The word Divide shares its major root with the word widow. The root vidua refers to a separation. In widow the meaning is obvious, one who is separated from the spouse. A similar version of the word was often meant to describe the feeling of bereavement that a widow would feel. The prefix, di, of divide is a contraction of dis, a two based word meaning apart or away, as in the process of division in which equal parts are separated from each other. Notice that the vi part of vidua is also derived from a two word, and is the same root as in vigesimal (two tens), for things related to twenty. An individual is one who can not be divided.
In a division problem such as 24 / 6 = 4 the number being divided, in this case the 24, is called the dividend and the number that is being used to divide it, the 6, is called the divisor. The four is called the quotient. If the quotient is not a factor of the dividend, then some quantity will remain after division. This quantity is usually called the remainder, although residue sometimes is used. The Treviso Arithmetic uses the word lauanzo for remainder. In Frank Swetz's book, Capitalism and Arithmetic he gives, "The term lauanzo apparently evolved from l'avenzo, meaning a surplus, or in a business context, a profit." Swetz also points out that in the 15th Century the term partition (partire in Latin) was synonymous with the word divisision.
In today's schools almost every grade school student learns to divide, so students may be surprised to learn that in the 16th century schools Division was only taught in the University. One of the first arithmetics for the general public that treated the subject of division was Rechenung nach der lenge, auff den Linihen vnd Feder by Adam Riese. Here is how the Math History page at St Andrews University in Scotland described it,
I think it is even more astonishing that the sitution described still existed in 1550 in Germany. Perhaps the earliest "arithmetic" to provide instruction in the local vernacular of the common people was the 1478 "Treviso Arithmetic", so named because it was printed in the city of Treviso (the author is unknown) just north of Venice. Frank J Swetz writes about the situation in Capitalism and Arithmetic (pg 10):
Early algorithms for division:
By the middle ages there seem to have been five approaches to the process of division.
The first was called the Galley, galea, or Scratch method. This method was efficient in a period of expensive paper and quill pens since it required less figures than other methods. Even the modern long division method requires more figures. The name Galley was used because the resulting pattern after the division left a picture that seemed to remind the early reckoning masters of the shape of a ship at sail. The term “scratch” has to do with the crossing out of values to be replaced with new ones in the process. The ease with which this could be done on a sand board or counting board made it a popular approach in the cultures of the East, and the method is believed to come from the early Hindu or Chinese. For example, Cajori writes, "It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure." He also comments on the popularity of this method, " For a long time the galley or scratch method was used almost to the entire exclusion of the other methods. As late as the seventeenth century it was preferred to the one now in vogue. It was adopted in Spain, Germany, and England. It is found in the works of Tonstall, Recorde, Stifel, Stevin, Wallis, Napier, and Oughtred. Not until the beginning of the eighteenth century was it superseded in England. " Addendum (I just found a note that says that the famous Arab mathematician, Al-Khwarizmi, whose book laid the foundation for western introduction to the now-common Arabic numbers, used this method in his writing. The Hindu mathematical calculations were usually done on a dust covered tablet, and they would wipe out numbers instead of scratching through them so that in the final result, only results appeared. The scratch-out method in Europe was simply an adaptation for paper or slates. )
Here is an image comparing how the galley method works shown beside the current US Model for long division, which the Italians called a danda.
In 2005 I acquired a German student "copy book" from 1804 which seems to show the Galley division method and the student's illustration of the ship around the work. (below right) (the link to this old pic seems to be broken, will replace when I'm back in Michigan where the copy book is stored. Until then I share this similar image from https://3010tangents.wordpress.com/2014/10/08/mathematics-in-art-throughout-history/. )
A second method that was sometimes taught was the process of repeated subtraction. The image below shows an example from Ray's New Practical Arithmetic published in 1877. I have seen this method in an English textbook as late as 1961 (Public School Arithmetic by Baker and Bourne). It also appears in a 1932 US publication of Practical Arithmetic, by George H. Van Tuyl, and perhaps in others .
This method of subtraction grew into what is taught frequently as an alternative use to many modern classrooms, and I believe is the standard method at some levels for a program called "Everyday Math". Instead of slowly subtracting one divisor at a time, the use of simple multiples is used to group subtract.
