Monday, November 11, 2013

Large numbers

We don't have enough experience with large numbers to hold them in our minds. A million, a billion, and a trillion are all just abstractions for "many" that we tend to equate with "more than I would care to count."

But they are hugely different!

Here's a simple illustration: count to ten. That's roughly ten seconds and easy to grasp. Have you got that length of time in your head? Okay now, how much is a million seconds?

One million seconds = 11.57 days, or more than a week and a half. That's a lot of seconds! A million seconds might measure the amount of time between laundry loads or of a pretty good vacation.

One billion is one thousand millions.
One billion seconds = 31.69 years. That's a big chunk of a person's lifetime. A billion seconds ago it was 1982 and John Belushi died. Michael Jackson released Thriller. Madonna's musical career began in earnest.

One trillion is one thousand billions or one million millions.
One trillion seconds = 31.69 millennia. One trillion seconds ago it was about 30,000 B.C. and earth was in the Pleistocene epoch. Glaciers covered 30% of earth's surface. Mammoths, mastadons, saber-toothed cats, and Neanderthals were about to become extinct.

Compare a laundry cycle to Madonna's career. Compare both of those to the time of the last ice age.

Example:
The current national debt in America is over $17 trillion dollars (usdebtclock.org).  
In 2012 PBS received $27 million from Congress.
I have heard some conservatives claim that cutting PBS's funding will restore balance to the debt (or at least reduce the burden somewhat, "sorry, Big Bird"). You'd need to cut half a million PBS's before you'd start really noticing any change to the debt.

3 comments:

hosander said...

Anna is still waiting for someone to tell her "the biggest number"

Linds said...

Thad, you are a very clear thinker and writer! What a cool way to put these numbers into perspective. My kids talk about big numbers all the time but their favorite is the googolplex. Is there a way to put the googolplex into perspective?

Thaddeus said...

A googol is 10^100, or a one with one hundred zeroes following it before the decimal.

A googol seconds is equivalent to... nothing.

The longest length of time we've ever measured is from the big bang until now (14 billion years or roughly 10^17 seconds). So a googol seconds is roughly the equivalent of... the combined ages of 10^83 of our universes. (remember: a trillion is 10^12, so we're talking about 100 billion trillion trillion trillion trillion trillion trillion universes a.k.a. sexvigitillion)

Even if we measured in units much smaller: nanoseconds, that only bumps up the exponent nine places (the universe is 10^26 nanoseconds old). Still quite a ways away from 10^100. Plus, you don't really know how big a nanosecond is anymore.

We could continue to go smaller with our units, but the problem then becomes, "how can we contextualize the inverse of a googol? (a googolth?)"

The problem is a human one. We experience things on certain scales. Dozens of donuts, scores of years, etc. We rarely encounter and experience both one of a thing and over a trillion of the same thing. Maybe the best we can do is to visualize a grain of sand vs. all the sand in the world: this only spans about 19 orders of magnitude (10^0 through 10^18), but even with that visual, I bet you still have a hard time thinking of all the sand in the world, since you typically interact with maybe a single beach or sand dune at a time. Without a solid experiential interaction, we are left to abstractions, for which our minds can only create crude analogies.

There are only around 10^80 atoms in the observable universe. Do you have any intuitive sense for how big an atom is? I don't. So a googol is basically out of the question to conceptualize.

But you asked about a googolplex!

That's a 1 with a googol zeroes following it or 10^10^100.

Such a huge number is basically indistinguishable from a googol in our puny minds, but it's mathematically soooo much huger! You couldn't even write the number out in standard notation. It would take all the matter in the visible universe to write it, let alone count.

But even still, a googolplex isn't even the largest number mathematicians have used! Graham's number is mind-bogglingly so much huger and it holds the record for the largest number ever used in a serious mathematical proof.