Numeration Systems

The ancient Mayans used a vigesimal, or base-twenty, numeration system in their mathematics. Each “digit” was a actually a composite of dots and/or lines, as such:

To represent numbers larger than twenty, the Mayans combined multiple “digits” the same way we do to represent numbers larger than ten. For example:

Based on the examples shown above, determine the place-weighting of each “digit” in the vigesimal numeration system. For example, in our denary, or base-ten, system, we have a one’s place, a ten’s place, a hundred’s place, and so on, each successive “place” having ten times the “weight” of the place before it. What are the values of the respective “places” in the Mayan system?

Also, determine the values of these Mayan numbers:

Digital computers use a numeration system with a base of two, rather than a base of ten as we are accustomed to using. It is much easier to engineer circuitry that counts in “binary” than it is to design circuits that count in any other base system. Based on what you know of numeration systems, answer the following questions:

How many different symbols (ciphers) are there in the binary numeration system?

What are the different place-weight values in the binary system?

How would you represent the number “seventeen” in binary?

In our base-ten (denary) numeration system, each character is called a “digit.” What is each character called in the binary numeration system?

If a digital meter display has four digits, it can represent any number from 0000 to 9999. This constitutes ten thousand unique numbers representable by the display. How many unique numbers could be represented by five digits? By six digits?

If an ancient Mayan ledger had spaces for writing numbers with three “digits” each, how many unique numbers could be represented in each space? What if the spaces were expanded to hold four “digits” each?

If a digital circuit has four bits, how many unique binary numbers can it represent? If we expanded its capabilities to eight bits, how many unique numbers could be represented by the circuit?

After answering these questions, do you see any mathematical pattern relating the number of “places” in a numeration system and the number of unique quantities that may be represented, given the “base” value (“radix”) of the numeration system? Write a mathematical expression that solves for the number of unique quantities representable, given the “base” of the system and the number of “places”.