History of Mathematical Symbols

Why Do We Use Symbols in Math?

Sometimes it is the little things that are the most important and you could lump mathematical symbols into this category. It is undeniable that symbols not only enhance understanding but also provide a universally perceivable manner in which to show a certain math function or illustrate a sequence. This is not a new concept. It has been around in math since ancient times. It was probably even around in one form or another during the stone age!
The fundamental need in math is to represent the relationship between a sign and the number or value it refers. Certain ideas and concepts can be clearly illustrated only by the creation and use of symbols. Measuring the relationship between numbers and representing the relationship symbolically not only serves to simplify the process but also gains a better understanding of the concept than a wordy description of the same. This is where the issue of languages comes in.
In more simple terms, a plus sign, a minus sign, a multiplication sign are all symbols. We need them for a very simple reason: we have to express what we are doing in a clear manner. When we are adding it would be ridiculous to always write out one plus on equals two when we could express this symbolically with 1 + 1 = 2. Imagine trying to perform calculus if you have to write a lengthy equation out in several paragraphs. Not only would such prose be voluminous, it would be confusing and prone to error. Plus, what language do you want to write it in? Remember, math is universal but languages are incredibly vast. Simply put, without proper symbols math becomes next to impossible. In fact, you could look at it this way: the symbols of math are reflective of a mathematical language.


Math is comprised of primarily two things: numbers and symbols. Symbols are found in simple math, algebra, geometry, calculus, statistics, etc. Symbols are essentially representative of a value. Decimals and fractions, for example, are symbols of parts of a whole. These symbols allow us to "work with" parts in a theoretical manner. Without symbols you simply could not perform math. Remember, much of math is abstract. How could you possibly perform simple algebra - much less calculus -without having the use of the symbol "X"? Could you even imagine trying to perform geometry without symbolic representations of triangles, squares and rectangles? It simply can not be done or if it was done it would be so laborious that it wouldn't be as efficient.
It is important to understand that the key to comprehending math is in the interpretation of the concept and not really in the nature or amount of symbols and the role they play. However to understand concepts one must essentially have a good grasp of the role and meaning of symbols and also be able to appreciate their usefulness in making math that much more simpler to understand and duplicate. The logic of signs and symbols in math is undeniable and is often stressed as a vital tool in making math a universal science.

 Because symbols are so common in math we sometimes take them for granted. The reason we take them for granted is that they make math so easy to perform (actually, they make math performable period) we do not really tip our hat to their true value. That does not seem like a great way to treat the very thing that makes expressing math possible. Without various symbols you would be forced to go back to counting your fingers and toes and you don't want to do that again do you?

Where and When Did the Symbols “+” and “–” Originate?

The symbols for the arithmetic operations of addition (plus; “+”) and subtraction (minus; “–”) are so common today we hardly ever think about the fact that they didn’t always exist.  In fact, someone first had to invent these symbols (or at least other ones that later evolved into the current form), and some time surely had to pass before the symbols were universally adopted.  When I started looking into the history of these signs, I discovered to my surprise that they did not have their origin in antiquity.  Much of what we know is based on an impressively comprehensive and still unsurpassed research (in 1928–1929) entitled History of Mathematical Notations by the Swiss-American historian of mathematics, Florian Cajori (1859–1930).

The ancient Greeks expressed addition mostly by juxtaposition, but sporadically used the slash symbol “/” for addition and a semi-elliptical curve for subtraction.  In the famous Egyptian Ahmes papyrus, a pair of legs walking forward marked addition, and walking away subtraction.  The Hindus, like the Greeks, usually had no mark for addition, except that “yu” was used in the Bakhshali manuscript Arithmetic (which probably dates to the third or fourth century).  Towards the end of the fifteenth century, the French mathematician Chuquet (in 1484) and the Italian Pacioli (in 1494) used “” or “p” (indicating plus) for addition and “” or “m” (indicating minus) for subtraction.
There is little doubt that our + sign has its roots in one of the forms of the word “et,” meaning “and” in Latin.  The first person who may have used the + sign as an abbreviation for et was the astronomer Nicole d’Oresme (author of The Book of the Sky and the World) at the middle of the fourteenth century.  A manuscript from 1417 also has the + symbol (although the downward stroke is not quite vertical) as a descendent of one of the forms of et.
The origin of the – sign is much less clear, and speculations range all the way from hieroglyphic or Alexandrian grammar ancestry, to a bar symbol used by merchants to separate the tare from the total weight of goods.

