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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]
An integer can be either zero, a nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined by the product of their positive amounts, combined with the sign derived from the following rule:
The problem is that multiplication by zero is not invertible: if we multiply by any nonzero value, we can reverse the step by dividing by the same value, but division by zero is not defined, so multiplication by zero cannot be reversed. More subtly, suppose we take the same equation and multiply both sides by . We get
Multiplying by 0 gives the zero vector: 0v = 0; Multiplying by −1 gives the additive inverse: (−1)v = −v. Here, + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field.
Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number by negative zero (−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case of arithmetic underflow. [31]
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.
This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear. The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule.
0 (zero) is a number representing an empty quantity.Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures.