Multiplication

One of the main properties of multiplication is the commutative property, which states in this case that 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 area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.

The product of a sequence, vector multiplication, complex numbers, and matrices are all examples where this can be seen.

These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of complex numbers.

This is because most computers historically were limited to small character sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as ⋅ or ×), while the asterisk appeared on every keyboard.

In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example,

A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication.

The construction of the real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.

Beginning in the early 20th century, mechanical calculators, such as the Marchant, automated multiplication of up to 10-digit numbers.

Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.

Methods of multiplication were documented in the writings of ancient Egyptian, Greek, Indian,[citation needed] and Chinese civilizations.

The Ishango bone, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the Upper Paleolithic era in Central Africa, but this is speculative.

[15][verification needed] The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus, was by successive additions and doubling.

Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables.

[citation needed] In the mathematical text Zhoubi Suanjing, dated prior to 300 BC, and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed Rod calculus involving place value addition, subtraction, multiplication, and division.

The Chinese were already using a decimal multiplication table by the end of the Warring States period.

Henry Burchard Fine, then a professor of mathematics at Princeton University, wrote the following: These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century and popularized in the Western world by Fibonacci in the 13th century.

The classical method of multiplying two n-digit numbers requires n2 digit multiplications.

Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers.

[20] In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of

[21] The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.

, which derives from the capital letter Π (pi) in the Greek alphabet (much like the same way the summation symbol

That is, One can similarly replace m with negative infinity, and define: provided both limits exist.

[27] Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions, and trigintaduonions, multiplication is generally not associative.

Peano arithmetic has two axioms for multiplication: Here S(y) represents the successor of y; i.e., the natural number that follows y.

The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including induction.

For instance, S(0), denoted by 1, is a multiplicative identity because The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers.

[citation needed] The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms.

[32] There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure.

Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements).