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# Science Fair Project Encyclopedia

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# 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, functions, and variables. See table of mathematical symbols for a systematic list of the notation. Mathematical expressions are evaluated according to a conventional order of operation which provides for calculation, if possible, of any expressions within parentheses, followed by any multiplications and divisions done from left to right, finally any additions or subtractions done from left to right.

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## Counting

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 rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick 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 both 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 René 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 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 electrical 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.

## The negative side of computerized standards

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.

Examples of abstract visualization which properly belong to the mathematical imagination, can be found, for example in these sample visualizations of mathematical models. The need for such models abounds, for example, when the measures for the subject of study are actually random variables and not really ordinary mathematical functions.