Wednesday, April 8, 2009

Wednesday Math, Vol. 66: Big numbers and a human scale

In everyday conversation about economic news, we deal regularly with millions and billions and now, even trillions. It can be hard to get your mind around these big numbers, to put two things in your mind and realize the small thing is a million times smaller than the big thing, or the very big thing is a billion times bigger than the very small thing. I want to try to put objects in perspective about just how big these things are, using money, time distance and volume.

How big is a million?

Time: A million years is unfathomable for humans. Even a million months is more than eighty thousand years, not a number we can get our heads wrapped around. It was about a million days ago that the city of Syracuse in Sicily was founded, around the year 740 B.C. Still not really a human scale.

A million hours ago is the year 1895. Oscar Wilde writes his last play, The Importance of Being Earnest, and is arrested for indecency when his relationship with a young man is made public. Babe Ruth and John Ford are born. It's getting closer to be something we can comprehend, but none of us expect to live a million hours.

A million minutes ago, I was in the first month of writing my blog, which is now two years old.

A million seconds ago, I wrote a post then took about a nine day break from blogging.

Money: A million dollars sounds like a lot of money, but it's definitely smaller than it used to be. Since the median income in the United States is between $40,000 and $50,000, most American workers can expect to earn a million dollars in salary in about twenty to twenty five years of work.

A million pennies is $10,000. That is what most Americans make in three months time or less.

Distance: A million miles is not a human scale, nor is a million kilometers. A million feet is about 189 miles. You might think of this as a human scale because it is some distance you might drive, but that puts it at a technological scale, not really a human scale. You can't put 189 miles in your mind's eye all at once, especially if you are measuring it in feet. A million inches is about sixteen miles, which is to say about the distance a person could walk in four or five hours. A million centimeters is ten kilometers, just a little longer than six miles. This is getting closer to a human scale.

Volume: With volume, it's easier to get the ideas of big and small in focus, because 10x10x10 = 1,000, so as we make a thing ten times bigger in every dimension, it becomes a thousand times more massive. I apologize to my American readers who might dislike the metric system, but when dealing with volume and powers of ten, the metric system shines and the English measurement system couldn't be more useless.

I have drawn a two dimensional representation of the three dimension cubic centimeter, known as a cc. The square in blue actually is a square centimeter, and so you can get the idea of the size. You can easily hold one cc between your thumb and forefinger. One cc of water at room temperature weighs one gram. Grams and cc's are used in the metric system to measure little stuff. If we get a cube of ten by ten by ten cc's, the metric system gives this a new name, the liter. Because of this relationship, a cc is sometimes also called a milliliter, or ml for short. A liter of water weighs one kilogram, or about 2.2 pounds in the English system.

Though we might think by looking at it that a cc is a size we understand, it's really tiny. Think about anything in your medicine cabinet that has a cap that is used for dosage, like mouthwash or cold medicine or Pepto-Bismol. The cap is usually one or two tablespoons, which means 15 or 30 cc's. Would you really notice if you got a short dosage of 29 cc's, or an overdose of 31 cc's instead of 30 cc's? Not likely.

A million cubic centimeters is a square box where every side is one meter in length, a little bit less than forty inches. If you were moving, this would be a very big box, and you should pack it carefully. If a cubic meter is filled with water, for example, it weighs over 2,200 pounds, a weight known as a metric ton.

A million cubic inches is a square box 8'4" on every side. It would be a very cramped prison cell.

How big is a billion?

Time: A billion seconds is a little less than thirty two years. If your heart beats 72 times per minute, a billion heartbeats is about twenty six and a half years. In industrialized nations, most people will get about three billion heartbeats. Four billion is very rare, and nobody gets five billion.

Money: I see no way to bring this into an amount most people will ever have to deal with personally. Even a billion pennies is ten million dollars. The median worker needs to work for centuries to see even a billion pennies.

Distance: Take the distances for a million and multiply them by a thousand each. None of these are on a truly human scale.

Volume: Here, we go back to our little friend the cubic centimeter again. Consider a racquetball or squash court, twenty feet tall, twenty feet wide and forty feet deep. Now, think about two of them side by side. They contain nine hundred million cubic centimeters, just barely under a billion cc's.

To try to think of a billion cubic inches, our cube would have to be slightly over eighty three feet on each side. This is still visible to the naked eye, something we could walk around in. A basketball arena has more than a billion cubic inches of volume.

How big is a trillion?

Volume: The Louisiana Superdome has about three and a half trillion cubic centimeters in volume. That's about one hundred billion capsful of Nyquil, enough to keep every man, woman and child on the planet knocked on their asses for about a week.

These are the numbers we now deal with when talking about money. Remember our tiny cubic centimeter, which weighs a gram if it is filled with water? There are about thirty three billion trillion water molecules in that little thing. Because the words start becoming useless when dealing with sizes like this, scientific notation becomes more useful and we say 3.3 x 10^22.

I'm going to talking about really big and really small numbers in the next few weeks of posts, but it's important to understand. Really big isn't infinite, and really small isn't infinitesimal. There is good evidence that the infinite and the infinitesimal do not exist in nature. Everything is measurable, though the scales are often very hard to fathom.


BobManDo said...

Matty Boy, Glad you are with us... I perhaps visited more often in you absence, with my imagination going to "bad" places and possibilities... I like the new examples you give on the biggies... You might have missed this link I posted earlier... It helped me get my head around a Trillion dollars:

"How Much is a Trillion"?



Matty Boy said...

Hi, Bob. I did check out the trillion video, and I didn't like the representation. I don't think it really puts things on a human scale, which is why I made this one instead.

dguzman said...

Pure genius.

This morning, I heard (on NPR) about these people who are looking for the largest prime number--and the guy talked about the current largest one being millions of numbers in length, and how that would take up over 20 miles if you wrote it out, ten digits per inch. It all just made me think of you.

Distributorcap said...

i LOVED this post - it was great --
if you stacked a million dollar bills on top of each other it would be around 360 feet high