For example, to divide 227 by 8, it is easy to see that 10 x 8 or eighty can be subtracted, so they might take out ten groups of eight repeatedly until the remaining part was too small to divide by 80. So after removing 20 groups of eight, there would be 67 remaining. At this point they might recognize that 8x8 is 64 so by removing 8 more sets of eight would leave only three remaining, so the quotient would be 28 with 3 remaining.
A video of this method is shown here.
(I have been informed that the "correct" term for this method is "partial quotients".)
A third method was called per repiego by parts, which I have seen in books into the 20th century. In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient. The problem below is modeled on a problem in the 1919 copyright A School Arithmetic, by Hall and Stevens.
A fourth method is presented in the Liber Abaci, by Fibonacci in 1202. After introducing how to divide by numbers of one digit,
and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit. A composed fraction might look like image at right. Fibonacci used the Arabic method of writing fractions from right to left, and this composed fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation, 67/75 with each part of the numerator being read over the product of all the denominators below or to the right.
The "composition" of 75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the fraction 1/75.
When he divides 749 by 75, he first uses only the first denominator, 3. The quotient of 749 by three is 249 with a remainder of 2. The 2 is placed as a numerator over the three, and the 249 is divided by the second number in the denominator (a five). 249 divided by 5 gives 49 with a remainder of four.
This remainder, 4, is placed as a second number in the numerator over the five in the denominator. Now the 49 is divided by the final number in the denominator (another five) and the quotient is 9 with another remainder of four. This four is placed over the final five and the nine is placed to the right as the quotient. Fibonacci then gives the answer of 749 divided by 75 as 9 and 4/5 + 4/25 + 2/75 or 9 74/75.
A fourth method, which is similar to what we would now called short division except that the student used a table of division or multiplication facts. The method was called per colona, by the column, or per tavoletta by the table, in reference to the table of facts used. An example of this method appears in another popular American arithmetic by Nicholas Pike, from 1826. The use of tables to aid in multiplication and division were a common practice from the 1400’s up to the early 20th century.
The fifth is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step. The image below shows a typical long division problem with the partial products crossed out and the resulting "Italian method" on the right.
The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 The Common School Arithmetic : prepared for the use of academies and common schools in the United States by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided.
In a Gresham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school."
Later as I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . (It seems when I just checked that the Met no longer allows that link, will replace ASAP) The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.
I recently found a site called
The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one
interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk, and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.
Recently (in 2015) James Tanton posted a short article about problems that are circulating on the internet such as (and this is the one he used) "What is the value of the following expression: 62 ÷ 2(3)+4, and then asked, "Is the answer 10 or is the answer 58?" (my personal choice for historical reasons explained below is 3.6)
I don't care to argue the possible choices, although Professor Tanton does a good job of that in his blog, but I'm more interested in the history of some symbols for division he mentions there, obelus, vinculum, and one he didn't, the solidus. In particular, I'm interested in how the usage may have changed over time.
The earliest of the three terms to appear was the vinculum, and it came to us from the Hindu or Arabic mathematicians between the seventh and twelfth century. Here is how it is described by Jeff Miller's excellent web page on the first use of math symbols
Now if you read Prof. Tanton's article, in which he ecstatically plugs the use of the vinculum, this is NOT what he is suggesting. The horizontal fraction bar made its way into western culture mostly on the back of Leonardo Fibonacci, who introduced both Arabic numbers, and some of their symbols. He referred to the fraction bar as "uirgula"; which has become the more modern word virgule, something like a wand or small rod. Unfortunately, today the virgule is a term interchangeable with the older term solidus, and you recognize it as the slanted fraction bar, as in 3/5 (and occasionally with an s like bend such as the current symbol for integration), but all that would come much later.
The use of the vinculum that has the professor so excited was introduced around 1452 by Nicholas Chuquet The word is from the diminutive of vincere, to tie. Vinculum referred to a small cord for binding the hands or feet often used to keep cattle from wandering too far afield as they grazed in common areas. The meaning in math is mostly unchanged from that original meaning. The vinculum notation was once used in much the same way we now use parenthesis and brackets to "bind together" a group of numbers or symbols. Where today we might write (2x+3)5 the early users of the vinculum would write . Originally the line was placed under the items to be grouped although a bar over the grouping became the more lasting usage (and still is in the symbols for the radical sign for roots, and the repeat bar for decimal fractions). The bar on top seems to have been first used by Frans van Schooten.
