Apr 10, 2017

Logarithms: The #STEM Story Behind Human Calculators

This is actually from the Historical Interlude for the 4th lesson of LASER Classroom's Bringing STEM to Light. However, with the recent Hidden Figures (#HiddenFigures) movie, it seemed that this interlude seemed timely for math teachers who might want to show how math integrates (#STEM at its best).


image by Y. Ng

The importance of logarithms is not always covered in many of today’s pre-Calculus classes. This “invention” from the 1600s is mathematically useful because it allows us to use a small scale to analyze values that cover a wide range, which permits us to see patterns and relationships better.
Compare these two figures, which show how wide range of data looks when graphed on a regular scale versus a logarithmic one. The regular scale hides the details of the actual y values and for x=1 through x=7, but the log scale allows us to see how those values compare with the y values.


More importantly, in the days before electronic computers, logarithms gave humans a faster and more accurate way to do multiple-digit multiplication and division. In fact, many scientific discoveries and technological tools depended heavily on logarithms, including Kepler’s calculations of planetary orbits, British navigational charts for exploration during colonial times, and artillery trajectory charts for Napoleonic campaigns.

Today, the logarithmic scale is still present in our everyday lives through scales for sound (decibels), acidity (pH), and earthquakes (Richter).


How were logarithms developed? Since ancient Babylonian times (2000 BC), humans have looked for methods to do multiplication and division of large numbers quickly and accurately. Until the relatively recent advent of computers and personal calculators in the 1960s and 1970s, people used methods include used using addition, subtraction, and lookup tables. In 1614, Scottish mathematician John Napier published the logarithm concepts that he noticed from studying arithmetic and geometric series. In 1620, the Swiss mathematician and clockmaker Jost Bürgi published tables of logarithm values that could be used in calculations.

In 1859, a French army officer named Amédée Mannheim created the modern form of the slide rule. This toolkit for multiplication and division of multi-digit numbers, reciprocals, powers, roots, and trigonometric values. It relied on the logarithm property that logx (y) + logx (z) = logx (yz). The slide rule uses two rulers marked with a log scale (rather than a regular scale). By lining up the two rulers, you can add two log values together, as shown in the left-hand side of the equation above. If you then read the answer off the log scale rulers, you get the product of the two values (the right-hand side of the above equation). Thus, the slide rule provides a way to do multiplication of large numbers by using edition.


Today’s “modern” technological advances were first developed using only this simple tool–not the complex electronic computers that we associate with calculations day. Rocket design and the development of the atomic bomb during World War II, determining the shape of DNA in the 1950s, missions into space and to the Moon and design of the aircraft (like the Boeing 747) in the 1960s and 1970s–all of these advances were done on slide rules, using logarithms!