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Published online 29 May 2008 | Nature | doi:10.1038/news.2008.866
Column: Muse
Why we should love logarithms
The tendency of 'uneducated' people to compress the number scale for big numbers is actually an admirable way of measuring the world, says Philip Ball.
I'd never have guessed, in the days when I used to paw through my grubby book of logarithms in maths classes, that I'd come to look back with fondness on these tables of cryptic decimals. In those days the most basic of electronic calculators was the size of a laptop and about as expensive in real terms, so books of logarithms were the quickest way to multiply large numbers (see 'What are logarithms'.
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Logarithms ARE beautiful. And so is the metric system! Metric facilitates thinking across vastly different scales from inifinitesimally small to infinitely big and everything in between. Why is the United States still stuck in the dark ages?
Hi, logarithm of a given number to a given base is the power or exponent to which the base must be raised in order to produce the given number. Going from a defined numerical point into a different numerical space and doing something and then going back is a powerful tool. The Fourier transform is a good example. Unfortunately the modern form of education where children use lab-tops in the school simple things such as e , mantissa are a forgotten language. I lot of the problems we have to solve deal with complex number, a week point of mathematics. We must develop new tools such as cellular automata a description of the Grieks Χαος, Khaos in my office. Regards Dr. Terence Hale
I agree. Logarithms are beautiful and needed in todays complex World. JT http://www.Ultimate-Anonymity.com
Paul Dunn: "Logarithms ARE beautiful. And so is the metric system! Metric facilitates thinking across vastly different scales from inifinitesimally small to infinitely big and everything in between. Why is the United States still stuck in the dark ages?" Apples and oranges, Mr Dunn. You seem to confuse logarithms and exponents - and the metric and US customary system of measurement accommodate exponential expression of large numbers equally well. The metric system allows everything to be expressed in orders of magnitude built around powers of 10, which is nice to have but not crucial. However, proponents of the metric system can take comfort in the fact that the United States officially adopted that system around a hundred years ago. We use it in medicine, in science, in military operations, and increasingly in commerce (for example, the "fifth" of a gallon which was formerly the usual size bottle of distilled liquor here in the US was replaced long ago by the 750 ml bottle. Why not a nice, round liter of liquor is a mystery to me. But the English have requested extra time to drink their beer in pint glasses, haven't they? You, same as us, count distance on the highway in miles (there was a brief attempt to impose metric units of land distance here, but it died late in the 1970s). Supply and demand are the culprits here. Eventually as globalization wraps its tentacles around us all, we'll go over to Systeme Internationale because we want to sell to other countries and buy from them. So rest easy, resistance (to the SI) is useless... resistance is useless.
What we call numbers and what they actually represent must be separated. Because we use base ten (an accident of morphhology ..10 fingers??) does not mean that the rest of nature has some strange affinity with the number ten. It is interesting that all natural constants (ie pi and e) are infact irrational numbers. They cannot be expressed as a finite digit. The real number system is fine for economic transactions (ill swap 4 oranges for 2 avocadoes). The ratios are linear. More oranges equals a proportional more avo's. Similarly the relation between the radius and diameter the circle, an intrinsic feature of nature, is a linear proportion yet our mathematics fails to absolutly quantify it. Logarithms capture the esence in nature of each sucessive number is a product of previous steps. Base ten logs.. each unit is 10 times the previous. In this way it captures the reltivesness of nature. Numbers to not exist absolutely, they are relative.
@Paul Dunn: I personally don't use the metric system because it is wrong, on so many points. Here are a few... When the French were deciding on the base measurement of length, they narrowed it down to several choices... A portion of the equator, a portion of an arc from the equator to the north pole, and the length of a pendulum with a one second period. They threw out the last because they thought the second was an arbitrary unit. They threw out the first because it would be too hard to define. So, they settled on the second, and spent many years, and the equivalent of many millions of dollars, to come up with an estimate that is wrong. They measured the farthest distance they could across France, added in England's measurement, and used astronomical observations to determine what percentage they had measured. They calculated it out, and were almost two kilometers short. So, the meter is almost two millimeters short; thereby, it is wrong, and arbitrary, and not any better than a yard. The length of a meter is now defined as the distance light will travel in a vacuum in so-many vibrations of a cesium atom. So, the meter is now defined by time, which the French said was arbitrary. The kilogram is defined as the weight of nearly pure water that can be contained in a cube with sides ten centimeters long. A gram is defined as one-thousandth of a kilogram... Or, a gram, which is s'posed to be the standard unit, is one millikilogram. And it goes on ad nauseum from there.
@ tim: having worked with both systems all my life long, I do prefer the metric system. Of course the basis of the units is arbitrary, as in any other system (can you still verify the length of the king's foot ?). A unit is, by nature, arbitrary, so, once accepted as a reference, it cannot be wrong. It is convenient when it can be reproduced everywhere whith accuracy, and when you can switch from one unit to any multiple without too much trouble or risk of mistake. The metric system has a property that is unique and makes it so easy to use, that the whole planet has signed the ISO convention (the USA signed in 189?) : you can change the order of magnitude very easily. 1000 grams in a kilogram, 1000 mm in a m. So every child of 9, knows that a dm3 of water weighs 1 kilogram, and can calculate in ten seconds that a cm3 of water would weight 1 gram, 1 m3 1 Mg. Can you tell me in ten seconds how many onces of water weighs a gallon, a cubic foot, a cubic inch ? Regards
Nail on the head Michel. @tim Can you honestly say you use standard because the French were off a bit in their calculations? The inch, or "thumb" in French is just that, the length of a thumb. And the foot, guess where that term came from? 12 inch is 1 foot. 3 feet is a yard. 1760 yards is a mile. Great system! lol. It is impossible to use decimals, and you are stuck using fractions. Fractions are either imprecise, or meaningless for small values. There is a good reason the whole world has switched to Metric. There is also a good reason america decided against it.
A picky point here: It's still logarithmic, but the energy release by an earthquake doesn't scale by 10 with each unit on the Richter scale. The amplitude of the shaking does, but the actual energy release goes up by a factor of 32 with a unit increase on the Richter scale (1000x for a 2 unit increase).