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?

The sum of two even numbers is even; the product is even.
The sum of two odd numbers is even; the product is odd.
The sum of an even and an odd number is odd; the product is even.
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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