Figure 1. The first use of the +and – symbols in print in Johannes Widman’s Behëde und Lubsche Rechenung auff allen Kauffmanschafft, 

and – symbols in print in Johannes Widman’s Behëde und Lubsche Rechenung auff allen Kauffmanschafft, Augsburg edition of 1526.

The first use of the modern algebraic sign – appears in a German algebra manuscript from 1481 that was found in the Dresden Library.  In a Latin manuscript from the same period (also in the Dresden Library), both symbols + and – appear.  Johannes Widman is known to have examined and annotated both of those manuscripts.  In 1489, in Leipzig, he published the first printed book (Mercantile Arithmetic) in which the two signs + and – occurred (Figure 1).  The fact that Widman used the symbols as if they were generally known points to the possibility that they were derived from merchants’ practices.  An anonymous manuscript—probably written around the same time—also used the same symbols, and it provided input for two additional books published in 1518 and 1525.
In Italy, the symbols + and – were adopted by the astronomer Christopher Clavius (a German who lived in Rome), the mathematicians Gloriosi, and Cavalieri at the beginning of the seventeenth century.
The first appearance of + and – in English was in the 1551 book on algebra The Whetstone of Witte by the Oxford mathematician Robert Recorde, who also introduced the equal sign as the rather longer than today’s symbol “═.”  In describing the plus and minus signs Recorde wrote: “There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse.”
As a historical curiosity, I should note that even once adopted, not everybody used precisely the same symbol for +.  Widman himself introduced it as a Greek cross + (the sign we use today), with the horizontal stroke sometimes a bit longer than the vertical one.  Mathematicians such as Recorde, Harriot and Descartes used this form.  Others (e.g., Hume, Huygens, and Fermat) used the Latin cross “†,” sometimes placed horizontally, with the crossbar at one end or the other.  Finally, a few (e.g., De Hortega, Halley) used the more ornamental form “.”
The practices of denoting subtraction were somewhat less fanciful, but perhaps more confusing (to us at least), since instead of the simple –, German, Swiss, and Dutch books sometimes used the symbol “÷,” which we now use for division.  A few seventeenth century books (e.g., by Descartes and by Mersenne) used two dots “∙∙” or three dots “∙∙∙” for subtraction.
Overall, what is perhaps most impressive in this story is the fact that symbols which first appeared in print only about five hundred years ago have become part of what is perhaps the most universal “language.”  Whether you do science or finances, in Kentucky or in Siberia, you know precisely what these symbols signify.

Followers soon adopted the notation for addition and subtracting. The fourteenth century Dutch mathematician Giel Vander Hoecke, used the plus and minus signs in his Een sonderlinghe boeck in dye edel conste Arithmetica and the Brit Robert Recorde used the same symbols in his 1557 publication, The Whetstone of Witte (Washington State Mathematics Council). It is important to note that even though the Egyptians did not use the + and – notation, the Rhind Papyrus does use a pair of legs walking to the right to mean addition and a pair of legs walking to the left to mean subtraction (see below)(Weaver and Smith).  Similarly, the Greeks and Arabs never used the + sign even though they used the operation in their daily calculations (Guedj, 81).