Dr Peterson at the Math Forum disagrees with calling the fraction bar a vinculum and has written, "I find no evidence, by the way, that it has ever properly been called a vinculum, which is a bar OVER an expression and serves to group it as parentheses do today. The fraction bar has something in common with that, but not enough in my opinion to justify the usage. With both vinculum and virgule used for other things, I just call it a fraction bar and am perfectly happy with that term!" (I'm OK with that, too.) Professor Tanton suggest that the vinculum, properly used, would eliminate questions about whether the answer to the question is 58, 10, (or 3.6).
The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking. Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger and the same name is applied to an editing symbol that looks like a little dagger, †. The root also gives rise to the word obelisk for a pointed pillar of stone.
The symbol(s) was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction, for instance, by the famed Adam Riese as early as 1525, although he did not use it exclusively, intermixing the standard horizontal subtraction bar. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659.
There has long been a controversy about whether the symbol was introduced to him by John Pell. Cajori in his famous book on mathematical notation says there is no evidence for this, but some later historians, Jacqueline A. Stedall for one, now think it quite probably was Pell's creation. Pell had been Rahn's teacher in Zurich and they communicated on the book. Pell was famous for vacillating over whether he would, or would not, let his name be used on information he shared with others.
Let me make it clear I am not an authority on math history and do not read German, but as I looked at the examples in Teutsche Algebra, I began to think that Pell/Rahn was not introducing this as a mathematical operator as it is now used. I could find no examples where the books used something like the expression in the problem in Prof. Tanton's blog. Instead it seems to be used exclusively for a shorthand in explaining the operations used.
Here is an image from page 76 of the Algebra, and it is using a method of teaching algebra by use of a 3 column format, which is certainly from the work of Pell. Each line contains a line number in the middle, instructions for what is being done to the equation in the left column, and the result in the right column. Today many solutions would simply show the sequence of equations in the right column.
The first two lines describe the given information. In the third line, the swirl is exponentiation and says that equation 1 has been squared on both sides. It is line 8 that provides the interesting note about the ÷ usage. The left column says equation 7 is divided by GG+1, but if you look at the right side, you will see that 7 ÷ GG+1 treats all the material to the right of the expression as if it were included in a parenthetical enclosure. Don't divide by GG and then add 1, but divide by the total quantity GG+1.
Now the two surprises here, for me, is that a) Rahn/Pell intends that the "÷" breaks the operation into two parts, the left and the right side as if they were enclosed in parentheses or marked with a vinculum. But the second, is that he doesn't use the expression as an operator in his expressions. Instead he uses the common horizontal division bar/vinculum common to others. So when did we begin to use "÷" as an operation with numbers. I do not have access to the great libraries that contain the early English arithmetics and algebras that eagerly adopted the obelus (it was almost never used anywhere except in English speaking countries), so I am hoping some of you who have more experience/access/knowledge can share so the rest of us will know. When did expressions like 62 ÷ 2(3)+4 first appear in arihtmetic/algebra books? (At the moment I suspect they are a 20th century creation.)
So what about the Solidus. The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for example 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.
The solidus was introduced as a fraction/division symbol first suggested in De Morgan's Calculus of Functions he proposes the use of the slant line or "solidus" for printing fractions in the text, as in 3/4. In 188 G. G. Stokes put this into practice. Cayley would write to Stokes, "I think the solidus' looks very well indeed . . . ; it would give you a strong claim to be President of a Society for the prevention of Cruelty to Printers."
Stokes, in explaining his choice, says that the slanted bar is already in use for fractions, and simply uses it to expand to algebraic division. Then he states an explanation of the operational use, "In the use of the solidus, it seems convenient to enact that it shall as far as possible take the place of the horizontal bar for which it stands, and accordingly that what stands immediately on the two sides of it shall be regarded as welded into one." He then gives examples that make clear that he intends that a / bc means . He even gives a method for a period stop to indicate that the grouping has ended, so a/b.c would mean
So when did this end. When did we make the switch to the confusion of PEMDAS or BEMDAS or whatever it is called in your country. Cajori (1929) suggests that when using division and multiplication, "there is at present no agreement as to which sign shall be used first." So it seems that the advent of memorized mnemonics independent of the symbol seems to have occurred later than that. Similarly in 1923 the National Committee on Mathematical Requirements of the MAA recommended that the ÷ and : for division be replaced with the / solidus "(where the meaning is clear}."
