These posts deal with the concept of counting and quantity. Counting on a computer may involve working with alternate radices besides the common base 10, including binary (base 2), octal (base 8), and hexadecimal (base 16). Counting theory is also important in computer science, particularly in combinatorics. Image Based On A Photo by Luis Quintero on Unsplash

Negative Numbers In Binary

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Now I shall delve into non-standard positional notations. In this article, I will examine systems that allow us to represent negative numbers in binary and use those negative values in computations. By altering the interpretation of one or more of the place values (or the radix) of a binary representation, we are able to represent negative values. In this post I’ll be covering sign-magnitude, the most intuitive method, the radix complement methods (ones’ complement and two’s complement), offset binary (also known as excess-k or biased), and base -2 (base negative two).

Radix Economy

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This article begins with a recap of where we are in the series in regards to the concept of counting. I review the definition of positional notation as outlined in the first article and then move on to reveal how we can calculate the number of digits a value will have in a given radix. In doing so I will go over two mathematical concepts relevant to this calculation: exponents and logarithms. I will then use logarithms to show how you can calculate the efficiency of a given radix, also called the radix economy, and answer the question, “What's the most efficient radix?”

Converting To Binary, Octal, and Hexadecimal

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This is the second article in a series whose intention is to have the reader able to understand binary, octal, and hexadecimal; three radices of great importance to contemporary computer theory. This article builds upon the previous article by outlining three important radices (binary, octal, and hexadecimal) that are useful in the field of computer science. I start with arbitrary base conversion using two methods. Then, a bit of background is given for why these bases are important, particularly binary. Finally, we perform radix conversion.

Understanding Radix

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This article puts forth a brief history of counting, which details how we arrived at some of the conventions we have today including the notion of radix. It then explores the concept of radix in positional numeral systems, in particular the concept of using radices of arbitrary values. With this foundation, it becomes a simple exercise to use binary, octal, and hexadecimal, each with a radix of two, eight, and sixteen respectively.