Lumberjack music

The logarithmic scale, which few of us think we understand and most of us kind of do...

First a math joke...what is another term for lumberjack music?

Logger rhythms...

Now on to the column...

How large a number can you comprehend?

The answer to that question really depends on how you define 'comprehend'. This may seem a little like what the meaning of 'is' is, but the distinction is important here.

We all can focus on one object (whatever that object is) and follow its movement and characteristics easily. Two objects are also fairly easy for the human brain; beyond three, we are very limited in our ability to track dynamic objects, unless we group them, such as the 11 players on an NFL team. For four or five objects, we must continually shift focus, as we are unable to track them all simultaneously.

But for static objects, most of us can easily count five or six (or more) on sight and perceive the difference in quantities at this level without taking the time to count them. So already, we can see that our mental ability to comprehend a number depends on whether we want to be able to follow these objects' activity or just count them quickly.

Beyond single digit numbers, we do fairly well in games that require us to remember the location of specific items (think matching games like concentration). Going on to three or four digit numbers, most of us can comprehend how tightly we need to control our money if we go on a three day trip with two-hundred dollars in spending money or how far a thousand miles is. Up to the scale of one million, we can see all the individual components as well as the collection (there is a wall with one million tiny dots on it at the Science Museum Of Virginia in Richmond; you can step back to the point of seeing the whole wall at once and still be able to see the dots...take this much farther, say 10 or 20 million, and to back up far enough to see the whole thing at once, the dots would blur together). Of course, we can think of 20 of these walls, but we are again on a different scale of comprehension. Thinking of a trillion (a million million), we would have to think of each dot being another wall of a million, at which point, we still have some idea of the scale, but not in the same way we did with a million.

Thinking of this another way, you can perceive in detail what a foot of road looks like, down to the imperfections in the pavement. A mile is not too difficult to walk and can be driven very quickly and easily; though the two modes of mobility will yield vast differences in how we think of the distance and in the detail we notice in our surroundings during the trip. How many of us have flown across country in a couple of hours? It doesn't seem that far, but driving it gives us a different impression of what those three thousand miles consist of and walking it yet again yields a different perspective. Carry that out to man going to the moon and it changes again (consider that the space shuttle orbits the Earth in about an hour and a half-far faster than a jet and orders of magnitude above driving or walking). Yet all of these distances can be understood in some manner.

This is the logarithmic scale at work. When we are dealing with one, two or three things, each is very significant. At a scale of one-hundred, the individuals matter less-think of how much time you save walking when you cut to the inside of a turn versus how little it matters which lane you are driving in around a particular bend on a hundred mile drive.

At a scale of one thousand, even a few dozen are less significant. There is an economic theory based on this fact that talks of the decreasing marginal value of a dollar; basically, when you are very poor, an extra dollar may mean eating or not eating, the next dollar may mean eating enough to not feel hunger pains, while another may mean feeling satisfied...on up to the point where a dollar means very little or, for those of greater means, a dollar on the side walk is not worth the time to pick it up (they say that if you based the value of Bill Gate's time on his net worth, the amount on the sidewalk would have to exceed 10 thousand dollars to be worth his time to pick it up).

So what exactly is the logarithmic scale?

Logarithms are numbers that you raise 10 (or some other number, like 2) to in order to get a specific number. So if you think of a normal chart (say the time it takes fifth graders to run a hundred yards), you may see a scale of evenly incremented numbers along the bottom...say 10 second intervals. But a logarithmic chart would change in scale as you go from left to right. This is useful with data that has multiple events (or items) early on, but which stretches out very far. If you were to chart income, you would see that the vast majority are clumped around some range (say 20 to 100 thousand per year), but a scale using increments of 10 thousand would have to be very long to reach the billions (10 thousand goes into a billion 100 thousand times), but using an evenly incremented scale that reaches a billion in the width of a sheet of paper (and is large enough print to decipher) would require increments of 5 million or so and the first part would have all of the data. So instead, we might start with 10 thousand, then double to 20, then 40, then 80, 160, 320, 640, 1.2 million, etc, and a billion would only require 18 increments. We might also do this for time based events: say we dropped a million pennies in a field from an airplane and gave a prize to anyone who found one (or more). If the field was relatively small, we might have several finds in the first minute, more in the second and so on. After some time (the length of which depends on the size of the field, height of the airplane and number of people looking), we may have the same number of finds in an hour or day as we did in the first minute. Unless we have an organized archeological team working, we will likely have some left in the field after the first group gives up looking. Likely, pennies will turn up years later (assuming people still return to look occasionally), at which point, charting by hours or days might even be pointless. Perhaps the field will be abandoned for a century or longer and records of the penny drop will be discovered by future generations who actually do initiate a dig at the site (though at this time, the person maintaining the chart may be long in the grave, we will assume that the chart is found, scanned and updated by some future Indiana Jones.

Does any of this matter to the average Joe? Probably not, though if he read this column, he may be able to sound slightly smarter by using the term 'logarithmic scale' properly to his buddies in the bar. And it serves as a prelude to another column I am posting on large numbers.

Finally, for those who understand logarithms and like bad humor, there is a math joke involving Noah's Ark (as well as some other poor puns) here:

http://www.lhup.edu/~dsimanek/noahfool.htm