Multiplication is one of the four basic operations of arithmetic (the other being addition, subtraction, and division). Multiplication operates on the set of real numbers such that for any real numbers multiplied together, a unique real number is determined. Multiplicative notation is defined as the system of symbols used in the operation of multiplication to represent numbers and actions. For the expression "2 x 3" the natural numbers "2" and "3" are multiplied together to produce a unique natural number, namely "6". In the expression "2 x 3" the operation of multiplication is denoted by "x", while the numbers "2" and "3" are called the "multiplicand" and "multiplier", respectively. Since multiplication is commutative on the real numbers (that is, the multiplicand and the multiplier can be interchanged) they are, also, both called factors.
Various symbols are used to indicate the multiplication of two numbers. The symbols "x" (lying cross), "⋅" (raised dot), and "*" (asterisk) are all used; so for numbers "a" and "b", "a x b", "a ⋅ b" and "a * b" are equivalent ways of expressing the multiplication of "a" times "b". Sometimes multiplication is represented with only adjacent parenthesis (i.e., by writing "(a)(b)", or simply "ab", to denote the multiplication of numbers a and b).
The use of particular symbols in mathematics is not arbitrary. Rather, the symbols in use today can be traced back along a definite historical path. In the next few paragraphs a brief background is given on the multiplicative symbols described above.
English mathematician William Oughtred (1574-1660), who gave free private lessons to pupils interested in mathematics, used the symbol "x" for times (multiplication). In 1631 he used the lying cross "x" as a symbol for multiplication in his book Clavis Mathematicae (Key to Mathematics). The symbol "x" appeared earlier in 1618 in an anonymous appendix to Edward Wright's translation of John Napier's Descriptio. However, Oughtred is believed to have written this appendix. The symbols were introduced as a way to make writing faster and easier, and to take up less written space for the new printing process of the times.
German mathematician Gottfried Wilhelm von Leibniz (1646-1716) advocated the raised dot, stating that the "x" looked too much like the unknown variable "x". In 1631 English mathematician Thomas Harriot (1560-1621) (posthumously) used the raised dot "⋅" in his book Analyticae Praxis ad Aequationes Algebraicas Resolvendas and in 1655 Thomas Gibson used the raised dot in Syntaxis mathematica. However, it is felt that neither author meant for these dots to represent multiplication. On the other hand, several research papers indicated that Harriot occasionally used the dot to denote multiplication, but admitted that its use was not accepted until Leibniz adopted it. Leibniz also used the cap symbol "" for multiplication. Today the cap symbol is used to indicate intersection in set theory. The asterisk "*" was used by Johann Rahn (1622-1676) in his 1659 book Teutsche Algebra.
To place variables side by side (sometimes called juxtaposition) in order to indicate multiplication was first found in a manuscript found buried near the village of Bakhshali, India. It is believed that the manuscript was written between the eighth and tenth centuries. Multiplication by juxtaposition was also found in fifteenth-century manuscripts, specifically by al-Qalasadi. In 1544 Michael Stifel (1487 or 1486-1567) used the notation in his Arithmetica integra and repeated the use in 1553. In 1637 the French mathematician René Descartes (1596-1650) also used juxtaposition.
The term powers is used with reference to the multiplication of equal terms. The nth power of "a", denoted as "an", is the product of "n" factors of "a". That is to say: "a2 = a ⋅ a" and "a5 = a ⋅ a ⋅ a ⋅ a ˙ a", and so forth. In the expression "an", "a" is called the basis and "n" the exponent. Moreover, when dealing with powers, both "a" and "n" customarily have integer values only. The physical sciences as well as mathematics make frequent use of powers. For instance, the volume (V) of a cube having sides of length "x" may be expressed as "V = x ⋅ x ⋅ x", or more compactly as the third power of x, "V = x3". If x = 5 centimeters, then the cube's volume is the third power of 5, or "V = 53 = 125 cubic centimeters". As powers grow larger, the need for this compact notation becomes even more apparent.
The factorial of a positive integer "n" is another multiplicative notation. Factorial, represented as "n!", is the product of all positive integers from 1 to n. It is generically written out as "n! = 1 ⋅ 2 ⋅ 3 ⋅...⋅ n", where by convention, "0! = 1". For example, "5! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 120".
Besides its use with respect to the real numbers, multiplicative notation is used to denote the products of complex numbers, vectors, functions, matrices, and tensors. The symbols described previously to denote multiplication of real numbers (i.e., lying cross, raised dot, asterisk, brackets, and juxtaposition) are also used to indicate multiplication of functions, complex numbers, etc. However, different symbols may denote different types of multiplication. A good illustration is vector multiplication. The lying cross, "x", is used to denote the 'cross product' of two vectors, resulting in another vector (ie., f x g = h). In contrast the dot symbol, "⋅", indicates the 'dot product' of two vectors (ie., f ⋅ g = h), which is a scalar quantity (i.e., a real number).
It is perhaps worth noting that multiplicative notation is encountered in both set theory and abstract algebra. These two branches of mathematics were created and extensively developed within the last two centuries, and have proven to be of tremendous use in unifying, heretofore, seemingly disparate mathematical disciplines. In set theory one encounters Cartesian products, denoted "A x B", which is the set of all ordered pairs of sets A and B. Multiplicative notation is also used in the definitions of groups and fields, which are central concepts in abstract algebra.
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