Rational numbers

Posted by on Jun 7, 2013 in Blog, Numbers | 0 comments

In a previous post, I discussed the basic concept of number, beginning with natural numbers, or counting numbers. I am pursuing a notion here that certain kinds of numbers go together with certain kinds of operation. In the case of the natural numbers, the fundamental operation is counting. We begin with no thing and add one thing to it. The addition is equivalent to a count of one. Intellectually, if we have a set of, say, three items in front of us, and we are going to count them, we begin with nothing (“0”), then count “1”, then “2”, then “3”. At each step, we are adding “1” to our existing count. Continuing on, ad infinitum, gives us the entire set of natural numbers. The set is the operation, applied iteratively. We might imagine that the name “1” means “addition to nothing”. Of course, we can consider addition of more than one element of a set; but at the end of the day, doing “+2” is equivalent to “+1” two times, and so on.

So, the set is equivalent to the operation; or said in plain language, the noun is the verb and the verb is the noun. In English, we could say that a given number, say, “5”, was a count or a counting, a sum or a summing up, an addition or an adding up. Likewise, we can count down: “10”, “9”, “8”, “7”, “6”, “5”, “4”, “3”, “2”, “1”, “0”, “Liftoff, we have liftoff!”, which is, of course, subtraction, the way I’ve written it down here. In practice with rocket ship liftoffs, it would actually be addition because time before $t=0$ is negative. The ground crew at the launch site would start that count down sequence by saying something like “It’s t minus 12 seconds and counting, minus 11, 10, …” and drop the “minus” along the way. The whole “count down” is conceptually backwards because they’re actually counting up in the usual way. If liftoff is at $t=0$, then events after liftoff, like booster separation, occur at positive time into the mission.

My point here is that inverting the sense of the operation, in this case, counting up to counting down, addition to subtraction, delivers a new set of numbers by extension: we get the negative natural numbers from the natural numbers by a simple inversion of the operation. Putting the whole set together yields the integers: we can count up ad infinitum from 0 and we can count down ad infinitum from 0 in the integers.

Given that we have a set of items that we have counted, say, 100 sheep, we can divide the set into subsets; for example, 60 white sheep and 40 black sheep. We might have 100 tosses of a fair coin and get 55 heads and 45 tails. We might have 2 dozen pieces of fruit and have 1 dozen apples and 1 dozen oranges. And so on. This gives us the notion of a ratio: 60 out of 100 sheep are black, 55 out of 100 tosses are heads, 1 out of 2 pieces of fruit are apples…. It is common notation to write this down as “60/100”, “55/100”, “1/2”.

Further, we can reduce these ratios to more basic ratios. If 60 out of 100 sheep are black, then we can create 10 equivalent subsets of 10 sheep each with 6 black and 4 white sheep. We can also create 20 equivalent subsets of 5 sheep in which we have 3 black and 2 white sheep. This practical operation is intellectually equivalent to the abstract operation of finding an irreducible fraction representation for the overall set and its subsets.

This operation of dividing a set into subsets is just the inverse of creating a larger set by creating new copies of an initial set. We could start with a set of 5 elements: 3 black and 2 white sheep, and count up 20 copies of this. Or, we can begin with a set of 100 sheep of which 60 are black and 40 are white, and divide it into 20 equivalent subsets. Either way. Same thing. Division and multiplication.

To be somewhat more specific, the form of division that I started with is partitioning. In partitioning, we begin with a set of some specific size, a, and partition it into some number, b, of subsets of equal size. The other form of division is called quotative; we begin with a set of some specific size, a, and form subgroups of a smaller size, c: the number of subgroups, b, is called the quotient of a and c. A set does not have to be partitioned into subsets of equal size, but this is the model that yields division. In quotative division, it is possible to have a remainder; that is, we may divide a into b subsets of size c, and have a non-zero remainder subset of size less than c “left over”.

