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.

A Fond Farewell aka What's in a Name?

It has been a while since I have posted. That does not mean that I have not been thinking. I have a very long list of ideas and a few partly written posts; I just have not had much time to write. Hopefully I will get a few more out on a regular basis, but I expect it to be more fits and starts as I work several at one time and occasionally finish multiples together. We will see....

On with this post...

What's in a name? that which we call a rose


By any other name would smell as sweet;

-Shakespeare, Romeo and Juliet

18 Feb 1930 – 18 Aug 2006

RIP, Pluto

On August 18, we marked the third anniversary of the passing of an old friend. On that date, five percent of scientists voted to pull the rug out under Pluto’s claim to planethood. This is not the first time this has happened. Ceres, discovered in 1801, was named a planet at one point. Later, the planets Pallas, Juno and Vesta (1802, 1804 and 1807, respectively) were added to the list. Over subsequent decades, several more bodies were found orbiting the sun between Mars and Jupiter, were named and given symbols in the manner of other planets. By the mid-1800’s, their number grew to the point that scientists had to acknowledge their definition of a planet was too liberal and had to be modified. At that point, the terms ‘asteroid’ and ‘minor planet’ came to be used for bodies in the asteroid belt, excluding the first four, which were still planets. Eventually (there is no definite time of death for them), the first four planetary homicides were committed by mankind. This took place through the delisting of them as planets in astronomical almanacs or alterations in nomenclature in such journals and in scientific observation.

Ceres, which went through several designations of planet, minor planet and asteroid (and was even given dual designation for a time), settled into what appeared to be permanent status as an asteroid until 2006. Ironically, it was in that year, simultaneous with the reduction in Pluto’s status, Ceres was once again elevated to minor planet. No one is quite sure if Ceres still carries the second designation as asteroid along with being a minor planet.

So why all the executions and exhumations? Scientists would have us believe it is all in the name of precision; we must be most careful and accurate in the way we classify. In fact, the careful precision is a necessary by product of practicality, born partly of shame from the lack of foresight that allowed us into this predicament.

If the total number of bodies sizeable enough to achieve a semi-spherical shape was in the single or low double digits, we could name them all planets and no one would say ‘boo’ (at least until we explored another solar system in which this was not the case). But, as going about things in this freewheeling manner results in a fairly large number of planets, with multiples sharing the same orbits, things get ugly in a hurry. So, we have to create constraints to bring the numbers back down. But is it necessary to rescind planetary status in order to do this?

It would be a simple enough thing to say, ‘this much and no more’, limiting planets to those bodies already bearing the definition. We could also say, ‘future discoveries must meet certain criteria to be designated planets’, allowing for minimal expansion, while grandfathering existing planets. All of these offend scientific sensibility because they rely on sentiment or tradition rather than a well-defined system of classification.

So there it is, in the name of consistency and scientific discipline, we must sacrifice our beloved nine (or ten or fourteen) planet solar-system and forever declare our planet to be one of eight, right? Well, no….

Many scientists claim that the agreed upon definition of planet excludes Earth, Jupiter and Neptune because they have not ‘cleared their neighborhood’ of other orbiting objects. This does not mean that they are not planets, as all eight planets are named in footnotes, but this could be done without even including a definition.

But all of this is beside the point. The criteria described for qualification as a planet is clearly worded to include the eight named planets and exclude Pluto, Eris, Ceres and the rest of the ‘dwarf planets’. This is quite an unscientific way to go about things; they just as easily could have created a definition to include some of the smaller planets (as some proposals did); it is also quite easy to imagine a scenario involving a body that all agree is a planet, but does not meet the criteria.

