Multiplication of Different Kinds of Numbers
Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).
- Integers
- is the sum of M copies of N when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by and . The same sign rules apply to rational and real numbers.
- Rational numbers
- Generalization to fractions is by multiplying the numerators and denominators respectively: . This gives the area of a rectangle high and wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.
- Real numbers
- is the limit of the products of the corresponding terms in certain sequences of rationals that converge to x and y, respectively, and is significant in calculus. This gives the area of a rectangle x high and y wide. See Products of sequences, above.
- Complex numbers
- Considering complex numbers and as ordered pairs of real numbers and, the product is . This is the same as for reals, when the imaginary parts and are zero.
- Further generalizations
- See Multiplication in group theory, above, and Multiplicative Group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ring. An example of a ring that is not any of the above number systems is a polynomial ring (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)
- Division
- Often division, is the same as multiplication by an inverse, . Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain x may have no inverse "" but may be defined. In a division ring there are inverses but they are not commutative (since is not the same as, may be ambiguous).
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