So I looked on my bookshelf and found a 1939 copy of The New Curriculum Arithmetics, Grade Seven. The authors are a professor of elementary education, a dean of a school of education, a superintendent of schools, and an elementary supervisor, surely folks who would be aware of the MAA recommendations, and yet, there was the ÷ all through the problem sets. What was not there was a section on order of operations, or any problems that went beyond " number ÷ number." No long strings of numbers and operations strung together.
Certainly the question was in the air, but unsettled in 1938 when Joseph A. Nyberg of Hyde Park HS in Chicago wrote in The Mathematics Teacher
Read the part in Italics again.... multiplication first, then division, without regard to the order. That is not what you are telling your students today (I hope). So maybe they were just working it out.... Nope, here is what N. J. Lennes had written in The American Mathematical Monthly in the article Discussions Relating to the Order of Operations in Algebra in February of 1917, 21 years earlier.
Better, right? then turn the page, and find
So there is our old friend the obelus used exactly as I suspect Pell and Rahn had intended (if they intended it to be used as an operator at all), and lower down the solidus in the manner that Stokes suggested, but apparently used in a way the users thought distinguished it from the use of the obelus. And you wonder why your students are confused?
I still have yet to resolve when the first use of the obelus appeared for division as an operator in an algebraic or arithmetic problem. Anyone who has more information, please share.
I will continue my search as time allows and when I find out more I will continue to update this post. Thank you for any information you can share.
I wrote about an experience that happened when I let my kids watch an old science fiction movie in class just before Christmas... The blog, and a followup requested by a teacher who admitted he wasn't really sure why the common "divide and multiply method worked... Here they are as a package...
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The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu(Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgeable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.
I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... 2/3 x 5/7 = 10/21. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divided by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...
I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the equivalent expressions that "15 dogs divided by 5 = 3 dogs." and just as naturally "15 dogs divided by 3dogs = 5" . Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then understand that 5 eighths divided by three eighths is just as clearly 5/3.
A few days after I wrote that blog I got a response that asked, more or less, "OK, why does the common algorithm work?"
This was my response
I want to make one comment about division of fractions that seems harder to visulaize than for general division, and then I hope to explain in simple terms just why "invert and multiply" works.
For every multiplication problem, there are two associated division problems; A x B = C begets C/A=B and C/B=A. Elementary teachers call these a "family of facts for C" (or did in the recent past.. educational language changes too fast for firm statments by a non-elementary teacher). So if we add units to one or both factors, appropriate units must be appended to the product. So how does this effect operations with fractions? Well if we have length, as in ANON's comment, then the division problem he states, "If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood" he is dividing length by length to get a pure scaler counting how many pieces (or fractions of a piece) will fit into another. In the case he gives, the answer would be only 14/15 of a piece... because the 2/3 unit length is not quite enough to provide a 5/7 unit length piece...
The multiplication associated with this operation is then 14/15 of 5/7 units = 2/3 units... What about the other division in this family of facts... 2/3 units divided by 14/15 (a scaler here, not a length)will give 5/7 units length. What is this situation describing? This seems the one most difficult for teachers and students alike. We all know what it means to divide a length into (by?) two pieces, but what sense does it make to divide it into 1/2 a piece.
We might try to make this clear to students by taking some common length (12 inches?) and see what happens if we divide it into (by) 8 pieces, then four, then two, then one, (each division is by half the previous number) and look at the pattern of lengths. 12/8=3/2; 12/4 = 3; 12/2 = 6; 12/1= 12... I am confident most students could identify the next numbers in the sequence, 12/ (1/2) = 24, and 12/(1/4) = 48.
At this point, using whole numbers as divisors, the pattern for "invert and multiply" seems obvious, but this is far from a why for all fraction problems.
Let's look at one more case where we sneak in a related idea at the elementary level. Given a problem like 3.5 divided by .04, the student is taught to "move the decimal places enough to make the divisor (.04) a whole number. What we do is another problem (350 divided by 4) that has the same answer (87.5)as the original. Another why does that work that is not often explained.
