It’s a decimal, not a period.
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Before I begin today’s podcast, I want to send out a special thank you to Melissa Rodgers and the family of Lloyd Rodgers. Lloyd Rodgers was a Professor Emeritus of music at California State University Fullerton, where he taught for 44 years. Sadly, he passed away in December of 2016.
Professor Rodgers is the composer of most of the incredible music that you hear throughout my podcasts. He was a brilliant musician and professor with a gripping post-minimalist style. As described on Wikipedia, his music was inspired by medieval mensural notation, Renaissance polyphony, and Baroque counterpoint. It is tonal and modal. For me, I would describe his musical compositions as unconventionally, geometrically, and abstractly eloquent. Melissa, his wife, wrote to me in an email that they were both kind of math and history geeks. Which really does my heart good.
I came across his music on a public domain platform while looking for music for my podcast. Apparently, Professor Rodgers was adamant about making his music public domain. Which in some way is not only incredibly altruistic but also impressively rebellious against the capitalistic structure of the music industry. I find that inspiring. I also find all of his music incredibly rousing and moving. It is creative and filled with depth and emotion beyond explanation and beyond words. And so I urge you to visit www.LloydRodgers.com. Through his music, he inspired hundreds of students, moved thousands of hearts, and made my podcast a better podcast. I am so grateful to the Rodgers family for letting me use Lloyd Rodger’s music. From the bottom of my heart, thank you very much.
Now that Pi day is over, I have only one question: Where did the decimal come from? Well, in the grand scheme of mathematics, it is a relatively new application.
The indication of a number smaller than a whole integer has been evident in mathematics for thousands of years long before the decimal even was used. That decimal has always fascinated me. When I was 13, my oldest brother, John, gave me my first calculator, which by the way, was a Sharp, EL-240H solar cell calculator. I would spend hours on the sofa dividing numbers on my new calculator to see which fractions would give me a repeating decimal.
Even the ancient calculators, the bi-quinary abacus, had fractions, which indicated fractions. On this abacus, they had whole numbers that allowed them to create large numbers and small numbers on the right side of the abacas that showed the fractions ⅓, ¼, and ½.
Some historians believe that John Napier first presented the decimal in the sixteenth century, which is somewhat untrue. Though he did use the decimal, it was sporadic and not consistently used with all of his mathematics. Though Napier did know something about decimals, he did not contribute to the symbolism of the dot between two numbers.
Decimals came about because of fractions. When another number in some cases divides a number, you have a remainder, which can be represented as a fraction. As a result, when our ancestors were dividing in base 60, they would list those fractions indicating the value following the whole integer.
For example, in Book III of the Almagest, written by Hypatia in the late fourth century, she had to show the length of one day as it related to the celestial arc of the sun. So, she created a division table to help her readers find an accurate answer. That division table resembled something like the multiplication charts that we use today. This chart helped her readers divide for specific values by 360, and she was able to find a result to nine values, which means she found her result to nine decimals.
Once we started using base 10, mathematicians continued to list fractions. However, mathematicians had no way of indicating a decimal value other than by listing fractions. Eventually, instead of listing fractions, mathematicians would list tables that the reader could refer to, much like Hypatia’s table.
The problem that arose by the fifteenth century was that there were too many fractions. And for books being printed on a printing press, they needed to find an efficient way to indicate these fractions without listing them all. And so, the decimal started to be used sporadically throughout medieval manuscripts, first in the forms of a comma or a vertical bar. The methods for using the comma and bar were used in medieval England, India, and China. However, the use of the comma and the vertical bar were sporadic and inconsistent.
Then, in July 1424, Al-Kashi wrote the Treatise on the Circumference, which was a masterpiece that showed the value of Pi to 16 decimal places. In this case, Al-Kashi used the decimal system but did not use the decimal for the demarcation between whole numbers and their fractions.
By 1492, a mathematician by the name of Pellos used the decimal in his work. However, it was not intentional. It was simply his way to abbreviate values.[i]
In 1522, Adam Riese presented a table in his book Rechenbuch, which means Arithmetic. In this table, called the Tabula Radicum quadratarum, also known as the Table of Square Roots, he presented a list of values that would follow the decimal when the value’s root is calculated. However, there was still no use of a decimal point.
It was not until 1530 that Christoff Rudolff, the mathematician who wrote the first German algebra book, specifically used the concept of the decimal. In his work, Exempel-Büchlin, Rudolff solved an example of compound interest. Even though Rudolff did not specifically use the decimal but instead a vertical bar, he still understood the intent of the bar, the comma, and the decimal that was used to separate the whole number from its remaining value.[ii]
Finally, in 1585, the Dutch mathematician Simon Stevin published a 35-page booklet called De Thiende, which translates to The Art of the Tenths. In De Thiende, he presented his decimal fractions. Stevin was a big advocate for using the decimal. It was Stevin’s goal to teach others how to do math with “integers without fractions.” In some of his works, he even stated that the government should adopt the decimal system.
Thus, by the sixteenth and seventeenth centuries, the decimal became a standard placeholder for values less than an integer.
What I love about the decimal is that unlike the period in a sentence, it does not indicate an end. When you place that little dot at the end of a sentence, the statement is over. The thought is completed. There is no more.
However, when you place that little dot after a number, on some level, it indicates that there is more to come and that the whole integer is not the end of the statement.
Case in point, for the value of Pi, after the number three, there is a decimal and then a list of fractional parts that go on for an infinite amount of values. When you see that decimal in math, you know that there are more fractional values to hope for and more numbers to come. That beautiful little dot, the decimal, indicates that there is no end but rather more accuracies to discover.
And with that said, I will be ending the run of this podcast Math! Science! History! not with a period, but rather with a decimal. I began this podcast 18 months ago, and this is my 50th Episode! and it has been a wonderful journey through the world of podcasting. We live in a wonderful age where so much knowledge is available to us in so many different forms. I have learned so much about my listeners and what kinds of stories you want to hear. Some of the best emails I received were from those of you who never thought you were a math person until you listened to my podcast and gave math a go. Those emails meant the world to me!
I said this in my first podcast, and I will say this in my last podcast, we are descendants of mathematicians. Math is part of our DNA, and we are all mathematicians. But, learning math is like playing an instrument. The more you do it, the easier it gets. So, I encourage you all to honor your inner mathematician and trust in your own intellect. Never stop learning math.
Hopeful things are coming after my decimal point, including new projects, which I will undoubtedly share with you. In the meantime, you can always find me at www.Twitter.com/GabBirchak and www.Instagram.com/GabrielleBirchak!
I assure you, there is more math, science, and history to come. Take care of yourselves, and always remember to seize the day! Carpe diem, my friends!
[i] David Eugene Smith, History of Mathematics, Volume II (New York: Dover Publications, 1958), 238.
[ii] David Eugene Smith, History of Mathematics, Volume II (New York: Dover Publications, 1958), 240.