Introduction

Consider a perfect meter stick. It is perfect in the sense that it measures to infinite precision, (and not just to millimeter precision). This will be our model of a constant base numeral system, (decimal, to be precise.)

• The meter stick is divided into 10 decimeters.
• Each decimeter is divided into 10 centimeters.
• Each centimeter is divided into 10 millimeters.
• And so on...

No matter which subunit you pick off the stick, you will always find it subdivided into exactly 10 finer subunits.

Now let's consider a slightly more messy example. Consider a perfect yard stick. This will be our model for a mixed-base numeral system.

• The yard stick is divided into 3 feet.
• Each foot is divided into 12 inches.
• Each inch is divided into 2 half-inches.
• Each half-inch is divided into 2 quarter-inches.
• And so on...

It is no longer the case that any subunit we pick off the stick will be subdivided into the same number of subunits. Feet are divided differently than inches, and inches are divided differently than yards.

Now consider the following case of an eccentric scientist. This will be our model for an as yet unnamed type of non-positional numeral system.

A scientist on a beach has a wooden pole 1 yard in length. They plan on using the pole to measure sea level. The scientist cuts notches into the pole as follows.

• The pole is divided into 3 feet.
• The first foot is divided into 12 inches.
• The second foot is divided into 10 decifeet
• The third foot is divided into 4 quarter-feet
• (There were going to be more divisions, but the scientist ran out of funding.)

In this example, like units are not divided in the same way. One foot was divided 12 ways, while another foot was divided 10 ways. Having these kind of arbitrary divisions in a numeral system destroys the notion of place value, so we can no longer call it positional.

This last example represents the highest level of generalization we will be considering in this topic. A rigorous, axiomatic description of these numeral systems will be provided.

Before we can talk about Markov Bases, we must first go back and examine how numbers are represented in each of the previous systems. Each measurement can be represented by a sequence of digits. For example,

• In the meter system, we can use 0.973... to represent a measurement of 9 decimeters, 7 centimeters, 3 millimeters, and some extra.
• In the yard system, we can use 0.1A01... to represent a measurement of 1 foot, 10 inches, 0 half-inches, 1 quarter-inch, and some extra.

Our third system broke the concept of place value, but we can still use digits to describe measurements. For example,

• We can use 0.17 to represent a measurement of 1 foot and 7 decifeet.
• We can use 0.0B to represent a measurement of 0 feet and 11 inches.

In a Markov base, each subunit reads the last few digits of it's address and feeds them into a function that determines how many finer subunits it will split into.

In this example, each subunit divides itself into a number of finer subunits equal to the last digit of its address plus 2.

For example,

• The subunit at address 0.112012 would be divided into 4 finer subunits since it's last digit is 2 and 2 + 2 is 4.
• The subunit at address 0.01230 would be divided into 2 finer subunits since it's last digit is 0 and 0 + 2 is 2.

In this topic, we will examine the properties of numeral systems like this, as well as Markov bases in general.

Navigation

Back to Menu

Next: Interval-Division Numeral Systems