 

 

The division and multiplication signs have an equally interesting past.  The symbol for division,¸, called an obelus, was first used in 1659, by the Swiss mathematician Johann Heinrich Rahn in his work entitled Teutsche Algebr. The symbol was later introduced to London when the English mathematician Thomas Brancker translated Rahn’s work (Cajori, A History of Mathematics, 140). Before the explanation of how the letter “x” became used to mean multiplication, a short biography must be presented for the man who has contributed so much, both directly and indirectly, to mathematical notation, William Oughtred. Oughtred lived in England during the late 1500’s and into the early 1600’s and was educated at Eaton and King’s College Cambridge. He then went on to teach some very studious pupils, one of whom was John Wallis, whose name will come up again in the history of mathematical notation (O'Connor and Robertson). Oughtred is credited with using 150 different symbols in his work, however, one of the few modern survivors is the “x,” meaning multiplication. Oughtred’s cross can be        seen below (Weaver and Smith).

It was not all smooth sailing for Oughtred, as he received some opposition from Leibniz, who wrote, "I do not like (the cross) as a symbol for multiplication, as it is easily confounded with x; .... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC.LM." (Weaver and Smith). It wasn’t until the 1800’s that the symbol “x” was popular in arithmetic. However, its confusion with the letter “x” in algebra led the dot to be more widely accepted to mean multiplication (Weaver and Smith). Oughtred’s name will appear again in the history of math, his contributions were significant and widespread.

Equality and Congruence

The contributions of Oughtred’s fellow countryman, Robert Recorde, are also notably profound. In his 1557 book on algebra, The Whetsone of Witte, Recorde wrote about his invention of the equal sign, "To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can be moare equalle" (Smoller).

A similar looking symbol, º, meaning “congruent,” was credited to Johann Gauss in 1801. He stated “-16º9(mod. 5),” which means that negative sixteen is congruent to nine modulo five (Cajori, A History of Mathematical Notation, 34). During the same time period, Adrien-Marie Legendre tried to employ his own notation for congruence. However, he was a bit careless because he used the “=” twice to mean congruence and once for equality, which, needless to say, angered Gauss. (Cajori, A History of Mathematical Notation, 34). Gauss’ notation stuck and that is what is still used today in number theory and other branches of mathematics.

Inequalities

Three British mathematicians, Harriot, Oughtred and Barrow, popularized the early symbols for “>” and “<”, meaning strictly greater than and strictly less than. They were first used in Thomas Harriot’s The Analytical Arts Applied to Solving Algebraic Equations, which was published in 1631 after he died (Weaver and Smith). In 1647, Oughtred used the symbol on the left to stand for greater than and the symbol in the middle for less than (see below). Then in 1674, Isaac Barrow used the notation on the right in his Lections Opticae & Geometricae meaning "A minor est quam B" (symbols below from Weaver and Smith).

           

Almost one hundred years later, in 1734, the French mathematician Pierre Bouguer, put a line under the inequalities to form the symbols representing less than or equal to and greater than or equal to, “£” and “³”(UC Davis, 2007). Bouguer’s notation, like variations of the British inequality signs, is still in use today.

Factorial

The factorial, like other symbols in math, has a multinational background, with roots in Switzerland, Germany and France. In 1751 Euler represented the multiplication of (1)(2)(3)…(m) by the letter M and in 1774 the German Johann Bernhard Basedow used “*” to mean 5*=(5)(4)(3)(2)(1).  It wasn’t until 1808, with Christian Kramp’s contributions, that the term n! meant n(n-1)(n-2)…(3)(2)(1) (Cajori, A History of Mathematical Notations, 72). 

Radical 

The radical sign, originating from Italy and Germany, has a Middle Eastern connection as well. Initially, it was used by the Italian mathematician Rafael Bombelli, who lived in the sixteenth century, in his l’Algebra. He wrote that “R.q.[2]” is the square root of 2 and “R.c.[2]” is the cube root of 2 (Derbyshire, 84). During this time, Arab mathematicians had the

symbol pictured at the left to mean square root, however it was not widely adopted elsewhere (Weaver and Smith). It wasn’t until the seventeenth century, with the help of Descartes, that the symbol that we still use today was employed (see below) (Weaver and Smith).