Of course, it is easy to criticize and much harder to offer solutions to the problem. This is where I go out on a limb as a non-scientist and offer my proposal:

First, I think the easiest way to go about this is to subdivide the category of planets (as is done with animals, flora, etc) into groups such as dwarf planets, minor planets, intermediate planets and major planets (and possibly superplanets, for use with some of the larger bodies discovered orbiting other stars). It might even be useful to have a multi-faceted definition including such descriptions as terrestrial, gas and ice, as well as double and triple. Using this, Pluto may be a triple or quadruple ice dwarf planet, while Earth would be a single intermediate terrestrial planet. Complicated? Yes, but it would be a consistent definition that would likely prove useful in extra-solar discoveries. It would also still be possible to define a group we call ‘historical planets’, which would include the nine I grew up with, along with Ceres and possibly Eris, which is the ninth largest known body orbiting the sun and largest in Pluto’s neighborhood (Kuiper belt); I would argue that this list is significant for more than sentimental reasons, as it outlines the expansion of our knowledge of the solar system.

JMHO, one of many I have (and believe make at least as much sense as those of the ‘experts’).

How hot is it?

How hot is it?

Some years ago, I was perusing a book my then-preschooler had picked up at a thrift store. This ‘science for kids’ book was likely written by someone with a stronger background in kids’ books than science. The text was easy enough to read and understand, but the facts and concepts were lacking.

Though years have passed and I have forgotten much of what was in the book, one thing still stands out in my memory. The author, trying to convey the intense heat of the planet Venus, made the statement that Venus was over 800 degrees Fahrenheit (which is true)-twice as hot as an oven and over six times as hot as the highest temperature on Earth (which is not true). Here’s why:

People think of temperature in terms of hot and cold, but in reality heat is a quantifiable metric, while cold is not. You can have any amount of heat and still be able to add heat; the coldest condition is the complete lack of heat known as absolute zero-if cold was ‘something’, you could still add more ‘cold’ at that point. This is where the problem comes in.

Fahrenheit is not an absolute scale. There are temperatures below 0F, so the zero is a point within the range of possible temperatures (not at the beginning). So, if it is 40 degrees in Maine and 80 in Florida, it is not twice as hot (or cold) in one place than the other. The problem is easier to see if you consider the case where it is -10 in Duluth and 30 in Sioux Falls; how do they compare? Is it -3 times as warm in Sioux Falls? The only way to express a ratio of temperature is to use an absolute scale like Kelvin or Rankine.

In any absolute scale, the 0 point is absolute zero. From there, it does not matter how large your increments are, they still maintain the same relative ratios. For instance, if you compared the two main absolute scales, Kelvin (using the increments from Celsius and subtracting 273.15) and Rankine (using the increments from Fahrenheit and subtracting 459.67), you would see that 100K=180R and 200K=360R. So, double the heat in one absolute scale and you double the heat in any other (note that 100K or 100C is the difference between water freezing and boiling, as is 180R and 180F). If we do the same with Fahrenheit and Celsius, we see that 100C=212F and 200C=392F; Celsius doubled, but Fahrenheit did not. Using smaller numbers would exaggerate this effect, while using larger ones would make the difference between 0C/F and absolute zero less significant; 10C=50F, 20C=68F, doubling C increases F by 36 percent; 1000C=1832F, 2000C=3632F, doubling C increases F by 98.3 percent-nearly the same.

So why does this matter? For the most part, I am just being difficult. But also, it really doesn’t make sense, especially when dealing with negative temperatures. And though I understand that science needs to be watered down to reach its audience, I hate the thought of starting kids of with a wrong concept of how things work.

In the end, though, it really matters when doing calculations. Where corrections are done in an experiment based on differences in heat, the ratios have to be based on an absolute scale. If heat accelerates a reaction in a linear manner, doubling of heat needs to mean double the quantity of heat and not just a number on a thermometer if the data is to be reliable.

Ideology and Pragmatism

As I get older (note the intentional non-use of ‘mature’), my body slows while my mind accelerates. I find I spend ever more time contemplating the time when I rule the world. If you didn’t know, it is a very demanding task and I am no longer sure I am up to it.

Take for instance the following scenario:

Imagine that you are an extreme believer in progressive taxation (this may be difficult for some, but remember this is just hypothetical). No matter how much the rich pay in taxes, they still have more than they need, while even with no taxation, there are those at the bottom who work hard and still have trouble keeping the lights on. You think that those who benefit the most from our society and economy have the greatest responsibility to give back to it.

