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My textbook says:

Clock arithmetic and modular systems were built upon ordinary numbers and involved familiar operations such as addition, subtraction, multiplication, and division.

So multiplication is a part of a system, but does that mean it is a system? It then states that an abstract system likely has properties such as: closure, commutative, associative, identity, inversive, and distributive. It does not define what these properties are. It also does not state what the requirements are to be classified as a system. There is no direct definition of a "system"

Then there is a question: "Is multiplication a system? Explain." What information do I need to provide to sufficiently answer this question?

I don't understand how to identify the properties of a system. What possible properties could a system have? Are a certain number of properties required in order to be classified as a system?

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    $\begingroup$ What is a "system" for you? Can you define this? In what realm? $\endgroup$ – DonAntonio May 21 '13 at 9:22
  • $\begingroup$ Look in your book, and find the definition of "system". Then, maybe you can answer the question yourself. If not, tell us the definition, so that we can understand the question. $\endgroup$ – bubba May 21 '13 at 9:26
  • $\begingroup$ "Clock arithmetic and modular systems were built upon ordinary numbers and involved familiar operations such as addition, subtraction, multiplication, and division." So multiplication is a part of a system, but does that mean it is a system? It then states that an abstract system likely has properties such as: closure, commutative, associative, identity, inversive, and distributive. It does not define what these properties are. It also does not state what the requirements are to be classified as a system. There is no direct definition! The course is called "math for teachers" (university)... $\endgroup$ – Shayna May 21 '13 at 9:44
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    $\begingroup$ Maybe the term "Algebraic Structure" would be more suitable in this context, then numbers are the objects of the system and multiplication is one of the operations which will result in result that is already an object in the system ( another number real or complex). $\endgroup$ – Arjang May 22 '13 at 4:24
  • $\begingroup$ Thank you very much! That is basically the answer I came to talking to the guys in the Mathematics chat room. I was confused because multiplication is an operation, yet can also be defined as a system of it's own. It is, the classification system for all systems. ;) $\endgroup$ – Shayna May 22 '13 at 5:08
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Multiplication is both a system and an operation. Multiplication is used as an operation within the properties which define systems. As http://en.wikipedia.org/wiki/Multiplication states, Multiplication has properties which include: a commutative property, distributive property, identity element, zero element, negation and an inverse element. Thus, multiplication is also a system. Multiplication is often used within other systems such as the modular arithmetic system, clock arithmetic, etc.

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