What do the two operations have in common.... multiplication by one. In each case we have a division (fraction) operation and we simply mulitiply the fraction by a carefully chosen version of one that will make it easier to do. If we view 3.5/.04 as a fraction, then every fifth grader knows that multipliying it by one will not change its value. This is the core of what we do to find equivalent fractions... to get 3/5 = 6/10 we multiply by one, but expressed as 2/2... The decimal division problem uses the same approach... we multiply 3.5/.04 by 100/100 to get another name for the same fraction, 350/4.
Now to explain "invert and multiply" we just use the same idea... dividing fractions is simply fractions which have fractions instead of integers in the numerator and denominator. We want to multiply by one in a way that the division problem will be easier. But the easiest number to divide by is one,... so why not pick a number that changes the denominator of the fraction over a fraction to be a one... that is, multiply by its reciprocal. So for 2/3 divided by 5/7 we can write
In today's schools almost every grade school student learns to divide, so students may be surprised to learn that in the 16th century schools Division was only taught in the University. One of the first arithmetics for the general public that treated the subject of division was Rechenung nach der lenge, auff den Linihen vnd Feder by Adam Riese. Here is how the Math History page at St Andrews University in Scotland described it,
"It was published in 1550 and was a textbook written for everyone, not just for scientists and engineers. The book contains addition, subtraction, multiplication and, very surprisingly for that period, also division. At that time division could only be learnt at the University of Altdorf (near Nürnberg) and even most scientists did not know how to divide; so it is astonishing that Ries explained it in a textbook designed for everyone to use."
I think it is even more astonishing that the sitution described still existed in 1550 in Germany. Perhaps the earliest "arithmetic" to provide instruction in the local vernacular of the common people was the 1478 "Treviso Arithmetic", so named because it was printed in the city of Treviso (the author is unknown) just north of Venice. Frank J Swetz writes about the situation in Capitalism and Arithmetic (pg 10):
From the fourteenth century on, merchants from the north traveled to Italy, particularly to Venice, to learn the arte de mercadanta, the mercantile art, of the Italians. Sons of German businessman flocked to Venice to study...
Early algorithms for division:
By the middle ages there seem to have been five approaches to the process of division.
The first was called the Galley, galea, or Scratch method. This method was efficient in a period of expensive paper and quill pens since it required less figures than other methods. Even the modern long division method requires more figures. The name Galley was used because the resulting pattern after the division left a picture that seemed to remind the early reckoning masters of the shape of a ship at sail. The term “scratch” has to do with the crossing out of values to be replaced with new ones in the process. The ease with which this could be done on a sand board or counting board made it a popular approach in the cultures of the East, and the method is believed to come from the early Hindu or Chinese. For example, Cajori writes, "It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure." He also comments on the popularity of this method, " For a long time the galley or scratch method was used almost to the entire exclusion of the other methods. As late as the seventeenth century it was preferred to the one now in vogue. It was adopted in Spain, Germany, and England. It is found in the works of Tonstall, Recorde, Stifel, Stevin, Wallis, Napier, and Oughtred. Not until the beginning of the eighteenth century was it superseded in England. " Addendum (I just found a note that says that the famous Arab mathematician, Al-Khwarizmi, whose book laid the foundation for western introduction to the now-common Arabic numbers, used this method in his writing. The Hindu mathematical calculations were usually done on a dust covered tablet, and they would wipe out numbers instead of scratching through them so that in the final result, only results appeared. The scratch-out method in Europe was simply an adaptation for paper or slates. )
Here is an image comparing how the galley method works shown beside the current US Model for long division, which the Italians called a danda.
The page the image is from has a nice step by step illustration of the process.
A second method that was sometimes taught was the process of repeated subtraction. The image below shows an example from Ray's New Practical Arithmetic published in 1877. I have seen this method in an English textbook as late as 1961 (Public School Arithmetic by Baker and Bourne). It also appears in a 1932 US publication of Practical Arithmetic, by George H. Van Tuyl, and perhaps in others .
This method of subtraction grew into what is taught frequently as an alternative use to many modern classrooms, and I believe is the standard method at some levels for a program called "Everyday Math". Instead of slowly subtracting one divisor at a time, the use of simple multiples is used to group subtract.
A video of this method is shown here.
(I have been informed that the "correct" term for this method is "partial quotients".)
