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In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a singular matrix where the determinant is 0. In this process, information is lost and cannot be regained. For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
However, the "existence of additive identity element" property is not satisfied; Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c). No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors.It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of ...
Multiplication distributes over addition: (+) = +. Identity. Any number multiplied by 1 is itself. Zero. Any number multiplied by 0 is 0. In the multiplication algorithm, the "tens" digit of the product of a pair of digits is referred to as the "carry digit".
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