The History of Mathematical Symbols

Tessa Gallant

History of Math

Galileo once said, “Mathematics is the language
with which God wrote the Universe.” He was correct in calling mathematics a language,
because like any dialect, mathematics has its own rules, formulas, and nuances.
In particular, the symbols used in mathematics are quite unique to its field
and are deeply rooted in history. The following will give a brief history of
some of the most famous symbols employed by mathematics. Categorized by
discipline within the subject, each section has its own interesting subculture
surrounding it.

__Arithmetic
__

Arithmetic most likely began with the need to keep track of
animals or bundles of food. It was a necessary tool utilized by our ancestors
to make it through the winter. Arithmetic is the most basic part of mathematics
and encompasses addition, subtraction, multiplication, and the division of
numbers. One category of numbers are the integers, -n,…-3,-2,-1,0,1,2,3,…n , where we say that n
is in . The capital letter Z
is written to represent whole numbers and comes from the German word, Zahlen,
meaning numbers (Gallian, 41). Two fundamental operations in mathematics,
addition, +, and subtraction, -, credit the use of their symbols to fourteenth
and fifteenth century mathematicians.
Nicole d' Oresme, a Frenchman who lived from 1323-1382, used the +
symbol to abbreviate the Latin “et”, meaning “and”, in his *Algorismus Proportionum.*

In
1489 the plus and minus symbols were printed in Johannes Widmann’s *Mercantile Arithmetic. *The German’s work can be seen in the
picture below (Washington State Mathematics Council).

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

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

__Radical__

The radical sign, originating from *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 *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)(16/5^{3}) – 4/239^{3 }+ (1/5)(16/5^{5}) - (4/239^{5})-…=
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). ^{2} 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.