Now suppose that studies have been done, economists have analyzed the data and the results are incontrovertible: eliminating taxes on large carat diamonds, luxury cars and private jets actually stimulates the economy to the extent that total tax revenue is higher than it was with the taxes and at the same time, more of the lower and middle classes will be hired into good paying jobs with benefits. Is it worth allowing the tax burden to become less progressive if it results in a net benefit for all?

There is evidence for economic growth through a reduction in taxes on investment and business, but there is also strong evidence for economic stimulus through tax cuts for, or payments to, those in the lower income brackets. The difference is in the specific structure of tax cuts or payments and the specific conditions of the economy. How ridiculous is a discussion of capital gains tax cuts at a time when there are no capital gains? But before we discuss which path is the ideal for the current circumstances, we need to decide how important the ideal is (perhaps for moral or cultural reasons) and how important bottom line results are. If we can all agree on this (or even agree to disagree to some degree), perhaps at that point we can discuss the merits of each side rather than each side just arguing a position they settled on years ago and have not considered since.

How about if you are dead set against the redistribution of wealth? This would include direct payments, as well as government benefits for the poor paid for by taxes on the rich or middle class. As far as you are concerned, each citizen should be responsible for himself and the strongest will survive and thrive.

Now, what if the numbers showed that a college education resulted in higher lifetime income and productivity? Let’s say that the increase in income is so great that it results in additional tax revenue exceeding the total cost of the education. In addition, the resultant increase in income also means an increase in economic activity that creates more jobs. What would you choose as the ideal role of government in the funding of education? Would you stick to your guns on the ideal of personal responsibility and leave educational funding up to the individual? Maybe you think we should take a middle road (much like we do today) by providing loans that allow each individual to get an education, but having them pay back the costs out of their greater earnings. Or on the other end of the spectrum, does logic and pragmatism prevail over ideals? Knowing that the greatest number of people would get a college degree and the greatest increase in long term economic growth would come from full government funding of higher education, do we sacrifice the belief in personal responsibility for the greater good?

The facts in this case are that a college education not only makes a marked difference in income, but that during slow economic times, the unemployment rate has an inverse relationship with education. It is also a fact that outside of loans, grants and scholarships, higher education is still heavily subsidized and a pure market solution would drop us so far below the rest of the industrialized world that we would be competing with developing nations over low skilled, dollar a day type jobs. There is also the likelihood that a free college education for everyone would result in a diminished return on investment. However, the debate always seems to center on fairness or opportunity and usually fails to address the net effect of any policy on the society and economy as a whole.

We can continue this line of questioning with the health care debate. Is our real concern the redistribution of wealth through subsidies? Is it government control over health care? Is it the threat to availability or quality? What are our major motivations or objections with regard to our position on health care reform? Like taxation, this is not a simple issue and there are data and anecdotal evidence to back up either side. There are also practical trade offs between the positions. But whatever our take, if we are to move forward in a positive manner, we need to honestly express our concerns and decide how much of our position is based on ideals and how much we will compromise these ideals to reach our end goal. Once there, we need to be open to the facts and how they frame the issue as a whole.

The same kinds of idealism vs pragmatism dichotomies can be played out with funding for the arts, first time homebuyer subsidies, alternative energy, etc, but the concept is the same. Anyone can find data to back up any position on any issue. There are statistics and misrepresented facts that can harden positions on both sides until there is no compromise, only winners and losers. In some instances, we end up with losers and losers in order to avoid all possible doomsday scenarios that have propagated into the argument. Why? Because we have created a zero sum game of politics and no one wants to be vulnerable by stepping out and doing the right thing.

None of this argues against ideals, I just argue for a better consideration of ideals. Do we want less disparity between classes or a better life for those at the bottom? Energy independence or cheap energy? More security or more convenience? Often these questions get lost in the rhetoric. Too often we fear the honest questions because they get to the heart of what we really want instead of what we claim to want. Sometimes they reveal the complexity of something that we really want to be simple.