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:

\log_{10} (AB) = \log_{10} (A) + \log_{10} (B)

Suppose little 1973 me had the task of multiplying 18793.26 by 54778.18. Using the multiplication algorithm would take a bit of time, but it’s feasible. But here is the time-saving technique we were taught: Let A = 18793.26 and let B = 54778.18. Now look up the logarithm of each of the numbers from a table. (Back then we would have relied on tables in the back of our textbooks, but the only book on my shelf that has such tables is my 1971 copy of the CRC Standard Mathematical Tables, 19th edition. The upcoming 2011 edition is here.)

Reading from the table for figures close to A:

\log_{10} (18790) = 4.27393    and     \log_{10} (18800) = 4.27416

Now if we linearly interpolate between these two figures, for greater accuracy, we obtain the approximation

\log_{10} (A) = 4.274005

Reading from the table for figures close to B:

\log_{10} (54770) = 4.73854    and     \log_{10} (54780) = 4.73862

Now if we linearly interpolate between these two figures, for greater accuracy, we obtain the approximation

\log_{10} (B) = 4.738605

Next, we use the property of logarithms mentioned earlier to estimate the logarithm of AB:

\log_{10} (AB) = \log_{10} (A) + \log_{10} (B) = 4.274005 + 4.738605 = 9.01261

The process of adding logarithms is very easy, and this is the point of the method. We’ve taken a relatively complicated problem (multiplying two numbers that have many digits) and converted it to a much easier problem (adding two numbers that have many digits). Now we have to convert the result back into the realm of the initial problem.

Next, we convert \log_{10} (AB) = 9.01261 to exponential form:

AB = 10^{9.01261} = 10^{0.01261} \times 10^9

Using a table of “anti-logarithms,” as they were called back then (i.e., a table of powers of 10), we read that:

10^{0.012} = 1.028    and    10^{0.013} = 1.030

Interpolating again, we get the approximation that

AB = 1.0292 \times 10^9

Using a hand calculator, the result is

AB = 1.029460579 \times 10^9

so the approximation using logarithms is correct to four significant figures.

The only way to really appreciate how much work is saved using logarithms is to actually multiply A and B by hand.

Besides the value in taking a little trip down memory lane (which is always useful for students, to inform them about how things were done in the past), there is a more general lesson that one can take from this little calculation technique.

IDEA: If you are having difficulty solving a mathematics problem, see if it is possible to transfer the problem into a different realm, where it is easier to solve a related problem, and then transfer the result back into the initial realm to obtain the solution to the original problem.

This is a valuable problem-solving idea. Another example of this idea is the use of Laplace transforms in solving certain differential equations. The idea is to convert a differential equation into an algebraic equation, solve the algebraic equation (which is easier than solving the differential equation directly), and then use an inverse transform to convert the resulting algebraic expression back into the realm of the original problem.

Pedagogically, it’s very useful to have the logarithm example of this post in your back pocket before you encounter Laplace transforms; once you realize they are both instances of the same basic idea, it helps you to understand the big picture in which Laplace transforms are writ, and it helps you to get the hang of the Laplace transform method.

There are lots of other instances of the same basic idea. There are lots of other integral transforms (Fourier transforms are just one type), and in signal processing one frequently switches back and forth from the time domain to the frequency domain. Integral transforms are also used in the computer software that converts raw data from medical imaging devices to the lovely images that doctors then peruse. The same ideas are used in analyzing crystal structure using X-ray diffraction, and more generally in quantum mechanics one often switches from configuration-space representations to momentum-space representations. (The crystallographers speak of “space” and “reciprocal space,” and also reciprocal basis, and reciprocal lattice.)

One also encounters the same idea in a technique for solving troublesome real improper integrals: One switches to the complex domain, evaluates a related contour integral using the techniques of complex analysis, then switches back to the real line to evaluate the real integral.

Back to the technique described in this post. The same idea can also be used to divide numbers with many digits, and to raise a number to another number; one just uses the appropriate properties of logarithms. Try it for yourself and see if you can get this to work!


