In school I learned that there are four basic arithmetic operations: addition, subtraction, multiplication, and division. I always wondered why modulus is not a basic arithmetic operation. Is there any explanation that is also understandable by a math noob?
Why is modulus not a basic arithmetic operation?
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0Who knows? Because modulus is defined in terms of the other four? Because modulus is first needed only much later? – 2012-09-24
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0It's a bit hard to define for all numbers - it is peculiar to the integers (or at least, the meaning of the modulus of two rational numbers is obscure.) – 2012-09-24
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0What is the question exactly? What is the operation you are thinking of? On what numbers? Do you want "mod" to act as a binary operation like "x mod y"? – 2012-09-24
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0@ThomasAndrews: One can get a perfectly good modulus operation on the reals by defining $a\bmod b$ to mean the smallest nonnegative number $c$ such that $a-c$ is an integer multiple of $b$. (It won't preserve multiplication the way integer modulus does, but sometimes you don't care about that). – 2012-09-24
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0As I said, it can be defined, but it isn't very intuitive. In particular, you need a separate notion of "integer" when you are dealing with rationals, for example. We don't, interestingly, often treat the integer quotient of reals as something which needs to be taught as a fundamental operation. – 2012-09-24
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0@ThomasAndrews: You can avoid "integers" if you say "is in the additive subgroup generated by $b$" instead of "is an integer multiple of $b$". I think it's intuitive enough, say for angles measured $\bmod 2\pi$. – 2012-09-24
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1@HenningMakholm You think when OP writes, "In school I learned that there are four basic arithmetic operations: addition, subtraction, multiplication, and division. " he is talking about school where the notion of "additive subgroup" is something he also learned? I think perhaps he is talking about the math you learn when you are 6-12 years old... – 2012-09-24
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0An "operation" like addition or subtraction takes _two_ arguments and outputs one number. The term modulus usually is used with complex numbers but with real numbers it is the same as absolute value. The absolute value is a function of _one_ number. – 2012-09-27