Logarithms keep Doc Brown in Perspective

In the movie "*Back to the Future*, Part III", Doc Brown and Marty McFly go back to the 1880s in the days of the Wild Wild West. There Doc Brown meets the romance of his life, a woman named Clara. In one scene, at a bar, a man tries to tell him that there are other women out there. Doc Brown seeks to dispel that notion from him by saying, with starry eyes, "She's one in a million. She's one in a billion. She's one in a googolplex! The woman of my dreams, and I've lost her for all time." Is "one in a googolplex" reasonable? Do all big numbers seem about the same because they are so huge?

We use these numbers all the time. The National Debt is $4 trillion. The population of the United States is about 281 million people. The Milky Way has 100 billion stars. We complain when senators talk about spending hundreds of thousands of dollars in junkets. But a thousand is small compared to a billion, say, which is far less than the US government proposes spending on a national missile defense. In 1935, Martin Sirotta defined two large numbers: a googol, which is 1 followed by a hundred zeroes, and a googolplex, which is one followed by a googol of zeroes. He defined them for his father, the mathematician Ed Kasner, when he was only 8. These last two numbers are far larger than we encounter in our daily lives. But even million and billion seem really huge.

So how do we handle these numbers? We can count the number of digits in the numbers, or the number of zeroes that follow the 1. For example, in the United States, a billion is 1,000,000,000, with 9 zeroes after the 1. A trillion is 1,000,000,000,000 in the United States, with 12 zeroes after the 1. That suggests one way of measuring these numbers. 9 and 12 are easier to deal with than 1,000,000,000 and 1,000,000,000,000. So we say that President George Bush is proposing a cut in taxes of about $1,000,000,000,000, or, in a sense, a level 12 cut. A level 7 cut would cut $10,000,000 from a small town's budget.

We use this type of notation when using scientific measurements. For example, 1 followed by 15 (which is 5 x 3) zeroes is represented by "peta". A petabyte is 1,000,000,000,000,000 bytes, or a quadrillion bytes, or a million GB (gigabytes) or a billion MB (megabytes). A petabyte has level 15; a typical .jpg format picture has about 100,000 bytes and is level 5, so, since 15 - 5 = 10, we conclude that a number of pictures of level 10 can fit in a petabyte of storage. That is 10 billion pictures! We could use that to store the family pictures of everyone in the United States.

That works well for 1 followed by all zeroes, but what about other numbers? 100,000,000 has 8 zeroes after the 1, but how many zeroes follow the 1 in 281,421,096? It doesn't apply. Yet definitely the number is between level 8 and level 9. It is between 100,000,000 and 1,000,000,000. If we were to assign a level to 281,421,906, where would it be? Would it be 8.2? 8.281? 8.4? How do we assign such a fraction?

Mathematicians a long time ago found a way to assign such small numbers to large numbers. The way to do it is with logarithms. First, assign a logarithm to the number 1 followed by n zeroes. It is defined to be n, so that the logarithm of 1,000,000,000, with 9 zeroes after the one, is 9. The logarithm of a trillion, 1,000,000,000,000, is 12. The logarithm of a googol is 100, since a googol is one followed by a hundred zeroes. The logarithm of a googolplex is a googol, 1 followed by a hundred zeroes, because a googolplex has that many zeroes after the one.

Before we assign other logarithms, we find the mathematical properties of these logarithms. For example, we know that 2 + 3 = 5. So what does that mean about 100, 1000, and 100000? We note that 100 x 1,000 = 100,000. So when logarithms are added, their corresponding numbers multiply. This means that since 6 + 9 = 15, that a million (1,000,000) times a billion (1,000,000,000) is a quadrillion (1,000,000,000,000,000).

So with this in mind, what is level 0.5, a half? Note that 0.5 + 0.5 = 1, so that a x a = 10, where a is the number whose logarithm is a half. This means that a squared is 10, so a is the square root of 10, which is 3.162, approximately. So the level, or logarithm of 3.162 is a half. How about 3.5? Since 3 + 0.5 = 3.5, therefore 1,000 x 3.162 or 3,162 is the number corresponding to 3.5. I.e., logarithm of 3,162 is 3.5. By using computations of this sort we learn that the logarithm of 2 is 0.301, the logarithm of 7 is 0.845, and so forth, and we can create a table of logarithms. In past times, people constructed logarithm tables to simplify multiplication, which can be involved if it has large figures. The idea is to find the logarithms of the numbers and add them instead; addition is easier. Then convert it back. Today of course the calculator makes this obsolete.

But it does not make the logarithm itself obsolete. In many areas of life, we already use logarithms. For example, earthquakes are measured by the Richter scale. An R6, or Richter 6, earthquake, is 10 times as stronger as an R5 earthquake, which in turn is 10 times stronger than an R4 quake. But that is not how we feel it. An R6 quake is usually a massive quake, causing some destruction. An R5 may cause destruction and will shake things up a bit, while an R4 will be felt but nothing much may happen. It feels like the gap from R4 to R5 is about the same as R5 to R6, when actually the ratios are 1 to 10 to 100. In 1989 an R6.9 quake hit San Francisco, causing about 60 deaths and damaging some highways and a few buildings. In 2001 an R7.9 quake hit western India, causing over 10,000 deaths and knocking down just about every building in the city of Bhuj. The R numbers tell us that the Indian quake was 10 times stronger than the San Francisco quake, since it is one magnitude higher.