About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. https://qedinsight.wordpress.com Further resources, and online tutoring, can be found at my other site http://www.qedinfinity.com
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19 Responses to A Practical Use For Logarithms, Part 2: How We Multiplied Large Numbers 40 Years Ago, And How Integral Transforms Use The Same Basic Idea

  1. Immensely informative. Instills interest in oneself before moving into the real working maths. Wish this should be taught first/made understood by the students before they are actually made to work out.
    Only in India students are not taught the practicality and I learnt the real applications only when I was 24 years old. Till then I frown at maths stating ‘Math n all its calculations are okkay. But when am I going to use it in Life?’
    Keep it up Sir. Good Job.

  2. mathew says:

    Very informative post professor, perhaps a sneak peek into your childhood for us.I worked out a few problems using the log and anti log tables(never heard of it) and I agree with you on doing tedious calculations to appreciate the power of logarithms.I took a course in Linear Algebra this summer and I found multiplying matrices of length 5 X 5 or 7 x 7 to find the jordan canonical form too laborious and asked myself the question of whether I’ll use this again but it has certainly enhanced my appreciation of using maple/matlab.Is there anything interesting from your childhood that involves matrices that we are not used to today?

  3. Yunus says:

    log (explanation) (good) = good

  4. Yunus says:

    Good illustration.

  5. Matt says:

    Seems to me that having logarithm tables isn’t much different than having a calculator. You start out by stating the importance of performing tedious calculations (like doing scales on the piano) and then go right in to using lookup tables to perform a calculation. You don’t find log tables anymore because it’s outdated technology.

    I guess you learn something about logarithms and linear interpolation in the process.

    Conrad Wolfram has another take on where computers fit in to teaching math:

    Regardless, thank you for this illustration. This practical example of logarithms went a long way in furthering my understanding.

  6. Markus says:

    Wow! I wish I had stumbled upon this blog while I was in college dong Laplace transformations. Really well explained and informative!

    Greetings from Finland!

  7. Justin says:

    Great post! Thanks for taking the time to write this. The first thing I thought of was how similar this idea was to the integral transform techniques so it’s great to see you mention them.

  8. tomcircle says:

    Great post, Prof. Thanks!

    I also used the log table in high school for 2 years (1971), then ‘advanced’ to ‘Slide Rule’ in undergrad years, by the time I graduated as engineer, the younger students switched to calculator by 1980 …

    P.S. Spotted small typo errors at the Anti-log
    section, should be 10^0.12 and 10^0.13
    You typed 0.012 and 0.013.

    • Thanks for your kind comments.

      Thanks also for pointing out that there is a typographical error; I’ve double-checked and the error appears one display earlier than the one you mentioned; I should have typed 10^0.01261 instead of 10^0.1261. I’ve made this correction now.

      Thanks again!

  9. tomcircle says:

    Would like to share the similar idea of “RMI” (Relationship-Mapping-Inverse) technique developed by a Chinese prof. Basically it is the same spirit of solving a problem in one area by using a method in another area of math.
    Like (log X.Y = log X + log Y)


  10. Mark says:

    I thoroughly enjoyed this Professor. Your introduction was most useful to me pointing out how logarithms apply to aspects of our everyday life and most of us (well me) are unaware of their applicability.

  11. Hey Prof – Thanks! Great article which I will take time to go back and fully digest. I was in high school in the mid-eighties just as log tables and slide rules were fast becoming a thing of the past, or already were a thing of the past! I am now ignominiously going back and relearning math through calculus (and probably beyond) on my own through MOOC’s and videos – but this time around I am learning it to understand it, and not just to pass tests. This article and articles like it are helping me frame a new subject area before going on to learn and work out the details. Thank you immensely!

  12. K Fleming says:

    The ‘idea’ presented here isn’t one I ever picked up on, even remotely, in learning LaPlace and Fourier transforms. The ‘why’ of such transforms and in moving between time and frequency domains to solve engineering problems was lost on me. No one’s fault–but in playing with a sliderule last night, I found this page, and feel it opened a very wide door. Maybe in my 60s I’ll now be able to learn what I only managed to ‘crunch’ 4 decades ago. Thanks!

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