A third method was called per repiego by parts, which I have seen in books into the 20th century. In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient. The problem below is modeled on a problem in the 1919 copyright A School Arithmetic, by Hall and Stevens.
divide 92467 by 168 or 4 x 6 x 7
4|92467
6|23116 …. groups of four and 3 units over
7| 3852 ….. groups of 24 (4x6) and 4 foursover
___550 groups of 168 and 2 twenty-fours over
The complete remainder is 2 (24) + 4(4) + 3 = 67
and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit. A composed fraction might look like image at right. Fibonacci used the Arabic method of writing fractions from right to left, and this composed fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation, 67/75 with each part of the numerator being read over the product of all the denominators below or to the right.
The "composition" of 75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the fraction 1/75.
When he divides 749 by 75, he first uses only the first denominator, 3. The quotient of 749 by three is 249 with a remainder of 2. The 2 is placed as a numerator over the three, and the 249 is divided by the second number in the denominator (a five). 249 divided by 5 gives 49 with a remainder of four.
This remainder, 4, is placed as a second number in the numerator over the five in the denominator. Now the 49 is divided by the final number in the denominator (another five) and the quotient is 9 with another remainder of four. This four is placed over the final five and the nine is placed to the right as the quotient. Fibonacci then gives the answer of 749 divided by 75 as 9 and 4/5 + 4/25 + 2/75 or 9 74/75.
A fourth method, which is similar to what we would now called short division except that the student used a table of division or multiplication facts. The method was called per colona, by the column, or per tavoletta by the table, in reference to the table of facts used. An example of this method appears in another popular American arithmetic by Nicholas Pike, from 1826. The use of tables to aid in multiplication and division were a common practice from the 1400’s up to the early 20th century.
The fifth is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step. The image below shows a typical long division problem with the partial products crossed out and the resulting "Italian method" on the right.
The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 The Common School Arithmetic : prepared for the use of academies and common schools in the United States by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided.
In a Gresham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school."
Later as I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . (It seems when I just checked that the Met no longer allows that link, will replace ASAP) The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.
I recently found a site called
The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one
interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk, and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.
The Agony and the Obelus, much Ado about Notation
I don't care to argue the possible choices, although Professor Tanton does a good job of that in his blog, but I'm more interested in the history of some symbols for division he mentions there, obelus, vinculum, and one he didn't, the solidus. In particular, I'm interested in how the usage may have changed over time.
The earliest of the three terms to appear was the vinculum, and it came to us from the Hindu or Arabic mathematicians between the seventh and twelfth century. Here is how it is described by Jeff Miller's excellent web page on the first use of math symbols
Ordinary fractions without the horizontal bar. According to Smith (vol. 2, page 215), it is probable that our method of writing common fractions is due essentially to the Hindus, although they did not use the bar. Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.
The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).
Several sources attribute the horizontal fraction bar to al-Hassar around 1200.
Now if you read Prof. Tanton's article, in which he ecstatically plugs the use of the vinculum, this is NOT what he is suggesting. The horizontal fraction bar made its way into western culture mostly on the back of Leonardo Fibonacci, who introduced both Arabic numbers, and some of their symbols. He referred to the fraction bar as "uirgula"; which has become the more modern word virgule, something like a wand or small rod. Unfortunately, today the virgule is a term interchangeable with the older term solidus, and you recognize it as the slanted fraction bar, as in 3/5 (and occasionally with an s like bend such as the current symbol for integration), but all that would come much later.
The use of the vinculum that has the professor so excited was introduced around 1452 by Nicholas Chuquet The word is from the diminutive of vincere, to tie. Vinculum referred to a small cord for binding the hands or feet often used to keep cattle from wandering too far afield as they grazed in common areas. The meaning in math is mostly unchanged from that original meaning. The vinculum notation was once used in much the same way we now use parenthesis and brackets to "bind together" a group of numbers or symbols. Where today we might write (2x+3)5 the early users of the vinculum would write . Originally the line was placed under the items to be grouped although a bar over the grouping became the more lasting usage (and still is in the symbols for the radical sign for roots, and the repeat bar for decimal fractions). The bar on top seems to have been first used by Frans van Schooten.
Dr Peterson at the Math Forum disagrees with calling the fraction bar a vinculum and has written, "I find no evidence, by the way, that it has ever properly been called a vinculum, which is a bar OVER an expression and serves to group it as parentheses do today. The fraction bar has something in common with that, but not enough in my opinion to justify the usage. With both vinculum and virgule used for other things, I just call it a fraction bar and am perfectly happy with that term!" (I'm OK with that, too.) Professor Tanton suggest that the vinculum, properly used, would eliminate questions about whether the answer to the question is 58, 10, (or 3.6).