Of course, we can multiply with natural numbers and integers. In that case, we treat all the elements as equivalent: sheep, coin faces, fruit, whatever: either 2 copies of 3 things gives 6 things or 3 copies of 2 things gives 6 things.

This concept of nouns and verbs together yields algebras. For example, a very basic form of algebraic structure is called a magma, denoted $M$. A magma has a single binary operation, “$\cdot$” such that for any two elements of $M$, $a,b \in M$, then also $a\cdot b\in M$. This single property of a magma under the binary operation “$\cdot$” is called closure. But I have argued here that there may be an even more primitive unary version of the binary operator “$\cdot$“, call it “latex +\$”, which is a generator of $M$, such that every element, $a \in M$, may be created by successive applications of $+$ on some single initiating null element that may or may not be in $M$.

This is rather like the ladder operators of quantum mechanics or the annihilation and creation operators of quantum field theory. Ladder operators move a quantum system up and down the spectrum of eigenvalues, and hence, up and down the spectrum of discrete energy or momentum values.Annihilation and creation operators decrease or increase the number of particles, aka resonances, of the field. Creation is a fundamental operation on the field that increases the count of discrete resonances by $1$. This immediately assumes that the resonances in a quantum field are countable, by definition; and likewise for the spectrum of eigenvalues of a quantum mechanical system.

Setting aside the question of whether or not every binary operation for any magma is associated with a more primitive unary operation of set generation, we can state that a magma is a “group-like” algebraic structure. If we add the associative property to a magma, we have a structure called a semigroup; that is, $\forall a,b,c \in M$, $(a\cdot b) \cdot c = a \cdot (b \cdot c)$. If the semigroup has an identity element “0” such that $\forall a \in M$ then $a \cdot 0 = 0 \cdot a = a$, then we have a monoid. Finally, if we include an operational inverse in the monoid, we have a group; that is, if $\forall a\in M, \exists -a | a \cdot -a = -a \cdot a = 0$. In English, we’d say that if for all elements of the monoid, there is an inverse element such that the result of the group operation on the element and its inverse yields the identity element.

I have written some of the last paragraph with a loose assumption of a commutative property for the identity and the inverse. With somewhat more rigor, one can distinguish between a left identity and a right identity, and between a right inverse and a left inverse. If we accept that the operation is commutative at the level of the semigroup, we have a semilattice. And, if we maintain commutativity all the way to the group, then the group is Abelian.

There is another pathway between a magma and a group. If the magma supports division, then it is a quasigroup. If we add an identity element to the quasigroup, we get a loop. Finally, we include the associative property to the loop, and we again have a group. The first path goes magma -> associativity -> semigroup -> identity -> monoid -> invertibility -> group. The second path goes magma -> divisibility -> quasigroup -> identity -> loop -> associativity -> group. A quick look at the two paths suggests a relationship between divisibility and invertibility; and that is the case.

A quasigroup is a set, $Q$, with a binary operation “$\cdot$“, such that

$\forall a,b\in Q, \exists x,y|a\cdot x=b \land y\cdot a=b$

which can be read as for any two elements of $Q$, call them $a$ and $b$, there are two other elements, call them $x$ and $y$, such that $a\cdot x=b$ and $y\cdot a=b$. Commutativity is not assumed, and we see both left and right operations. If commutation did hold, then $x=y$; however, do not assume that as yet.

Since $x$ and $y$ exist, by the definition of the quasigroup, we can introduce a notation for them: $x = a\backslash b$ and $y = a / b$. These are left and right division, respectively. This begins to give us the concept of an identity. We can define a left multiplication operator, $L(x)$ as

$L(x)a = x \cdot a$

and a right multiplication operator, $R(x)$ as

$R(x)a = a \cdot x$

And since we are assured of division in the quasigroup, we can also define inverse operators $L^{-1}(x)$ and $R^{-1}(x)$ as

$L^{-1}(x)a = x \backslash a$

and

$R^{-1}(x)a = a/x$

Hence, $L(x)L^{-1}(x) = L^{-1}(x)L(x) = R(x)R^{-1}(x) = R^{-1}(x)R(x) = 1$. These operations correspond, for example, to $x(x/a) = a$. So, left multiplication and division by any quasigroup element define an identity, and so on.

