SOLUTION: Hello my teacher has me all confused about finite numbers and how to make sets closing in addition, subtraction and multiplication and division. I just really need to know the whol

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Question 776556: Hello my teacher has me all confused about finite numbers and how to make sets closing in addition, subtraction and multiplication and division. I just really need to know the whole concept of it
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Hopefully I can help, and not confuse you further.

A set closed to addition means a set with an operation called addition defined in such a way that the result of adding two elements of the set is always an element of the set.
A set closed to addition could be {0,1} with addition defined as follows
0%2B0=0
0%2B1=1
1%2B0=1
1%2B1=0.
You could make other two-element sets like that one by changing the names of the elements to something like A and B, or EVEN and ODD, or you could use shapes as a triangle and a circle.
That is an example of clockwork arithmetic, where you "count" around in a circle and end up at the beginning.
The example above is a 2-hour day clock.
Most people are familiar with addition on a 12-hour clock, where 12 (o'clock) plus 1 is 1, 12+2=2, 11+2=1, and so on.

You could have addition and multiplication closed on the same set.
For example, we could add to the previous example
0%2A0=0
0%2A1=0
1%2A0=0
1%2A1=1


(You could also make up fancy operation names and use invented symbols for the operators. No need to call them "addition" using a + for a symbol, and "multiplication" using a middle dot, or a little cross. Your operations could be SILLY, with S as a symbol and FUNNY with F as a symbol. Then you could say that 1 SILLY 1 is 0, in symbols 1S1=0, instead of 1%2B1=0 and 1 FUNNY 1 is 1, In symbols 1F1=1, instead of 1%2A1=1. That could really confuse people).

If you want larger sets, you can use the idea of the remainders from division by a number. For example, take 7. You could use the set of possible remainders, {0,1,2,3,4,5,6} and define
0%2B0=0, 0%2B1=1, 0%2B2=2, ..... , 0%2B6=6
1%2B0=1, 1%2B1=2, ... , 1%2B5=6, red%281%2B6=0%29
.............................................
5%2B0=5, 5%2B1=6, red%285%2B2=0%29, red%285%2B3=1%29, ..., red%285%2B5=3%29, red%285%2B6=4%29
6%2B0=6, red%286%2B1=0%29, red%286%2B2=1%29, ..... , red%286%2B5=4%29, red%286%2B6=5%29
and for multiplication:
0%2A0=0, 0%2A1=0, 0%2A2=0, ... , 0%2A5=0, 0%2A6=0
1%2A0=0, 1%2A1=1, 1%2A2=2, ... , 1%2A5=5, 1%2A6=6
2%2A0=0, 2%2A1=2, 2%2A2=4, ... , red%282%2A4=1%29, red%282%2A5=3%29, red%282%2A6=5%29
.....................................................................
6%2A0=0, 6%2A1=6, red%286%2A2=5%29, red%286%2A3=4%29, red%286%2A4=3%29, red%286%2A5=2%29, red%286%2A6=1%29
Some of the results (in red) look unusual, but we are just calculating clock-style, with remainders.
Adding and multiplying just the remainders is what is called modular arithmetic.
With an infinite set of numbers 6%2B6=12, but if after 6 comes 0, there is no 12; you never get to 12. When dividing 12 by 7, the remainder is 5, and 5 stands in for 12. In fact, 5 stands in for all the numbers that have 5 for a remainder when divided by 7, and we say that they are "congruent modulo 7".
Similarly, with infinite numbers, 6%2A5=30, but since 30 divided by 7 gives a remainder of 2, in the modular arithmetic example with a 7-element set, red%286%2A5=2%29.

Addition and multiplication is all we need to replicate in finite sets the structure of the infinite number sets.
Subtraction is a mirage. It does not really exist. Subtracting a number means just adding the opposite.
For "addition", there is a neutral element, a "zero" that added to any other element, equals that other element. In the examples above that "zero" is 0. (Making it otherwise, I would confuse myself).
When two of the other numbers add up to that "zero", those two numbers are opposites. So 5 and 2 are opposites. Then subtracting 5 is the same as adding 2 and vice versa.
There is no such thing as division, either. It is just the reflection of multiplication. Dividing by a number is just multiplying times its reciprocal.
For multiplication, there is also a neutral element that in the example above was called 1. All the elements of the set above, except the "zero" have a "reciprocal" element, and when an element and its reciprocal are multiplied, the result is the multiplicative neutral element: 1.
The reciprocal of 1 is 1 itself. The reciprocal of 6 is 6 itself. 2 and 4 are each other's reciptrocals, and 3 and 5 are each other's reciprocals.

EXTRA:
The nicest example of that modular arithmetic is the divisibility by 9 test for multiplication.
Calculating 568%2A347 with pencil and paper is hard work and you could make a mistake, but ther is an easy test that lets you detect most mistakes.
I know that 568 has a remainder of 1 when divided by 9, and 347 has a remainder of 5 when divided by 9. So 568%2A347 must have a remainder of 1%2A5=5 when divided by 9.
What makes itthe test easy is that finding the remainder when dividing by 9 is just a question of adding the digits.
For 568, 5+6=11, and I can keep adding the digits of that result before adding the final 8, so 1+1=2 and then 2+8=10, and finally 1+0=1.
For 347, you could add 3+4+7=14 and 1+4=5. Otherwise, you could add the 3 and 7 first to get 3+7=10, then add the digits of 10 1+0=1, and then add the 4 in the middle 1+4=5.
When we multiply and find the product to be 197,096, we can add 1+7+0+6=14, and 1+4=5, and we do not need to add the nines, because adding nine does not change the result. So we get the expected remainder, which means that we either got the right multiplication result, or we are off by a multiple of 9.