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Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is also not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because 0 is the additive identity, subtraction of it does not change a number.
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. Multiplying any number by 0 results in 0, and consequently division by zero has no meaning in arithmetic.
The plus and minus symbols are used to show the sign of a number. In mathematics, the sign of a real number is its property of being either positive, negative, or 0.Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign.
Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition, subtraction, multiplication, and division. [1]
It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. Specifically: The negation of 0 is 0, and; The negation of a negative number is the corresponding positive number. For example, the negation of −3 is +3. In general,
Finally let 0 be some object not in or , for example the ordered pair (0,0). Then the integers are defined to be the union P ∪ P − ∪ { 0 } {\displaystyle P\cup P^{-}\cup \{0\}} . The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero.
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The natural numbers including 0 form a commutative monoid with monus, with their ordering being the usual order of natural numbers and the monus operator being a saturating variant of standard subtraction, variously referred to as truncated subtraction, [7] limited subtraction, proper subtraction, doz (difference or zero), [8] and monus. [9]
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