Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes

We define an extension of parity from the integers to the rational numbers. Three parity classes are found — even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure.

The natural numbers {\mathbb{N}} split nicely into two subsets, the odd and even numbers

\displaystyle \mathbb{N}_O = \{ 1, 3, 5, 7, \dots \} \,, \qquad \mathbb{N}_E = \{ 2, 4, 6, 8, \dots \} \,. \nonumber \ \ \ \ \ (1)

As the sets get larger, the ratio of odd to even numbers tends to {1}, so we can say informally that there are the same number of odd and even integers. Similar arguments apply to the integers {\mathbb{Z}}, which split into two subsets

\displaystyle \begin{array}{rcl} \mathbb{Z}_O &=& \{\dots\ -3, -1, +1, +3, +5, \dots \} \\ \mathbb{Z}_E &=& \{\dots\ -4, -2,\ \ 0, +2, +4, \dots \} \,. \end{array}

Parity

The distinction between odd and even numbers is called parity. The even/odd concept is defined only for the integers. Is there is a natural way to extend the concept of parity to larger sets of numbers.

What characteristics might one require of such an extension?

  1. The sum of two even numbers is even; the product is even.
  2. The sum of two odd numbers is even; the product is odd.
  3. The sum of an even and an odd number is odd; the product is even.
  4. An odd number plus {1} is even; an even number plus {1} is odd.

If the concept of parity is extended to larger sets of numbers, some of the properties indicated above may have to be sacrificed. For rational numbers, we might define a number {q=m/n} to be even if the numerator {m} is even and odd if {m} is odd. But then {\frac{1}{4}+\frac{1}{4} = \frac{1}{2}}, meaning that two odd rationals would add to yield another odd one.

A three-way split

There is a simple way of separating the rational numbers into three subsets:

\displaystyle \begin{array}{rcl} \mbox{Even:\ \ } \mathbb{Q}_E &=& \{ q\in\mathbb{Q} : q = \textstyle{\frac{2m}{2n+1}\ \mbox{for some}\ m,n\in\mathbb{Z}} \} \\ \mbox{Odd:\ \ } \mathbb{Q}_O &=& \{ q\in\mathbb{Q} : q = \textstyle{\frac{2m+1}{2n+1}\ \mbox{for some}\ m,n\in\mathbb{Z}} \} \\ \mbox{None:\ \ } \mathbb{Q}_N &=& \{ q\in\mathbb{Q} : q = \textstyle{\frac{2m+1}{2n}\ \mbox{for some}\ m,n\in\mathbb{Z}} \} \,. \end{array}

The term none is an initialism for neither odd nor even.

These three sets are mutually disjoint and {\mathbb{Q} = \mathbb{Q}_E \uplus \mathbb{Q}_O \uplus \mathbb{Q}_N}. It is immediately obvious that {\mathbb{Z}_E \subset \mathbb{Q}_E} and {\mathbb{Z}_O \subset \mathbb{Q}_O}, confirming that the definition of parity for the rationals is an extension of the usual meaning for the integers. We see that the even and odd rationals respect the four `rules of parity’ listed above.

The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. We can list all rationals in {(0,1)} in a sequence where, for each {n} in turn, all (new) numbers {m/n} with {m < n} are listed in order. For {n_\mathrm{max}=8} we have

\displaystyle  \bigl\{ \textstyle{ \frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4}, \frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{5}{6}, \frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}, \frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8}} \bigr\} \,.

A Mathematica program was written to count the proportion of rationals in each parity class in the interval {(0,1)}, with denominators less than or equal to {n}, for a range of cut-off values {n \le n_\mathrm{max}}. As {n} increases, the ratios of numbers with parity even, odd and none all tend to the limit {\frac{1}{3}} (see Figure). Colloquially, there are an equal number of rationals with parity even, odd and none.

Parity ratio {r} for rationals {m/n} of parity even (solid blue line), odd (dashed red line) and none (dotted black line) for n <= 20

2-Adic valuation and the “degree of evenness”.

All multiples of 2 are even, but some are more even than others.

The {p}-adic valuation — or {p}-adic order [Katok, 2007] — of an integer {n} is the function

\displaystyle \nu_p(n) = \begin{cases} \max\{k\in\mathbb{N} : p^k \mid n \} & \mbox{for\ } n \neq 0 \\ \infty & \mbox{for\ } n = 0 \end{cases}

This is extended to the rational numbers {m/n}:

\displaystyle \nu_p\left(\frac{m}{n}\right) = \nu_p(m) - \nu_p(n) \,.

It is easily proved that, for any rationals {q_1} and {q_2},

\displaystyle \nu_p(q_1+q_2) \ge \min\{\nu_p(q_1),\nu_p(q_2)\} \,,

with equality holding if {\nu_p(q_1) \neq \nu_p(q_2)}.

We shall be concerned exclusively with the case {p = 2}. We note that {\mathbb{Q}_P = \{q\in\mathbb{Q} : \nu_2(q) \ge 0 \}} and {\mathbb{Q}_E = \{q\in\mathbb{Q} : \nu_2(q) > 0 \}}. The “degree of evenness” of a number can be expressed in terms of the 2-adic valuation. For an integer {n}, the 2-adic valuation is the largest natural number {k} such that {2^k} divides {n}. It is normally written {\nu_2(n)} or {\mbox{ord}(n)}. For even integers, {\nu_2(n)>0}; for odd integers, {\nu_2(n)=0}. By convention, {\nu_2(0)=\infty} (since zero is divisible by every power of {2}).