Music is measured in logarithmic terms, too. Middle C has twice the frequency of low C but high C is not three times the frequency; it is 4 times the frequency of low C. In music, these double intervals sound so much alike that they are regarded as the same pitch. Going up an octave does not change the pitch, only whether it is a high or low tone. This is a logarithmic scale, too, but here the ratios are based on two, rather than ten. This means that the frequency of F#, which is midway between two C's, is the square root of 2, not ten. It is 1.414, or close to 7/5. Low ratios of tones produce harmony, so efforts have been made to reconcile the low harmonic intervals (a fifth is 3/2, a third is 5/4 and so forth) and logarithms to base 2, but logarithms are irrational numbers and won't fit. The best compromise is the well-tempered scale, where the logarithms of the pitches are evenly spaced out, so that a piano, even when perfectly tuned, will always seem somewhat out of tune.

The brightness of stars in the sky are measured in a logarithmic scale, but it is inverted. The brightest stars in the sky are called "first magnitude" stars, and the faintest are called "sixth magnitude". By definition, a 1st magnitude star is 100 times brighter than a 6th magnitude star. Because ordinal numbers are used like this, the scale is inverted. Larger magnitudes mean fainter stars. If stars get brighter, then they acquire zero or even negative magnitudes. A star 100 times brighter than a 1st magnitude star is of magnitude 1 - 5 or -4. Venus, for example, has magnitude -4 and is bright enough to cast shadows.

We can measure other things in logarithms, too. How about population? A P3 town is rather small, or about 1,000 people. That's about the size, say of Louisa, Virginia. A P4 town has 10,000 people, and a P5 city has 100,000 people, about the size of Charlottesville, Virginia. A P6 city (metropolitan area) city has a million residents and is a medium-sized city like Rochester, New York. Los Angeles has about 15 million people and so is a P7.2 city, and New York is about P7.3. The largest city in the world is Tokyo, Japan with 26.4 million people, a P7.4. We have yet to see a P8 city; i.e., one bigger than P7.5 so that it rounds to P8, but with population growth like it is, we may see one soon. Of course we have P8 states like California and the United States as a whole is P8.4.

So anything that is measured in people can be measured on this P scale. There are about 200,000 Unitarian Universalists in the United States, making it a P5.3 group. A P3 convention is a large one, with a thousand attendees. A committee meeting of ten people is a P1 gathering, and so forth. Even people that get killed can be measured in this scale. Using a W for this scale, we can say that a murder is W0 (one person), and the Japanese attack on Pearl Harbor was about W3.3, but it led to a war that killed 50 million people, which makes that war, World War II, a W7.7 war. The Civil War killed 600,000 people and so is W5.7, and Vietnam for the US was about W4.6. A book called "The Statistics of Deadly Quarrels", by Lewis F. Robinson, describes all wars in recent history of levels W7, W6 and W5; also see John Redford's article.

If one works in a given universe, such as the world or the United States, then one can reverse this notation. About one out of a thousand people in the United States is belongs to Toastmasters International, so this is a Q3 organization. A Q2 group would have about 2,810,000 members in the US, about the size of Girl Scouts of America. Astrologers number 20,000 and so are about Q4, and astronomers, as 2,000, are Q5, a fact which I think should be the other way around, but this is what is really the case. The P level and the Q level of a group always add to the level of the population of the US, or 8.4. If your universe is the entire world with 6 billion people, the sum would be 9.7.

Even love and romance can be rated this way. This brings us *Back to the Future* and Doc Emmett Brown. Is it reasonable that his beau would be one in a million, a billion, a googolplex? We can use the same scale here. A woman who is one in a million would be an L6 (here I use L, meaning "love"). A one-in-ten-million person would be an L7, and one who is one in a billion would be an L9. It follows that if she were one in a googol, she would be an L100, and if she were one in a googolplex, she would be an L-googol. That is simply way out of this world. There are only a few billion people on this planet. Even one in a billion seems farfetched. There can be only about six L9's in the entire world. So I wouldn't even go that far, Dr. Brown.

But one in a million is much more reasonable. Incredible as it may seem, one should be able to find a partner who is one in a million, an L6. Suppose, for example, that you are a member of Toastmasters International, and that you insist that your spouse be a Toastmaster too. Then that means to you any Toastmaster is automatically one in a thousand, since only one in a thousand people is a Toastmaster. Such a prospect would already be an L3. Then if one goes to a large Toastmaster event, such as a Regional Conference (with 1000 attendees or so) and searches the conference for the best partner possible, then such a partner would be one in a thousand in one in a thousand, or one in a million, an L6. In terms of L numbers, 3 + 3 = 6. So finding such an L6 should be fairly easy, and if Doc Brown's girl friend is indeed the best out of that Western town of a thousand, and if he really values that town because it is way out of our time, then indeed, she probably was indeed an L6 for him, one in a million. Fortunately, he winds up with her at the end of the movie.

Dr. James V. Blowers

2001 February 14