The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking. Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger and the same name is applied to an editing symbol that looks like a little dagger, †. The root also gives rise to the word obelisk for a pointed pillar of stone.
The symbol(s) was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction, for instance, by the famed Adam Riese as early as 1525, although he did not use it exclusively, intermixing the standard horizontal subtraction bar. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659.
There has long been a controversy about whether the symbol was introduced to him by John Pell. Cajori in his famous book on mathematical notation says there is no evidence for this, but some later historians, Jacqueline A. Stedall for one, now think it quite probably was Pell's creation. Pell had been Rahn's teacher in Zurich and they communicated on the book. Pell was famous for vacillating over whether he would, or would not, let his name be used on information he shared with others.
Let me make it clear I am not an authority on math history and do not read German, but as I looked at the examples in Teutsche Algebra, I began to think that Pell/Rahn was not introducing this as a mathematical operator as it is now used. I could find no examples where the books used something like the expression in the problem in Prof. Tanton's blog. Instead it seems to be used exclusively for a shorthand in explaining the operations used.
Here is an image from page 76 of the Algebra, and it is using a method of teaching algebra by use of a 3 column format, which is certainly from the work of Pell. Each line contains a line number in the middle, instructions for what is being done to the equation in the left column, and the result in the right column. Today many solutions would simply show the sequence of equations in the right column.
The first two lines describe the given information. In the third line, the swirl is exponentiation and says that equation 1 has been squared on both sides. It is line 8 that provides the interesting note about the ÷ usage. The left column says equation 7 is divided by GG+1, but if you look at the right side, you will see that 7 ÷ GG+1 treats all the material to the right of the expression as if it were included in a parenthetical enclosure. Don't divide by GG and then add 1, but divide by the total quantity GG+1.
Now the two surprises here, for me, is that a) Rahn/Pell intends that the "÷" breaks the operation into two parts, the left and the right side as if they were enclosed in parentheses or marked with a vinculum. But the second, is that he doesn't use the expression as an operator in his expressions. Instead he uses the common horizontal division bar/vinculum common to others. So when did we begin to use "÷" as an operation with numbers. I do not have access to the great libraries that contain the early English arithmetics and algebras that eagerly adopted the obelus (it was almost never used anywhere except in English speaking countries), so I am hoping some of you who have more experience/access/knowledge can share so the rest of us will know. When did expressions like 62 ÷ 2(3)+4 first appear in arihtmetic/algebra books? (At the moment I suspect they are a 20th century creation.)
So what about the Solidus. The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for example 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.
The solidus was introduced as a fraction/division symbol first suggested in De Morgan's Calculus of Functions he proposes the use of the slant line or "solidus" for printing fractions in the text, as in 3/4. In 188 G. G. Stokes put this into practice. Cayley would write to Stokes, "I think the solidus' looks very well indeed . . . ; it would give you a strong claim to be President of a Society for the prevention of Cruelty to Printers."
Stokes, in explaining his choice, says that the slanted bar is already in use for fractions, and simply uses it to expand to algebraic division. Then he states an explanation of the operational use, "In the use of the solidus, it seems convenient to enact that it shall as far as possible take the place of the horizontal bar for which it stands, and accordingly that what stands immediately on the two sides of it shall be regarded as welded into one." He then gives examples that make clear that he intends that a / bc means . He even gives a method for a period stop to indicate that the grouping has ended, so a/b.c would mean
So when did this end. When did we make the switch to the confusion of PEMDAS or BEMDAS or whatever it is called in your country. Cajori (1929) suggests that when using division and multiplication, "there is at present no agreement as to which sign shall be used first." So it seems that the advent of memorized mnemonics independent of the symbol seems to have occurred later than that. Similarly in 1923 the National Committee on Mathematical Requirements of the MAA recommended that the ÷ and : for division be replaced with the / solidus "(where the meaning is clear}."
So I looked on my bookshelf and found a 1939 copy of The New Curriculum Arithmetics, Grade Seven. The authors are a professor of elementary education, a dean of a school of education, a superintendent of schools, and an elementary supervisor, surely folks who would be aware of the MAA recommendations, and yet, there was the ÷ all through the problem sets. What was not there was a section on order of operations, or any problems that went beyond " number ÷ number." No long strings of numbers and operations strung together.
