Theory of counting

This text is copied from wikipedia and written in Encoder.sv.

Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers starting from two), or for well-ordered objects, to find the ordinal number of a particular object, or to find the object with a particular ordinal number. Counting is also used (primarily by children) to demonstrate knowledge of the number names and the number system. In mathematics the term counting or enumeration also means finding the number of elements of a finite set.)

Forms

Counting can occur in a variety of forms. Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is often used to count objects that are present already, instead of counting a variety of things over time.

Counting can also be in the form of tally marks, making a mark for each number and then counting all of the marks when done tallying. This is useful when counting objects over time, such as the number of times something occurs during the course of a day.

Counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations. The most naive finger-counting uses unary notation (one finger = one unit) , and is thus limited to counting 10. Other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary (base 2 place-value notation), it is possible to keep a finger count up to 1023 = 210 - 1.

Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.


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