 

  

Descartes, who lived in the early part of the 1600’s, turned the German Cossits “Ö” into the square root symbol that we now have, by putting a bar over it (Derbyshire, 92-93). 

 Infinity

The symbol “¥” meaning infinity, was first introduced by Oughtred’s student, John Wallis, in his 1655 book De Sectionibus Conicus (UC Davis). It is hypothesized that Wallis borrowed the symbol ¥ from the Romans, which meant 1,000 (A History of Mathematical Notations, 44). Preceding this, Aristotle (384-322 BC) is noted for saying three things about infinity: i) the infinite exists in nature and can be identified only in terms of quantity, ii) if infinity exists it must be defined, and iii) infinity can not exist in reality. From these three statements Aristotle came to the conclusion that mathematicians had no use for infinity (Guedj, 112).  This idea was later refuted and the German mathematician, Georg Cantor, who lived from 1845-1918, is quoted as saying; “I experience true pleasure in conceiving infinity as I have, and I throw myself into it…And when I come back down toward fitineness, I see with equal clarity and beauty the two concepts [of ordinal and cardinal numbers] once more becoming one and converging in the concept of finite integer” (Guedj, 115). Cantor not only accepted infinity, but used aleph, the first letter of the Hebrew alphabet, as its symbol (see below) (Reimer). Cantor referred to it as “transfinite” (Guedj, 120). Another interesting fact is that Euler, while accepting the concept of infinity, did not use the familiar ¥ symbol,

but instead he wrote a sideways “s”.

א

Constants

One of the most studied constants of all time, p, the ratio of the circumference of a circle to its diameter, 3.141592654, has been long studied and closely approximated. It was originally written by Oughtred as p/d, where p was the periphery and d was the diameter.  In 1689, J. C. Sturmn, from the University of Altdorf in Bavaria, used the letter e to represent the ratio of the length of a circle to its diameter; however it did not catch on. Pi was introduced again in 1706 by William Jones. Jones was looking over the work of John Machin and found that he used p to mean the ratio of circumference to diameter. In Jones’ book, Synopsis Palmariarum Mathesos, he praises his intelligence by calling him “the Truly Ingenious Mr. John Machin” whom states “in the Circle, the Diameter is to the Circumference as 1 is to 16/3 -4/239 –(1/3)(16/5³) – 4/239³  + (1/5)(16/5⁵) - (4/239⁵)-…= 3.14159…= p”   (Arndt, Haenel, 166). In subsequent years Johann Bernoulli used “c” to represent pi and Euler used “p” in 1734 and then “c” in 1736 to represent the constant. Then Euler changed his mind again, and later in 1736 used p in his Mechanica sive motus scientia analytice exposita and then cemented it into mathematical culture with his 1748 work entitled Introductio in analysin infinitorum. (Arndt, Haenel, 166).

Another important mathematical constant is e, 2.718281828. This irrational number, meaning the base of natural logarithms, as studied by John Napier, was originally called M by the English mathematician Roger Cotes, who lived from 1682 to 1716 (Trinity). Newton first used the exponential notation a² to mean “e”, and Leonhard Euler replaced the “a” with an “e” most likely because e comes after a in the procession of vowels (Trinity). His “e” appeared in Mechanica and was later used by Daniel Bernoulli and Lambert (Cajori, A History of Mathematical Notations, 13). Euler’s choice of letter went down in history.

The square root of negative one is another important constant, with a simpler, less varied background. Again Euler’s approach to notation has been wedged in time. In 1777 he published Institutionum calculi integralis, where i is the square root of negative one, and has been undisputed ever since (UC Davis).

The mathematical symbols discussed here have long and convoluted pasts, quarreled over by different mathematicians spanning the ages, and some revised at a later date. Certain representations came into existence through mercantile records and others were born out of necessity to provide mathematicians with convenient shorthand for repetitious calculations. Although their creators have perished with the passage of time, their notations are still prevalent today and continue to play an integral part of our mathematical world.