# Mathematical notation

**Mathematical notation** is used in

mathematics, and throughout the

physical sciences,

engineering, and

economics. The complexity of such notation ranges from relatively simple

symbolic representations, such as

one and

two; to conceptual symbols, such as

+ and

*dy/dx*; to equations,

functionss, and

variables. See

table of mathematical symbols for a systematic list of the notation.

It is believed that a mathematical notation was first developed, at least 50,000 years ago, in order to assist with

counting. Early mathematical

ideas for counting were represented by collections of

rockss, sticks,

bone,

clay,

stone,

wood carvings, and knotted

ropes. The

tally is a timeless way of counting. Perhaps the oldest known mathematical texts are those of

ancient Sumer. The

Census Quipu of the

Andes and the

Ishango Bone from

Africa all used the

tally mark method of accounting for numerical concepts.

## Geometry becomes analytic

The mathematical viewpoints in geometry did not lend themselves well to counting. The

natural numbers, their relationship to

fractions, and the identification of

continuous quantities actually took millennia to take form, much less allow for the development of notation. It was not until the invention of

analytic geometry by

Rene Descartes that geometry became more subject to a numerical notation. However, some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs, for example. The power and authority of the custom of geometrical style of Theorem and Proof was even followed by the great

Isaac Newton's

Principia Mathematica,though he did not use geometry to invent his concepts, but instead blazed a new trail through the invention of

calculus to understand the

System of the World.

## Counting is mechanized

After the rise of Boolean algebra and the development of

positional notation, it became possible to mechanize simple circuits for counting, first by mechanical means, such as gears and rods, using

rotation and

translation to represent changes of

state, then by electical means, using changes in

voltage and

current to represent the analogs of quantity. Today, of course,

computers use standard circuits to both store and change quantities, which represent not only numbers, but pictures, sound, motion, and control.

## Computerized notation

The rise of expression evaluators such as calculators and slide rules were only part of what was required to mathematicize civilization. Today, keyboard-based notations are used for the e-mail of mathematical expressions, the Internet shorthand notation. The wide use of

programming languages, which teach their users the need for

rigor in the statement of a mathematical expression (or else the compiler will not accept the formula) are all contributing toward a more mathematical viewpoint across all walks of life.

There is a part of mathematics which is not algebraic, but which seems to use a different faculty of the mind. For those people with such minds and imaginations, like Isaac Newton's, if they are to benefit from the wide availability of mathematical devices, then they will need to be served by more

graphical,

visual,

aural,

tactile, and

temporal modalities in notation, as a first step.