# Category: Core Knowledge

Core theories of programming are as varied as they are deep. The primary one, of course, is the theory of computation. This theory rests heavily on discrete mathematics and formal languages and includes automata, computability, and complexity theories. Learning about these theories in conjunction with mathematical logic will help you gain a deep understanding.

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Mathematics

### Negative Binary Numbers

A non-standard positional notation is one where the value of each position isn’t necessarily a straightforward power of the radix. I am also including when the radix is not a positive integer (such as -2), even though mathematically the representation is consistent with standard positional notation. 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).

Mathematics

### Binary (Base-2) And Its Operations

This article continues the trend of the previous articles and begins with a history of binary. After that, I briefly reiterate why binary is used in modern electronic devices as covered in the previous article, and go into more depth regarding binary “sizes” (bit, byte, kilobyte, etc.) Then I move on to important elements of binary arithmetic, and the operations of addition, subtraction, multiplication, and division. I cover two operations often found in computing processors, the shift operators, and their mathematical meaning. Finally, I briefly cover Boolean logic operations.

Mathematics

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?”

Mathematics

### Converting To Binary, Octal, and Hexadecimal

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.

Mathematics

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, and 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.

Series

This series intends to have the reader able to understand binary, octal, and hexadecimal; three radices of great importance to contemporary computer theory. By the end of this series, you should be able to read and convert integer values into binary, octal, and hexadecimal, perform arithmetic operations on all three representations, understand basic Boolean operations, and otherwise have a further appreciation of the power of binary.

## Recent Posts

### Negative Binary Numbers

A non-standard positional notation is one where the value of each position isn’t necessarily a straightforward power of the radix. I am also including when the radix is not a positive integer (such as -2), even though mathematically the representation is consistent with standard positional notation. 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).

### Binary (Base-2) And Its Operations

This article continues the trend of the previous articles and begins with a history of binary. After that, I briefly reiterate why binary is used in modern electronic devices as covered in the previous article, and go into more depth regarding binary “sizes” (bit, byte, kilobyte, etc.) Then I move on to important elements of binary arithmetic, and the operations of addition, subtraction, multiplication, and division. I cover two operations often found in computing processors, the shift operators, and their mathematical meaning. Finally, I briefly cover Boolean logic operations.

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

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.