數學符號
维基百科，自由的百科全书
- 另見數學符號表
數學符號不只被使用於數學裡，更包含於物理科學、工程及經濟學內。此一符號的複雜度可由較簡單的符號表示，如數字1及2；函數符號sin和+，至概念性符號，如lim和dy/dx；至等式及變數。
目录 |
[编辑] 定義
書寫的工具記述如下，但一般現今的工具大部份為紙和筆，或者是電腦螢幕和鍵盤，以及黑板和粉筆。數學符號的一關鍵要點為對數學概念的系統性依附，此亦記述如下。(其他相關概念還有：主語、邏輯論證、信服、數理邏輯和模型論等。)
[编辑] 表示式
表示式是能被估算的一連串符號。例如，若符號代表字，則表示式依一定的運算順序來估算。在電腦科學中，此一規則被編譯器所執行。更多有關表示式的估算，請見電腦科學主題：熱情計算、惰性計算及評估運算子。
[编辑] 精確語意
- See Abstract models vs models in mathematics
Precision is necessary so that we can know what we are investigating. Suppose that we have statements, denoted by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the symbols refer to those denoted objects, perhaps in a model. The semantics of that object has a heuristic side and a deductive side. In either case, we might want to know the properties of that object, which we might then list in an intensional definition.
Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols. This mathematical notation might include annotation such as
- "All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function"
- "A mapping from the real numbers to the complex numbers"
[编辑] 歷史
[编辑] 計數
一般相信數學標記在50,000年前開始發展，以協助計數。早期用作計數的數學概念是石塊、樹枝、骨頭、黏土、木雕、繩結的彙集。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.
[编辑] 幾何變得解析
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, and even longer to 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. Some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs. Moreover, the power and authority of geometry's theorem and proof structure greatly influenced non-geometric treatises, Isaac Newton's Principia Mathematica, for example.
[编辑] 計數數械化
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, computers use standard circuits to both store and change quantities, which represent not only numbers, but pictures, sound, motion, and control.
[编辑] 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.
For some people, computerized visualizations have been a boon to comprehending mathematics that mere symbolic notation could not provide. They can benefit from the wide availability of devices, which offer more graphical, visual, aural, and tactile feedback.
[编辑] 表意符號
In the history of writing, ideographic symbols arose first, as more-or-less direct renderings of some concrete item. This has come full circle with the rise of computer visualization systems, which can be applied to abstract visualizations as well, such as for rendering some projections of a Calabi-Yau manifold.
Examples of abstract visualization which properly belong to the mathematical imagination, can be found, for example in computer graphics. 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.
[编辑] 另見
[编辑] 註記
- Florian Cajori, A History of Mathematical Notations (1929), 2 volumes. ISBN 0-486-67766-4
[编辑] 外部連結
- Mathematics as a Language at cut-the-knot
- Earliest Uses of Various Mathematical Symbols
- Mathematical ASCII Notation how to type math notation in any text editor.
