# euclid

1. ### Euclid Algorithm Mathematical Induction

Hello, This is my second post on this form, so please let me know if I should move it to a more suitable section or use better formatting. I have this exercise in the book Algorithms by R. Sedgewick and K. Wayne: Use mathematical induction to prove that Euclidâ€™s algorithm computes the...
2. ### How is quadratic equation derived from Euclid?

Referring to Book 2 of Elements, given a segment AB, find a point C between A and B such that the lengths of segments satisfy |AC| > |CB| as well as the following proportionality relation: |AC| / |AB| = |BC| / |AC| Please show me how exactly this can be translated into the quadratic equation...
3. ### Euclid's Proof of the Infinitude of Primes

So I'm going to refer to this link that the explanation is from: https://primes.utm.edu/notes/proofs/infinite/euclids.html Theorem. "There are more primes than found in any finite list of primes." (Chris K. Caldwell) Proof. "Call the primes in our finite list p1, p2, ..., pr. Let P be any...
4. ### Euclid's Fifth Postulate

Even after reading about Euclid's fifth postulate, I can't understand why it couldn't have been proven for Euclidean space. All of the geometers that lived between Euclid's time and Lobachevsky's worked only within the parameters of euclidean space (in which the postulate holds). So why could...
5. ### Volume One: Geometry without Multiplication

Volume One: Geometry without Multiplication I propose to write a textbook on the subject of Euclidean geometry. It will be divided into two volumes, Geometry without Multiplication and Geometry with Multiplication. In the future I may write a third volume about solid geometry. Geometry...
6. ### Why isnâ€™t Euclidâ€™s Elements taught anymore?

I recently posted a paper at Research Gate titled, â€œHow Math Can Be Taught Better.â€ https://www.researchgate.net/publication/282947903_How_Math_Can_Be_Taught_Better My section on geometry includes an appalling example of a geometry textbook totally screwing up a proof that would have...

8. ### Euclid Lemma/Division Algorithm

Proof by induction with existence and uniqueness of The Division Algorithm. m= qn + r 0 <= r < n
9. ### Hello everyone

Hello everyone! Not sure where to begin, I love math, and science. But I believe the way that I learned math isn't the best way to have learned. Unfortunately i never paid to much attention in geometry class, which now i realize was kinda important. I have been wanting to go back and start...
10. ### Question about Euclid prop 1.9

I have just started working through some of the easier propositions in Euclid's Elements. I have a question about Proposition 1.9. I hope someone will find the it interesting enough to refer to Euclid and follow my question. It's easy to see how 1.9 works intuitively, but there seems to be a...
11. ### Euclid's element (P14)

Hi, I was reading proposition 14 (Euclid's Elements, Book I, Proposition 14) of Euclid's elements and there is only one thing which I find weird: Why do we need postulate 4 to conclude that â€œthe sum of the angles âˆ CBA and âˆ ABE equals the sum of the angles âˆ CBA and âˆ ABD.â€ Why...
12. ### Euclid's division of algorithm

Hi, I am not able to understand how to apply the euclid's(a=b*q+r )division formula and solve the below problems. can you please help me solve the problem. 1. Prove that the product of 3 consecutive positive integers is divisible by 6? 2. For any positive integer N, prove that ( NÂ³ - N...
13. ### Euclid's algorithm

How does Euclid's algorithm lead to the equation am + bn = 1? a and b are relatively primes and m and n are integer number.
14. ### extended euclid algoritm

can i prove by induction on b that , for all positive integers a and b such that a > b, the triple (d; s; t) returned by EEgcd(a; b) indeed satis es d = s  a + t  b. I donot have a clue how to prove this. How i should start? Any help is appreciated
15. ### Euclid's Elements

Is there an edition of Euclid's Elements that doesn't include a commentary that interrupts (for lack of a better word) the original text? It doesn't matter if it is in Greek.
16. ### Inverse Modulo Using Euclid's Algorithm.

Hi all, I'm trying to learn how to do Inverse modulus by hand, by following a worked example from a book. I'm hitting a wall. Here are the steps for (223 MOD 660) 660 = 223 x 2 + 214 ... (1) 223 = 214 x 1 + 9 ... (2) 214 = 9 x 23 + 7 ... (3) 9 = 7 x 1 + 2 ... (4) 7 = 2 x 3 + 1 ... (5) 1 = 7...
17. ### ? >= log b/ log ? ( euclid's algorithm)

let ? be a positive integer. show that there exist integer a,b with a>b>0 and ? >= log b/log ? such that Euclid's algorithm on in put a,b performs at least ? divisions. ? = (1+sqrt 5)/2. proof. using the sequence of Fibonacci numbers F_n. Let a= F_ (?+2) and b= F_( ?+1). If running the...
18. ### Proofs, Euclid's misery

I need to prove that the converse of the alternate exterior angle theorem is logically equivalent to the euclidean parallel postulate :cry: for the love of god, I hope you guys know what your doing. This kind of math ends worlds.
19. ### Pythagorean Theorem in Euclid's Elements

Hi, In one of my books I found that Euclid in his Element's (Book I) cited eight ways how to prove it (in this book I have eight pictures containing "proof"). I have Euclid Elements (GreenLion, Heath translation) but I can only find one proof. To be well understood: I'm not looking for ANY proof...
20. ### euclid's division lemma!!

1. use euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m