Multiplication began as disciplined repeated addition
Among arithmetic’s four elementary operations, multiplication yields a product, written with a cross, a dot, juxtaposition, or a programming asterisk. For whole numbers it means adding one factor to itself as many times as the other factor counts—an idea that then generalizes to negatives, fractions, and reals.
Language muddies which factor is multiplier and which multiplicand: three times four, or three groups of four, treats 3 as the multiplier, while three multiplied by four makes it the multiplicand. The commutative property settles the matter, since adding three copies of four equals adding four copies of three. Factors multiply and terms add, a distinction that keeps the vocabulary straight. Systematic extensions then define products for negative integers, fractions and real numbers.
Pictures help. Whole-number products count objects laid out in a rectangle, and a rectangle's area is the same whichever side you measure first. Multiplying measurements creates a new kind of quantity with a derived unit, as metres times metres gives square metres, the business of dimensional analysis. Division undoes multiplication: because 4 times 3 is 12, 12 divided by 3 is 4.
More elaborate products, of sequences, vectors, complex numbers and matrices, bend the familiar rules; matrix multiplication and some vector products are not commutative. The symbols have their own history. Edward Wright introduced the cross in 1618, and Gottfried Leibniz adopted the raised dot in 1698 because the cross was easily confused with the letter x. The dot is now standard in the United States, and programming languages use an asterisk.
Countries that write decimals with a comma tend to multiply with a cross or a dot. Britain and Ireland once did the opposite, using a raised dot as the decimal point so it would not vanish into ruled lines and a full stop for multiplication. After the Ministry of Technology ruled in 1968 that the full stop would mark decimals, and SI units spread, that habit survives mainly in traditional journals such as The Lancet.
Source: Multiplication
