What is big?

Dear Diary,

Let's talk about length scales again. Consider the number

414,503

This number has a 'scale' of 10^5 since we can rewrite it as 4.14503 x 10^5. Let's look at  it on a graph:

To illustrate my point, I've also plotted the number 4.1 x 10^5.  It's almost the same! You can barely tell the difference!  But trust me!  Now, I only put the y-axis in steps of 10^5 because that's all we care about--it's our scale.

Let's really drive this pedantic point home by adding something way less than our scale--like the number 301 (=3.01 x 10^2).  It's of order 10^2.  The sum of those two numbers is

414,804

I've plotted it on our graph.  Again, you can barely tell the difference!  In fact, if you're not pulling any tricks, you can't!  Big numbers don't care about little numbers.  Sometimes, it's a good approximation to just concentrate on big numbers because if you added in smaller numbers, you'd basically get the same answer.

Let's look at the flip side.  10^8 is a huge number--it's on a bigger scale.  Let's add it to our first number:

100,114,804

On our new 10^8 scale, this number basically looks like plain old
10^8 (you couldn't tell the difference between the two on a graph with a scale of 10^8--or realistically, the y-axis are marked by 10^8).  I probably could have just put up a one with eight zeros after it, I certainly wouldn't have checked. This number is basically infinite on our old length scale, it's too big for our puny 10^5-scale to handle!

Lastly, for a real world example, consider you, a person!   Then consider the moon.  You can not handle the moon!   From our analogy above, it's too big a number to add to you without it just appearing that there is only the moon (I'm thinking of it crashing into you, sorry). You + moon = moon!  Only things on the same scale 'matter'...or we need to consider a different scale.


[Editor's note:  Perhaps in a future post, the author will muse on chaos.  This would be an example system where small numbers have big impacts on large ones.  The point remains, find the big numbers in your theory and you'll get a pretty good answer even if you left out some small ones.]