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First we prove the base cases b = 0 and b = S(0) = 1 (i.e. we prove that 0 and 1 commute with everything). The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a.
Addition is commutative, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if a and b are any two numbers, then a + b = b + a. The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition".
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x.One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.
[7] Arithmetic is closely related to number theory and some authors use the terms as synonyms. [8] However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality. [9] Traditionally, it is known as higher arithmetic. [10]
[1] [2] For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings . The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity) [ 3 ] when there is no possibility of confusion, but the identity ...
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The addition of two numbers is expressed with the plus sign (+). [6] It is performed according to these rules: The order in which the addends are added does not affect the sum. This is known as the commutative property of addition. (a + b) and (b + a) produce the same output. [7] [8]
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