Work in progress

Working on a flash or presentation based math tutorial, but thought I would paste one of my early attempts at conveying one way a conceptually based math brain sees things...

Visualizing percentages

The big little numbers and their exclamatory friend

aka...exponents and arrows and the math nerds who love them.

Is 4000 a big number? How about 1.2 million? 632, 415, 906, 255? What about 10¹⁵? As this is obviously a trick question, even the non-mathematic are likely guessing that the last of these is the biggest. Ten to the fifteenth power is a sixteen-digit number (as compared to the 12 digits of the next largest number mentioned above). Knowing that still doesn’t give a fair representation, though. Which seems larger: 1,000,000,000,000,000 or 10¹⁵?

Of course, neither really means anything to us.

Multiple exponents can make even larger numbers seem small. How big is (10³)x(10³)? Or (10³)³? How about 10^3³ (the ^ means to the power of, so 10^3=10³ and 10^3³ just means raise 10 to the 3³ power)? How do these two number compare to the numbers at the end of the last paragraph?

First, 1,000,000,000,000,000 and 10¹⁵ are the same; anytime you raise 10 to a power, the result is a one with that number (the power) of zeroes after it. When multiplying a number to a power by the same number to a power, add the exponents ((10³)x(10³)=10⁶ or 1,000,000). When raising a number to a power and then raising the result to a power, such as (10³)³, you multiply the exponents. So (10³)³ yields 10⁹ or 1,000,000,000. Finally, raising a number to a power that is raised to a power, the power in the exponent must be done first (following order of operations), so 10^3³ is 10²⁷ or 1,000,000,000,000,000,000,000,000,000. So minor changes in the use of these little numbers (the exponents) or movements of the parentheses can make a huge difference in the scale of the result. We can also represent unwieldy numbers in very small notations through careful use of exponents. In addition to the notations used above, the up arrow, ↑, indicates an exponential use of powers, such as 10^(3↑↑3) (this is Knuth’s up-arrow notation, for those who care) means 10^(3^(3^3)), which is 10^(3^27) or 10^7625597484987. This number is a one with over 7 trillion zeroes after it, which I represented with four digits, three arrows and a pair of parentheses (and only then because I wanted to keep using 10 to a power and used threes to keep the final exponent small enough to show here. The addition of a third arrow does not mean a third cubing of the 7 trillion digit number; instead, it means to do the double arrow operation three times: 3↑↑↑3= (3↑↑(3↑↑3), which is 3 cubed 7625597484987 times (a number far too large to represent in any form other than that already used here). Multiple arrows, even with small numbers, will yield numbers that have minimal, if any, use.

Taking our extreme number game a step further, Graham’s number is a number that makes creative use of the up-arrow notation. First, calculate 3↑↑↑↑3; this will be called g(1). To calculate g(2), calculate 3↑↑….↑↑3, with number of arrows between the 3’s being equal to g(1). Of course, remembering what 3 up arrows did to the threes in the previous example and the fact that an additional arrow means to do the triple arrow operation 3 times (3↑↑↑(3↑↑↑3), which is unwieldy when reduced to the already impressive double arrow notation. So, if g(2) is a pair of 3’s with g(1) arrows, this number is not only large beyond imagination, we cannot even comprehend how large the result is in comparison to g(1). Moving on, Graham’s number is g(64)….that’s right, a mind blowing ratio between mind blowing numbers 64 TIMES! Ridiculous!

So, The Math Factor (broadcast on KUAF, an NPR associate, and available on podcast) had a contest to see who could express the largest number on the fly. The contestants (Agustín Rayo against Adam Elga) took turns one-upping each other until Rayo introduced us to the number now known as Rayo’s Number. Simply put (sort of), Rayo’s Number is the smallest number larger than the largest finite number that can be expressed in the language of first order set theory (common math and logic symbolism) using no more than a googol (one with one hundred zeros after it or 10^100) symbols. A bit of a cheat, but the winner of the contest.

Math Factor Website

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

State of the Union Perspective

Check out this sampling of State of the Union quotes:

"When I visited this chamber last year as a newcomer to Washington, critical of past policies which I believe had failed, I proposed a new spirit of partnership between this Congress and this Administration and between Washington and our state and local governments...seldom have the stakes been higher for America. What we do and say here will make all the difference to auto workers in Detroit, lumberjacks in the Northwest, steelworkers in Steubenville who are in the unemployment lines, to black teen-agers in Newark a nd Chicago; to hard-pressed farmers and small businessmen and to millions of everyday Americans who harbor the simple wish of a safe and financially secure future for their children...To understand the State of the Union, we must look not only at where we are and where we're going but where we've been. The situation at this time last year was truly ominous...The last decade has seen a series of recessions...If we had not acted as we did, things would be far worse for all Americans than they are today...The economy will face difficult moments in the months ahead. But, the program for economic recovery that is in place will pull the economy out of its slump and put us on the road to prosperity and stable growth by the latter half of this year...Our current problems are not the the product of the recovery program that's only just now getting under way, as some would have you believe...because our economic problems are deeply rooted and will not respond to quick political fixes, we must stick to our carefully integrated plan for recovery...-The only alternative being offered to this economic program is a return to the policies that gave us a trillion-dollar debt...and unemployment...Now the budget deficit this year will exceed our earlier expectations. The recession did that. It lowered revenues and increased costs...We look forward to the enactment of a responsible Clean Air Act to increase jobs while continuing to improve the quality of our air...The Federal Government will still subsidize 95 million meals every day. That's one out of seven of all the meals served in America. Head St art, senior nutrition programs, and child welfare programs will not be cut from the levels we proposed last year. More than one-half billion dollars has been proposed for minority business assistance. And research at the National Institutes of Health will be increased by over $100 million. While meeting all these needs, we intend to plug unwarranted tax loopholes and strengthen the law which requires all large corporations to pay a minimum tax..."

Sounds like what we heard a few nights ago? This was from Reagan's 1982 State of the Union speech. I have my opinions on this (I agree with both of them for the most part; whether good or bad, most policy on such a large takes time for the impact to be seen), but the main thing that has stood out for me in the current debates is the lack of consistency. Most of my friends and relatives have different views than I do and I respect that. However, I find myself questioning the sincerity of many of the arguments I hear when they run contrary to those they exhibited under a different administration. If deficits are bad, then they are bad under either party; if large health care entitlements are bad, it doesn't matter who proposes it; if a president is held to account for a failed terrorist attack, this should apply to all presidents. Again, I am not trying to convince anyone of any point of view, only asking that all stop to consider whether they truly believe in a philosophy or principle or if they are merely trying to help their team by attacking the other side.

All of this is not said because of offense taken at the humor; I think both sides need to be able to laugh at themselves and politics as a whole. The context merely brought up something that I had been wanting to address for some time now.