Understanding Suitable Partitioning in Constructing New Number Systems

[Ryuzaki]

I have just started working through a book on higher algebra. I'm just at the beginning, where the authors introduce the notation and talk about the various number systems.

I found this particular paragraph confusing:- "The basic idea in the construction of new sets of numbers is to take a set, call it S, consisting of mathematical objects, such as numbers you are already familiar with, partition the set S into a collection of sets in a suitable way, and then attach names or labels, to each of the subsets. These subsets will be elements of a new number system."

What does the author mean, when he says a "suitable way" here? Does it mean, that I can partition in any way that I find suitable, or are there requirements to be met, for any number system that is constructed by me?

For instance, I'm familiar with the set of natural numbers. So, can I construct S={1,2,3,4,5,6,7,8,9,10} and call it a subset of a new number system? Can I go as far as to say that this subset is the only element of my new number system?

 

[lavinia]

He may be referring to equivalence relations on the integers. The equivalence classes are subsets and these can be multiplied and added if defined properly.

One example is remainder arithmetic. Two numbers are equated if their remainders after dividing by some fixed number are equal. For instance, there are five equivalence of classes of numbers whose remainders are equal after dividing by 5. You can check that adding and multiplying remainders is well defined on these equivalence classes and defines a new number system.

 

[HallsofIvy]

"Appropriate ways" means "ways that will give the result you want"!

For example, given the natural number (1, 2, 3, ...) we can define the set of all pairs (a, b) of natural numbers and then define the "equivalence relation", (a. b)~ (c, d) if and only if a+ d= b+ c. That "partitions" the set of pairs into "equivalence classes". All the pairs in a given class are equivalent to one another. We then define a operations on the equivalence classes. If X and Y are equivalence classes, we define X+ Y by "choose a pair (a, b) from equivalence class X and a pair (x, y) from equivalence class Y. Then X+ Y is the equivalence class that contains the pair (a+ x, b+ y). Of course one would have to show that this is "well defined". That is, show that if you had chosen different pairs from the equivalence classes you would get the same result. We can do the same thing to define multiplication: to multiply XY, choose (a, b) in X and (x, y) in Y. XY is the equivalence class containing (ax+ by, bx+ ay).
This is where the "suitable way" comes in. That last, rather peculiar, definition of multiplication comes from thinking of the pair (a, b), if a> b, (one can show that if (a, b) and (c, d) are in the same equivalence class and a> b, the c> d.) as representing the number a- b. So we are using (a- b)(x- y)= ax-bx- ay+ by= (ax+ by)- (bx+ ay).

(There is one equivalence class that contains all pairs of the form (a, a) where both members are the same. That equivalence class is the number "0". And if, in (a, b), a< b, the equivalence class represents the negative integer -(b- a).) Once we have the integers, we can do a similar thing to construct the rational numbers. Take the set of all pairs, (a, b) where a and b are integers and b\ne 0. We say that two such pairs, (a, b) and (c, d) are equivalent if and only if ad= bc. You can see that this is, algebraically, the same as \frac{a}{b}= \frac{c}{d}. But we don't write them as fractions because we haven't yet defined rational numbers (fractions).

 

[Ryuzaki]

Thank you, lavinia and HallsofIvy for your replies!

I think I understand it now. So to summarise, we find an "operation" that satisfies the conditions of an equivalence relation, to perform on the elements of a known number system, and the resulting set of numbers forms a subset of the new number system, each subset being known as an equivalent class. This "operation" is chosen in a "suitable manner", meaning that it is (i) well-defined (ii)fits the need (using appropriate equivalence relations for constructing the set of integers or rational numbers). Is this correct?

 

[Ryuzaki]

I'm going to take the silence as a 'Yes'. Thank you for the help!