‘What is a number?’ is a question with various answers depending on who you ask. This question is also vital to mathematical logic, and as a consequence, to all modern computation. In the early 1900s, the field of mathematics experienced multiple crises, as various paradoxes arose that proved all mathematical knowledge to be unfounded. The solution to these crises was the birth of Set Theory. Read More

Set Theory involves elementary ‘sets’, which can be used to define numbers and provide a language that consistently underpins mathematical systems. There are infinite ways of defining numbers using sets; however, the Ackermann Coding is the most suitable for the mathematical operations carried out by computer microprocessors.

Juan Ramirez of Binary Proj-x has developed a faster way to perform such operations based on the Ackermann Coding. His work has the potential to create microprocessors that are more efficient and cheaper to run.

Positive whole numbers – or so-called ‘natural numbers’ – can be expressed in binary form. This binary form denotes the unique set of powers of 2 that sum to it. So, since 7 is the sum of 4, 2, and 1, it can be written as: 2^{2} + 2^{1} + 2^{0}. We can then represent 7 as this set of powers: 2, 1 and 0. As each number’s binary form is unique, each number can be expressed as a specific set in this form. Expressing numbers in this way allows us to operate on them as sets, instead of sequences as they are traditionally encoded.

Ramirez’s novel method involves summing these set forms in an efficient and simple way. Using this approach, he has designed a patent-pending ‘fast-adder’ technique, which allows computer microprocessors to perform addition and multiplication. Ramirez’s technique outperforms other fast-adders and multipliers in terms of computational complexity, energy consumption, chip size and material costs.

The addition and multiplication of natural numbers that he describes can be extended to real numbers – those that have finite and infinite decimal fractions. While natural numbers are finite sets, real numbers are infinite sets. The simple and linear fast adder he proposes can operate approximations of real numbers, called rational numbers.

As well as being able to construct numbers and addition through this set building approach, finite mathematical objects such as functions, permutations, and even groups, are encoded as natural numbers, with important consequences in the classification and representation of these.

Real number functions, matrices, and sets of real numbers are also included in Ramirez’s mathematical framework. He also incorporates further mathematical objects such as rings, fields, and linear spaces, each finitely and infinitely.

Ramirez’s approach provides us with an optimal representation of all numbers and mathematical objects using set theory. Along with its broad applications in pure mathematics and data structures, his fast-adder offers the potential to upgrade the processors that almost all of our modern technology relies upon. Not only this, but Ramirez’s technique provides the blueprints on how to do so.