Math Notations & Symbols

Rhetoric Math to Symbolic Math

The development of mathematical notations & symbols can be divided into many stages. We would look at only the beginning and final stages for ease of understanding.

Symbols & notations in math can look scary & intimidating. They are just a short hand for words & phrases. Once you know their name & purpose, they will lose their mystery.

The beginning stage was the "rhetorical” stage is where calculations are described & performed by words and no symbols are used.

The "symbolic" stage is where comprehensive systems of notation supersede rhetoric.

Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, accelerated the use of symbolic math.

In chapter 3804 we talk in detail about the transition from Rhetoric Math to Symbolic Math.

Importance of Symbols

Math has thousands of different concepts, laws and theorems, and the human mind simply cannot remember or state them all “as is”. This is why mathematicians from the past decided to create symbols to represent the various units and concepts.

Symbolic math helped very precise expression of relations and mathematical statement, cutting out all details not necessary to solve a problem. It also enabled easy and even mechanical manipulation of math statements.

Mathematician A N Whitehead said - By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.

As we will see elsewhere, symbolic math can free our mind to think out-of-the-box, without asking whether such an expression was even possible. It is out of such questions that the discipline of math developed.

Standardization & Acceptance of Symbols

But it also required the invention of many symbols & signs. These inventions took many years before they gained general acceptance and there was some logic in the final shapes of the symbols.

According to a recent count, there are almost 400 commonly used mathematical symbols that cover numbers, basic operations, theorems, postulates, algebra, geometry, calculus, logic and everything in between.

Each of these symbols also has a name which is used when reading out. A list of some of the common symbols is given in chapter 1609.

Let us look at the history of some familiar symbols.

Numerals

We are so used to seeing numbers in their numeral form, that we forget that numerals are the first symbols we learn in math. Each language & number system has its own set of numerals.

The most familiar are the numerals 0, 1, 2 …. Up to 9.

Equal “=”

Robert Recorde (c. 1512 – 1558), a Welsh physician and mathematician, invented the equals sign (=) and also introduced the pre-existing plus sign (+)in 1557.

The equal sign “=” has two identical parallel bars because it represents how two objects or units are “the same” or balanced. Recorde is said to have commented that "nothing could be more equal".

He used the "=" symbol in writing an equation in his treatise "Whetstone of Witte" published in 1557.

Plus “+” & Minus “-“

Originally these symbols were used to indicate excess & shortage. The "+" sign is supposed to have evolved from the Latin term "et" meaning "all".

With time, the symbol evolved into the familiar “+” shape, first documented in the 15th century in an arithmetic text by Johannes Widmann.

The "-" sign has been in use right from the time of Egyptian and Greek mathematicians.

These signs are also used in representing positive & negative numbers.

Approximately Equal "≈"

The approximately equals sign, ≈, was introduced by British mathematician Alfred Greenhill (1847 - 1927).

Decimal Point or Comma "." & ","

The decimal system was introduced by Simon Stevin from Netherlands 1585 through his book titled De Thiende (meaning The Tenth). The system needed ways to separate the fractional part from the whole number part of a number. Many symbols were used for this purpose.

Decimals as they look today were used by John Napier, a Scottish mathematician who also developed the use of logarithms for carrying out calculations. The modern decimal point became the standard in England in 1619.

However, many other countries in Europe and others like South Africa still use the decimal comma.

Multiplication “X”, "Sin" & "Cos"

William Oughtred (1574-1660) introduced the "X" for multiplication. It replaced the (dot) "." which was previously used.

He also introduced the abbreviations "sin" and "cos" for the trigonometric functions sine and cosine, and "::" as the symbol for proportion.

Division “÷”

The symbol is also called "obelus". Johann Rahn introduced the division sign (÷) and the “therefore” sign in 1659.

Natural Logarithm "e", Sigma (Σ), Imaginary unit 'i" & Function "f(x)"

Leonard Euler was responsible for inventing “e” for the base of the natural logarithm, sigma (Σ) for summation, “I” for the imaginary unit, and the functional notation f(x).

The Square Root sign √

Christoph Rudolff introduced the square root symbol in his book “Coss” in 1525 . It was derived from the Latin term “radix,” meaning “root”.

The Infinity Symbol ∞

The infinity symbol, or lemniscate, was introduced by English mathematician John Wallis in 1655. The symbol depicts an unending, non-repeating loop, signifying the mathematical concept of infinity.

Some Exceptions

There are a few symbols which are used in different countries with different interpretations.

In the US, the dot "." is used as a symbol for multiplication.

In Many European countries the comma "," is used as the decimal separator instead of the point.

Math Notations

Along with math symbols, math also has formalised math notations, which are a way of writing symbols to convey certain meaning.

The Cartesian Coordinates

The cartesian coordinates as usually indicated in the form (x,y). The simple notion of assigning pair of numbers (x, y) to the points in a Euclidean plane bridged the gap between Geometry and Algebra simplifying the solution of many problems and proofs which otherwise would be tedious using the concept of Euclid Geometry alone.

Rational Numbers

The representation of rational numbers with a numerator & denominator with a separating line is another example of notation. This makes fraction operations very intuitive.

Leonhard Euler was also responsible for many of the notations currently in use: the use of a, b, c for constants and x, y, z for unknowns.

Math Syntax

For reading out a math statement which uses a lot of symbols, mathematicians have also developed a syntax so that the meaning of a statement is clear.

Syntax is an arrangement of the words while reading a statement.

An expression like 2(x+3) is read out as “two times the sum of x and three” and not “two times x plus three”.

Mathematics also follows conventions like BODMAS to make sure that the symbols are written in the correct order and manner to prevent a wrong calculation.

Grammar of Math

All these rules & conventions can be called the "grammar" of math.