Radix Minus One

William John Holden

  1. Introduction
  2. Proof
  3. Binary
  4. Programming
  5. Networking
  6. Unary


Some time ago I noticed an interesting property of binary arithmetic, described in my Math Tricks for IP Subnetting paper, where:

2^(n+1) - 2^n = 2^n*2(-1) = 2^(n+1) / 2 = 2^n

I wondered if such a rule could be applied for any radix. It seems intuitive:

10^3-(10-1)*10^2 = 1000-900=100=1000/10

Surprisingly, yes, this is very easily provable, and in binary this gives a very unique property.


The proof is trivial. For arbitrary radix r we hypothesize:


And verify using basic algebra:



Binary provides several unique properties given r-1=1, which can be phrased in many ways:

Additional means to express this in spoken language that might suggest further applications.


This mathematical feature provides a very useful trick for computers: bit shifting. Bit shifting manifests in C-style programming languages as the << and >> operators. These two operators allow a programmer to shift bits to the left or right, which respectively raises or lowers the number by a factor of two equal to the number of shifts shifted.

To multiply an integer value by four, a programmer may use:
x << 2

To divide by sixteen, the programmer could write:
y >> 4

(Note that most programming languages will silently ignore overflows and loss of precision.)


IT professionals working in networking may also find this property useful when subnetting. Adding or removing a bit to a subnet mask will result in a network that is half or twice the original network size.

To divide a /24 network in half, simply extend the length of the subnet mask by one bit to /25. To divide into four subnets, make /24 into /26. CIDR notation makes this obvious, but it is not as apparent using dotted-decimal subnet masks.


I might add that this rule does not apply for Unary, where radix r is equal to one, because r-1=0. However, Unary does not have a means to express zero, thus this exception illustrates invalidity of Unary as a number system.