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Multiplication is to addition as addition is to what?
Published online by Cambridge University Press: 16 February 2023
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When children are first taught about multiplication of the natural numbers, it is usually presented as repeated addition. Later, learning about raising to an exponent is presented as repeated multiplication. Then the following analogy is obvious: addition is to multiplication as multiplication is to raising to an exponent. An interesting question is to ask what happens if we go in the other direction. That is, multiplication is to addition as addition is to what? In this paper, we will answer this question, and show that there are several possible operations that could be used to answer the question. Some of them will be presented in connection with algebraic semirings. A semiring is a set, together with two operations, semiring addition and semiring multiplication, satisfying certain properties. We will define several semirings, such that the semiring multiplication is ordinary addition. In each case, the semiring addition will then be an answer to the question in the title of this paper.
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- © The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association
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