Here we see that the definition of an identity element in the algebraic structure is equivalent to the existence of an inverse operation. In other words, the introduction of the inverse operation is logically equivalent to the existence of an identity element.

Anyway, back to the rational numbers… Writing them in the form “$a/b$” with $a$ and $b$ integers, suggests that all the rational numbers could be laid out in the form of a table, as follows:

Table of rational numbers

This table has the numerators of each fraction in the rows and the denominators in the columns. Equivalence classes of fractions are indicated by color; for example, 1/1 = 2/2 = 3/3 and so on. The arrows show a counting plan that demonstrate that the rational numbers are countable. The plan weaves diagonally through the table and moves directly along the first row and column when it gets to them. Since the elements of the table are countable, it is apparent that a natural number can be associated with each entry in the table according to a simple rule. In this way, it can be seen, if unexpected, that there are no more rational numbers than there are natural numbers.

One could extend this table in a four-fold manner by allowing the numerators and denominators to be negative. The table shown above would then be the upper right (+,+) quadrant, in the usual convention for a Cartesian coordinate system. This would introduce another set of equivalence classes; for example, the (+,+) “1” class (that is, 1/1, 2/2, 3/3, …) would also be equivalent to the (-,-) “1” class (-1/-1, -2/-2, -3/-3,…). Doing this would not impact the countability of the entire set in any way; we could simply count each of the four corresponding elements in each of the four quadrants at each of the steps shown in the table above. The entire set is as countable as before.

Hence, a rational number is an ordered pair of integers. Each ordered pair has a magnitude and a sign. Ordered pairs that have the same magnitude and sign are equivalent. Of course, the magnitude of the rational number is not equivalent to the magnitude of the 2-vector in the Cartesian plane that would be associated with the four-fold table just described. For example, $1/3$ would have a very different magnitude than $3/1$, yet the magnitude of $(1,3)$ and $(3,1)$ would be identical. These are different concepts. What should stand apparent so far is the extent to which the rationals are composite constructs. Of course, the natural numbers are composite constructs as well; and it should come as no surprise that more complex sets of numbers are even more composite. The natural numbers are essentially repetitions of some basic operation; namely, $+1+1+1+1$ or $++++$ or $||||$. To call this natural number “4” simply obfuscates its origin. Likewise, to make it seem in any way mysterious that “3+1 = 1+3 = 2+2 = 4” is an equal obfuscation. Once we introduce an inverse operation, “-“, then every natural number becomes a member of an equivalence class; for example, “++++ = +-+-++++ = +++–+++ = …” ad infinitum.

In terms of the algebraic structures discussed above, the rationals are certainly a group under addition; since they commute, they are an Abelian group. The rationals are also an Abelian group under multiplication. This brings us to the concept of a ring, an Abelian group under addition in which multiplication is associative, and also distributive over addition; that is,

$\forall a,b,c\in R, a\cdot(b \cdot c) = (a \cdot b) \cdot c$

and

$\forall a,b,c\in R, a\cdot (b + c) = a \cdot b + a \cdot c$

The idea of a ring leads on to the ideal of a field, which is a ring in which all the non-zero elements have a multiplicative inverse. Another way to consider the concept of a field is as a ring whose non-zero elements are an Abelian group under multiplication, as just mentioned above. Hence, the rational numbers are also a ring and a field.

But the main point of this post is that, yet again, we encounter a set of numbers that are composite; the rational numbers are composed of two integers through the operation of forming a ratio. Since the integers themselves are composite, the rational numbers are doubly composite. Perhaps the only number that isn’t composite is $0$, which is arguably no number at all.