Certainly the question was in the air, but unsettled in 1938 when Joseph A. Nyberg of Hyde Park HS in Chicago wrote in The Mathematics Teacher
Read the part in Italics again.... multiplication first, then division, without regard to the order. That is not what you are telling your students today (I hope). So maybe they were just working it out.... Nope, here is what N. J. Lennes had written in The American Mathematical Monthly in the article Discussions Relating to the Order of Operations in Algebra in February of 1917, 21 years earlier.
I still have yet to resolve when the first use of the obelus appeared for division as an operator in an algebraic or arithmetic problem. Anyone who has more information, please share.
I will continue my search as time allows and when I find out more I will continue to update this post. Thank you for any information you can share.
Division of Fractions by the Alien Method (and followup)
I wrote about an experience that happened when I let my kids watch an old science fiction movie in class just before Christmas... The blog, and a followup requested by a teacher who admitted he wasn't really sure why the common "divide and multiply method worked... Here they are as a package...
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The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu(Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgeable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.
I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... 2/3 x 5/7 = 10/21. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divided by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...
I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the equivalent expressions that "15 dogs divided by 5 = 3 dogs." and just as naturally "15 dogs divided by 3dogs = 5" . Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then understand that 5 eighths divided by three eighths is just as clearly 5/3.
A few days after I wrote that blog I got a response that asked, more or less, "OK, why does the common algorithm work?"
This was my response
I want to make one comment about division of fractions that seems harder to visulaize than for general division, and then I hope to explain in simple terms just why "invert and multiply" works.
For every multiplication problem, there are two associated division problems; A x B = C begets C/A=B and C/B=A. Elementary teachers call these a "family of facts for C" (or did in the recent past.. educational language changes too fast for firm statments by a non-elementary teacher). So if we add units to one or both factors, appropriate units must be appended to the product. So how does this effect operations with fractions? Well if we have length, as in ANON's comment, then the division problem he states, "If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood" he is dividing length by length to get a pure scaler counting how many pieces (or fractions of a piece) will fit into another. In the case he gives, the answer would be only 14/15 of a piece... because the 2/3 unit length is not quite enough to provide a 5/7 unit length piece...
The multiplication associated with this operation is then 14/15 of 5/7 units = 2/3 units... What about the other division in this family of facts... 2/3 units divided by 14/15 (a scaler here, not a length)will give 5/7 units length. What is this situation describing? This seems the one most difficult for teachers and students alike. We all know what it means to divide a length into (by?) two pieces, but what sense does it make to divide it into 1/2 a piece.
We might try to make this clear to students by taking some common length (12 inches?) and see what happens if we divide it into (by) 8 pieces, then four, then two, then one, (each division is by half the previous number) and look at the pattern of lengths. 12/8=3/2; 12/4 = 3; 12/2 = 6; 12/1= 12... I am confident most students could identify the next numbers in the sequence, 12/ (1/2) = 24, and 12/(1/4) = 48.
At this point, using whole numbers as divisors, the pattern for "invert and multiply" seems obvious, but this is far from a why for all fraction problems.
Let's look at one more case where we sneak in a related idea at the elementary level. Given a problem like 3.5 divided by .04, the student is taught to "move the decimal places enough to make the divisor (.04) a whole number. What we do is another problem (350 divided by 4) that has the same answer (87.5)as the original. Another why does that work that is not often explained.
What do the two operations have in common.... multiplication by one. In each case we have a division (fraction) operation and we simply mulitiply the fraction by a carefully chosen version of one that will make it easier to do. If we view 3.5/.04 as a fraction, then every fifth grader knows that multipliying it by one will not change its value. This is the core of what we do to find equivalent fractions... to get 3/5 = 6/10 we multiply by one, but expressed as 2/2... The decimal division problem uses the same approach... we multiply 3.5/.04 by 100/100 to get another name for the same fraction, 350/4.
Now to explain "invert and multiply" we just use the same idea... dividing fractions is simply fractions which have fractions instead of integers in the numerator and denominator. We want to multiply by one in a way that the division problem will be easier. But the easiest number to divide by is one,... so why not pick a number that changes the denominator of the fraction over a fraction to be a one... that is, multiply by its reciprocal. So for 2/3 divided by 5/7 we can write
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