It’s a decimal, not a period.

Gabriellebirchak/ March 29, 2021/ Ancient History, Classical Antiquity, Middle Ages, Post Classical, Uncategorized

PODCAST TRANSCRIPTS

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.

Roman Biquinary Aba­cas, by Gabrielle Birchak.

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.

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