If we write a rational number {q} in the form {2^k(2m+1)/(2n+1)} with {k\in\mathbb{Z}}, then {\nu_2(q) = k}. Odd rationals have order {0} and rationals with no parity have negative 2-adic order. In particular, half-integers have 2-adic order equal to {-1}.

The 2-adic order clearly identifies the parity classes of the rationals, and it provides a means of partitioning them into finer-grain parity classes. The resulting partition reveals a wealth of algebraic structure. For all {k\in\mathbb{Z}}, we define the set of all rational numbers with 2-adic valuation {k}:

\displaystyle \mathbf{Q}_k = \{q\in\mathbb{Q} : \nu_2(q) = k \} \qquad\mbox{and}\qquad \mathbf{Q}_\infty = \{\ 0\ \} \,.

The union of all the {\mathbf{Q}}-sets comprises the entire set of rationals

\displaystyle \mathbb{Q} = \{\ 0\ \} \uplus \biguplus_{k=-\infty}^\infty \mathbf{Q}_k \,.

We illustrate the subsets {\mathbf{Q}_k} in ihe Figure below. The vertical axis is the 2-adic valuation {\nu_2}.

Dyadic rational numbers

A dyadic rational is a number that can be expressed as a fraction whose denominator is a power of two. The usual definition of the dyadic rational numbers [Bajnok, 2001, p. 122] is

\displaystyle \mathbb{D} = \left\{ \frac{z}{2^m} : z\in\mathbb{Z}, m\in\mathbb{Z} \right\} \,.

Note that the integers are included in the set of dyadic rationals. A convenient alternative definition is

\displaystyle \mathbb{D} = \{ 2^k(2\ell-1) : k\in\mathbb{Z}, \ell\in\mathbb{Z} \} \uplus \{\ 0\ \} \,,

since all the numbers of the form {2^k(2\ell-1)} are in {\mathbf{Q}_k}. Moreover, the expression of each number in this form is unique.

Partition of the rational numbers. The vertical axis is the 2-adic valuation {\nu_2}. Each (dense) subset {\mathbf{Q}_k} is represented by a horizontal dotted line. The sets {\mathbf{D}_k} are indicated by the marked points in {\mathbf{Q}_k}. The totality of these comprises the dyadic rationals {\mathbb{D}}.

By analogy with the definition of the {\mathbb{Q}_K}-sets, we construct a countable infinity of subgroups of {\mathbb{D}}:

\displaystyle \mathbb{D}_K := \{\ 0\ \} \uplus \biguplus_{k\ge K} \mathbf{D}_k \,.

Particular cases of the {\mathbb{D}}-sets include

\displaystyle \mathbb{D}_{-\infty} = \mathbb{D} \,, \qquad \mathbb{D}_{-1} = \textstyle{\frac{1}{2}}\mathbb{Z} \,, \qquad \mathbb{D}_0 = \mathbb{Z} \,, \qquad \mathbb{D}_1 = \mathbb{Z}_E \,, \qquad \mathbb{D}_\infty = \{\ 0\ \} \,.

There is an infinite chain of subgroups starting with {\mathbb{D}_\infty} and extending through all the {\mathbb{D}_K} groups to the full group of dyadic rationals:

\displaystyle \mathbb{D}_\infty = \{\ 0\ \} \ \trianglelefteq\ \ \cdots\ \trianglelefteq\ \ \mathbb{D}_{2}\ \trianglelefteq\ \ \mathbb{D}_{1}\ \trianglelefteq\ \ \mathbb{D}_{0}\ \trianglelefteq\ \ \mathbb{D}_{-1}\ \trianglelefteq\ \ \mathbb{D}_{-2}\ \trianglelefteq\ \ \cdots\ \trianglelefteq\ \ \mathbb{D} \,.

Challenge: Readers familiar with the theory of {p}-adic numbers may wish to show that {\mathbb{Q}_P} is the ring of rational-valued 2-adic integers, {\mathbb{Q}\cap\mathbb{Z}_2}, and the dyadic rational numbers may be expressed as

\displaystyle \mathbb{D} = \mathbb{Q} \cap \bigcap_{p\ \mathrm{odd}} \mathbb{Z}_p \,.

This article is a condensation of part of a paper [Lynch & Mackey, 2022] recently posted on arXiv. In a following post, we will show that the three parity classes have equal natural density in the rationals.

Sources

[1] Conway, J. H., 2001: On Numbers and Games. CRC Press, 242 pp. ISBN: 978-1-5688-1127-7.

[2] Dummit, David S. and Foote, Richard M., 2004: Abstract Algebra. John Wiley & Sons, Inc., 932pp. ISBN: 978-0-4714-3334-7.

[3] Katok, Svetlana, 2007: p-adic Analysis Compared with Real. Student Math. Lib., Vol. 37. Amer. Math. Soc., ISBN: 978-0-8218-4220-1.

[4] Lynch, Peter & Michael Mackey, 2022: Parity and Partition of the Rational Numbers. arXiv .

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