Basic Mathematical Notations

Mathematics seems so complete and packed with already complicated abstractions that we sometimes forget the fact that the most essential parts of math, symbols and notation, also started as a blank and share the entire history with math. They did not happen to exist from the top for one to conveniently play with numbers, or technically numerical values. Early mathematicians had to come up with their own notation and symbols and often tried to standardize them as means of communication for abstract operations in the arithmetic, algebra, and geometry. As it took many years (it is an understatement) to establish Hindu-Arabic number system in 8th to 9th century, considering that the first numeral system was invented by Sumerians in 3400 BC, the historical process by which the notation has reached current fashion is an exhaustively long journey filled with creativity and stubborn struggles.

The attempts to systematize the notation came from several directions across the world, and each of small contributions facilitated the study of math, especially in arithmetic and algebra, and eventually built the foundation for advanced topics.

The movement to what we now see as “the most basic” notations involved three stages. The first was rhetorical notation, and the second syncopation-based notation. They were the earliest, and probably the easiest approach to denote arithmetic relations, operations, and values. The two were used from ancient to 15th to 16th century, basically until the first symbolic notation emerged.

The rhetorical notation was a series of drawings and descriptive words to indicate an operation, and generally used in ancient mathematics like Egypt and Mesopotamia. Also in ancient Greek mathematics, there was no major move to systematize notation, and mathematicians mostly relied on natural language. Well, except that Aristotle denoted general number quantities by capital letters, and Euclid adopted this method in geometric algebra where he specified segments by letters with respect to their lengths.

The first syncopation-based notation appears in Diophantus’ work. A letter or syllable of the words denoted the unknown and its power.

math symbols


Image taken from

However, Diophantus’ notation had limitations of lacking generality and clarity because when a problem had more than one unknown, he had to individually point out first unknown, second unknown, and so on.

India joined this hot trend of the syncopation method by adopting the abbreviations of the words as signs. Addition was represented by yu from the word yuta, subtraction by xa from xaya, mutiplication by gu from guna, division by bha from bhaga, square root by mu from mula, and equality by pha from phalah.

In order to solve the issue Diophantus had faced, India interestingly used “colorful” notation for unknowns. The second unknown appears as ca from the word calaca meaning ‘black,’ the third unknown as ni from nilaca meaning ‘blue’, the fourth unknown as yellow, the fifth as red, and so on. The notation of powers was a combination of signs of a square and a cube, which were va and gha respectively. So, for example, the fourth power was va-va.

Mathematical Notations from Florian Cajori

Image: A History of Mathematical Notations by Florian Cajori.

Despite the tediousness, the syncopation method could have been an easy way to represent the abstraction at the time and communicate within a community, but apparently it was so inseparable to culture and language that unification across the world couldn’t possibly be achieved.

It is actually surprising that full symbolism of basic arithmetic and algebra was achieved not very long ago. Almost all math was written in the rhetorical and syncopation methods before late 15th century. After experimenting with creative notations, mathematicians started to acknowledge the simplicity and efficiency of the symbols, and symbolism grew into another topic of math. They continued to experiment with various graphic marks, which they modified, sometimes changing them completely, until they found the most successful symbols to use. Apart from the continual progress in study of math, it was a slow, collaborative process rather than an individual’s eureka-y invention that everyone happily agrees to follow. Thus, regarding the fact that progress in developing notations was not in the same speed as other branches of math, it should not be a huge surprise that the symbol of equality that we use today was not used in print before 1757 (basically around the time when everybody was talking about trigonometry and differential analysis!), when the Welsh mathematician and physician Robert Recorde designed the symbol ===== to avoid writing “is equal to” over and over in his book Whetstone of Witte.

Nicole Oresme first used the plus sign + in 1360, which was an abbreviation for et, meaning ‘and’ in Latin, and in 1489, Johannes Widmann introduced the minus sign – and used both + and – in his work, Mercantile Arithmetic. Now they are known to be the most widely used arithmetic symbols. A German mathematician named Christoph Rudolff introduced the radical symbol for square root in 1525, and Albert Girard came up with the radical signs for nth root in 1629. Nicolas Chuquet introduced first potential notation of exponent in 1484, and Rene Descartes in his book The Geometry in 1637 simplified the notation of powers, unknown variables, and constants, which converges to modern fashion. Superscript letters or numbers were applied to denote the exponentiation like xy, a2, and a3. The unknown variables were denoted by the small letters, x, y, z, w from the end of the alphabet, and known constants by a, b, c, d, e from the beginning of the alphabet. The symbols bleached out the vagueness of the verbal expression and also decisively, or I would say economically, facilitated the process of formulating problems and prepping the operations for effective solution. It is truly marvelous that every part of math, even including all tiny details I took for granted and overlooked, is a written form of humanity’s most abstract ideas to explain the world a little better.



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