Euclid's Algorithm

 Been looking at Euclid's algorithm, a procedure on two numbers that yields their

greatest common factor. Euclid lived sometime around 300 BCE. He is credited

with elaborating the standard developement of geometry-Elements - with a text still pertinent

today. (Indeed I remember working through the whole thing in High School, a

wonderful little work). This algorithm is straightforward, but it appears to be the

bugbear of many university students as an instance of 'discrete' mathematics...

The video below takes us through with integers, but one needs to remeber that

at the time, the decimal numbers that we find normal were not in use in Greece. The

mathematical imagination at work had to be visual!!

                                                              

source: Wikipedia

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You

What did numbers look like for Euclid?

Meta AI

Euclid, a Greek mathematician, lived in the 3rd century BCE and wrote the famous book "Elements," which is one of the most influential works in mathematics. In Euclid's time, numbers were represented using Greek letters, with each letter having a specific numerical value. This system was known as the Greek numeral system.

Here's how it worked:

Alpha (α) = 1

Beta (β) = 2

Gamma (γ) = 3

Delta (δ) = 4

Epsilon (ε) = 5

Zeta (ζ) = 6

Eta (η) = 7

Theta (θ) = 8

Iota (ι) = 9

Kappa (κ) = 10

Lambda (λ) = 20

Mu (μ) = 30

Nu (ν) = 40

Xi (ξ) = 50

Omicron (ο) = 60

Pi (π) = 70

Rho (ρ) = 80

Sigma (σ) = 90

Tau (τ) = 100

Upsilon (υ) = 200

Phi (φ) = 300

Chi (χ) = 400

Psi (ψ) = 500

Omega (ω) = 600

Using this system, numbers were represented by combining letters. For example:

11 = ΙΑ (Iota-Alpha)

25 = ΚΕ (Kappa-Epsilon)

100 = Ρ (Rho)

This system was used for basic arithmetic operations like addition and subtraction, but it was not very efficient for more complex calculations. The Greek numeral system was eventually replaced by the Hindu-Arabic numeral system, which is the one we use today (0-9).

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You

What is discrete mathematics?

Copilot

Discrete mathematics is the study of mathematical structures that can be considered “discrete” (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than “continuous” (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numbers, calculus, or Euclidean geometry. It deals with countable sets, which can be either finite or infinite. Research in discrete mathematics has grown due to the development of digital computers, which operate in “discrete” steps and store data in “discrete” bits. Concepts from discrete mathematics are useful in computer science, including computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Although the main objects of study in discrete mathematics are discrete, analytic methods from “continuous” mathematics are often employed as well.

Usingthe web

Discrete mathematics is the study of mathematical structures that can be considered “discrete” (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than “continuous” (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numbers, calculus, or Euclidean geometry. It deals with countable sets, which can be either finite or infinite. Research in discrete mathematics has grown due to the development of digital computers, which operate in “discrete” steps and store data in “discrete” bits. Concepts from discrete mathematics are useful in computer science, including computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Although the main objects of study in discrete mathematics are discrete, analytic methods from “continuous” mathematics are often employed as well123