Multiplication

In its simplest form, multiplication is the sum of a list of identical numbers. For example, the product 7 × 4 is 7 + 7 + 7 + 7. The numbers being multiplied are called the multiplicand and multiplier or the factors.

Notation

Multiplication can be denoted in several equivalent ways. All of the following mean, "5 times 2":

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:5 imes 2

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:5cdot2

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:(5)2, 5(2), (5)(2), 5, 2,

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:5*2

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The asterisk (*) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. Frequently, multiplication is implied by Juxtaposition rather than shown in a notation. This is standard in algebra, taking forms like

Related Topics:
Asterisk - FORTRAN - Juxtaposition - Algebra

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:5x and xy.

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This is potentially confusing if variables are permitted to have names longer than one letter. The notation is not used with numbers alone: 52 never means 5 × 2.

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If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums).

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Thus, the product of all the natural numbers from 1 to 100 can be written 1 cdot 2 cdot ldots cdot 99 cdot 100. This can also be written with the ellipsis vertically placed in the middle of the line, as 1 cdot 2 cdot cdots cdot 99 cdot 100.

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Alternatively, the product can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet.

Related Topics:
Letter Π (Pi) - Greek alphabet

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This is defined as:

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: prod_{i=m}^{n} x_{i} := x_{m} cdot x_{m+1} cdot x_{m+2} cdot cdots cdot x_{n-1} cdot x_{n}.

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The subscript gives the symbol for a dummy variable (i in our case) and its lower value (m); the superscript gives its upper value.

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So for example:

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: prod_{i=2}^{6} left(1 + {1over i} ight) = left(1 + {1over 2} ight) cdot left(1 + {1over 3} ight) cdot left(1 + {1over 4} ight) cdot left(1 + {1over 5} ight) cdot left(1 + {1over 6} ight) = {7over 2}.

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One may also consider products of infinitely many terms; these are called infinite products.

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Notationally, we would replace n above by the infinity symbol (∞).

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The product of such a series is defined as the limit of the product of the first n terms, as n grows without bound.

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That is:

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: prod_{i=m}^{infty} x_{i} := lim_{n oinfty} prod_{i=m}^{n} x_{i}.

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One can similarly replace m with negative infinity, and

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:prod_{i=-infty}^infty x_i := left(lim_{n oinfty}prod_{i=-n}^m x_i ight) cdot left(lim_{n oinfty}prod_{i=m+1}^n x_i ight),

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for some integer m, provided both limits exist.

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