Multiplication by juxtaposition
Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.
The multiplication of whole numbers may be thought as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the value of the other one, the multiplier. Normally, the multiplier is written first and multiplicand second, (though this can vary by language.)
For example, 4 multiplied by 3 (often written as and spoken as "3 times 4") can be calculated by adding 3 copies of 4 together:
Here 3 and 4 are the "factors" and 12 is the "product".
One of the main properties of multiplication is the commutative property: adding 3 copies of 4 gives the same result as adding 4 copies of 3:
Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.
The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.
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