful that any human being in history has yet successed in such a task in any tongue or nomenclature. It is about as hopeless as walking a million miles, if not even more so.

The internet contains more than a billion webpages. As you can see a billion is roughly on the scale of world wide.

How long would your faucet have to run at full force to waste a billion gallons of water ? about 951 years, or almost an entire millenium.

Think, if someone had left the faucet running at the birth of christ, all the way to modern times only 2 billion gallons would have been wasted !

A billion is seemingly inexhaustable by human standards. Yet a billion is only just the 2nd in the illion series ! Yes after billion comes ...

ONE TRILLION

1,000,000,000,000

A Trillion is so large that it is difficult to think of examples to bring home it's immensity. First off we know that our faucet would have to run for a million years to waste a trillion gallons. All of human history only spans about 7000 years but the time period involved is about 136 times the span of recorded history ! The human species is believed only to have emerged as recently as 500,000 years ago. A million years ago, various species of pre-humans were competing for dominance. Since a span of a million years is so beyond human comprehension anyway, it totally fails to bring home how much water a trillion gallons is.

To bring home just how much water a trillion gallons is, we need a much bigger faucet. Imagine Niagara falls ! It is stated that roughly 150,000 gallons of water pour over the falls every second ! At this rate it would still take just alittle over 77 days and 3 hours for a trillion gallons to spill over the falls ! That's over 2 months !!

Imagine starring at the falls steadily, day after day, until 2 months pass. That is a lot of time, considering just how fast the water is flowing.

Yet as big as a trillion is, we still here mention of it sometimes when they speak of the "trillion dollar US debt". In otherwords, the wealth of countries can be measured in the trillions of dollars !

The nearest star to us (besides the sun), alpha centuari , is over 26 trillion miles away ! Again, that really says more about our universe than about the number itself.

It has been cited that the human body contains somewhere around 50 trillion cells , and that there are roughly a trillion fish in the world [1] .

Our world is literally full of trillions. But what about even larger "illions" ? Let's now pass one step beyond the well known illions ...

ONE QUADRILLION

1,000,000,000,000,000

After a trillion, there is a very sharp decrease in the need for very large numbers in most situations. Numbers larger than a trillion mostly arrise in science. Because of this it is common for people to ask, " what comes after a trillion ? ". A thousand trillion, is a quadrillion.

A quadrillion is also a million billion, which is pretty insane.

Alright, so how long would it take Niagara Falls to use up a quadrillion gallons ? It would take alittle over 210 years !

210 years ago, the United States was still a very young country. Imagine, from the declaration of independence to today, only about a quadrillion gallons have gone over the falls.

Here is another interesting figure. The great lakes have a volume of about 6 quadrillion gallons [2] . They say there is a quadrillion grains of sand on the beach.

The entire financial assets of the earth probably only amount to about a quadrillion US dollars.

The Known universe is only about 473 quadrillion seconds old.

It's about 586 quadrillion miles from one end of the milky way to the other.

Okay, so what's next ? ...

ONE QUINTILLION

1,000,000,000,000,000,000

When I was a kid, a quintillion seemed to be some kind of cut off point for large numbers. illions larger than this just don't get used very much, you almost never hear of them outside of lists.

It would take Niagara up 210,000 years ( about 30 times all recorded history ) to use up a quintillion gallons of water.

The entire earth itself contains about 326 quintillion gallons of water [3] .

The distance from the Milky way to Andromeda is 2 million light years, which is about 11.73 quintillion miles.

A single drop of water contains 1.7 quintillion water molecules [4] !

After quintillion come the far less familiar illions, which only arrise in science. Yet the staggering scales of our universe make it impossible to ignore these kinds of numbers. They are an undeniable part of our reality.

But once we pass a quintillion, we are beyond our everyday experiences. The next article continues the journey into even larger collections than those encountered in newspaper articles. It's time to start talking about some "really big numbers" ...

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Source Material :

[1] http://pages.prodigy.net/jhonig/bignum/indx.html : jhonig's website, entry for a "trillion".

[2] http://www.great-lakes.net/lakes/ref/lakefact.html : Great Lakes facts and figures page.

[3] http://science.howstuffworks.com/question157.htm : Blog where figure is stated.

[4] http://www.nps.gov/archive/rocr/naturecenter/Educational/aquaticecology.htm : The figure is quoted from the bottom paragraph of this page under the heading "cool facts about water".