A Practical Use For Logarithms, Part 2: How We Multiplied Large Numbers 40 Years Ago, And How Integral Transforms Use The Same Basic Idea
(Click here for Part 1.)
A common argument for the use of technology is that it frees students from doing boring, tedious calculations, and they can focus attention on more interesting and stimulating conceptual matters. This is wrong. Mastering “tedious” calculations frequently goes hand-in-hand with a deep connection with important mathematical ideas. And that is what mathematics is all about, is it not?
The desire to free students from boring technical matters is a false dichotomy: Mastering technique and deep conceptual understanding go hand-in-hand, and there is absolutely no reason why one can’t work on both in tandem. This is what music students do: To learn to play a musical instrument, one must spend a certain amount of time every day on theory and technique, and a certain amount of time every day practicing pieces of music, developing musicality, and so on. Trying to take a short-cut by not doing scales every day is deadly for a music student; can’t we see that the same kind of short-cut is deadly for a mathematics student, too?
A case in point is some of the algorithms we used to learn 40-odd years ago that have now been relegated to the slag heap. For instance, when I was in high-school (could it have been elementary school?) I learned an algorithm for extracting the square root of a number; nowadays, this is never taught, because we can quickly determine the result to many decimal places with hand-calculators, which were not available to students or teachers back then. Another example is the use of trigonometric tables. But the example I want to talk about in this post is the use of logarithm and anti-logarithm tables to facilitate the multiplication, division, and exponentiation of numbers, particularly large numbers.
So take yourself back, back, back, … back to a time when little me and my little classmates had no hand calculators. Let me show you the technique we learned to multiply large numbers, and then we’ll make a connection to higher mathematics.
The technique depends on a property of logarithms:
A common argument for the use of technology is that it frees students from doing boring, tedious calculations, and they can focus attention on more interesting and stimulating conceptual matters. This is wrong. Mastering “tedious” calculations frequently goes hand-in-hand with a deep connection with important mathematical ideas. And that is what mathematics is all about, is it not?
The desire to free students from boring technical matters is a false dichotomy: Mastering technique and deep conceptual understanding go hand-in-hand, and there is absolutely no reason why one can’t work on both in tandem. This is what music students do: To learn to play a musical instrument, one must spend a certain amount of time every day on theory and technique, and a certain amount of time every day practicing pieces of music, developing musicality, and so on. Trying to take a short-cut by not doing scales every day is deadly for a music student; can’t we see that the same kind of short-cut is deadly for a mathematics student, too?
A case in point is some of the algorithms we used to learn 40-odd years ago that have now been relegated to the slag heap. For instance, when I was in high-school (could it have been elementary school?) I learned an algorithm for extracting the square root of a number; nowadays, this is never taught, because we can quickly determine the result to many decimal places with hand-calculators, which were not available to students or teachers back then. Another example is the use of trigonometric tables. But the example I want to talk about in this post is the use of logarithm and anti-logarithm tables to facilitate the multiplication, division, and exponentiation of numbers, particularly large numbers.
So take yourself back, back, back, … back to a time when little me and my little classmates had no hand calculators. Let me show you the technique we learned to multiply large numbers, and then we’ll make a connection to higher mathematics.
The technique depends on a property of logarithms:
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