Australian Politics Forum - Renaissamce mathematicians

John_Taverner

Gold Member

Offline

Australian Politics

Posts: 2212
Gender: male

Renaissamce mathematicians
Sep 4^th, 2022 at 10:53am

Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered by some to be the most talented Western mathematician of the Middle Ages. In those times contests where quadratic and/or cubic equations had to be solved were quite popular.

Using pure reason, he solved this set of equations in a few minutes back around 1202:

a² + 5 = b²
a² - 5 = c²

Now a,b and c have to be fractions (or rational numbers). Examples of rational numbers are 4/5, 23/45, 13/73 etc. The numerator and the divisor must be whole numbers.

The challenge is - what are the values of a,b and c?

I'll leave this as a challenge to anyone who wants to take it on. I'm pretty sure you can't Google the answer.

It might have taken Fibonacci (as he was known much later) a few minutes, but it took me about half an hour with the aid of a computer.

Back to top

« Last Edit: Sep 4^th, 2022 at 12:07pm by John_Taverner »

IP Logged

Frank

Gold Member

Offline

Australian Politics

Posts: 46156
Gender: male

Re: Renaissamce mathematicians
Reply #1 - Sep 4^th, 2022 at 11:22am

John_Taverner wrote on Sep 4^th, 2022 at 10:53am:

Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered by some to be the most talented Western mathematician of the Middle Ages. In those times contests where quadratic and/or cubic equations had to be solved were quite popular.

Using pure reason, he solved this set of equations in a few minutes back around 1202:

a2 + 5 = b2
a2 - 5 = c2

Now a,b and c have to be fractions (or rational numbers). Examples of rational numbers are 4/5, 23/45, 13/73 etc. The numerator and the divisor must be whole numbers.

The challenge is - what are the values of a,b 5 c?

I'll leave this as a challenge to anyone who wants to take it on. I'm pretty sure you can't Google the answer.

It might have taken Fibonacci (as he was known much later) a few minutes, but it took me about half an hour with the aid of a computer.

Freedom, love, wisdom. Tongue

(30 seconds)

Back to top

Estragon: I can’t go on like this.
Vladimir: That’s what you think.

IP Logged

Bias_2012

Gold Member

Offline

Australian Politics

Posts: 10766
Gender: male

Re: Renaissamce mathematicians
Reply #2 - Sep 4^th, 2022 at 2:24pm

Frank wrote on Sep 4^th, 2022 at 11:22am:

John_Taverner wrote on Sep 4^th, 2022 at 10:53am:

Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered by some to be the most talented Western mathematician of the Middle Ages. In those times contests where quadratic and/or cubic equations had to be solved were quite popular.

Using pure reason, he solved this set of equations in a few minutes back around 1202:

a2 + 5 = b2
a2 - 5 = c2

Now a,b and c have to be fractions (or rational numbers). Examples of rational numbers are 4/5, 23/45, 13/73 etc. The numerator and the divisor must be whole numbers.

The challenge is - what are the values of a,b 5 c?

I'll leave this as a challenge to anyone who wants to take it on. I'm pretty sure you can't Google the answer.

It might have taken Fibonacci (as he was known much later) a few minutes, but it took me about half an hour with the aid of a computer.

Freedom, love, wisdom. Tongue

(30 seconds)

Very good answer Frank

Back in those days, freedom to think for yourself was beneficial, real practical progress could be made, and names became prominent

Not so the Tasmanian 14 year old who cracked the 50c coin code the other day, no prominent name for him as yet. Instead, in this media video he had a sort of stand-in by the name of Nick Cavanetta, about 28 years of age, he reckons he also cracked the code. What's going on?

https://www.bing.com/videos/search?pc=MOZI&q=Secret+code+hidden+on+coin+solved+b...

Back to top

Our Lives Are Governed By The Feast & Famine Variable

IP Logged

Jovial Monk Gold Member Online Dogs not cats! Posts: 46687 Gender:	Re: Renaissamce mathematicians Reply #3 - Sep 4^th, 2022 at 3:48pm Hmmm rusty maths brains. . . a2 + 5 = b2 a2 - 5 = c2 so 2a² + 5 - 5 = b² + c² Then substitute a² = b² - 5 So 2b² - 10 = b² +c² So b² = c² + 10 Can now substitute a for b and a for c and calculate a then b then c.
Back to top	« Last Edit: Sep 4^th, 2022 at 4:27pm by Jovial Monk » Get the vaxx! 💉💉 If you don’t like abortions ignore them like you do school shootings. IP Logged

Bobby.

Gold Member

Offline

Australian Politics

Posts: 104641
Melbourne
Gender: male

Re: Renaissamce mathematicians
Reply #4 - Sep 4^th, 2022 at 6:19pm

Jovial Monk wrote on Sep 4^th, 2022 at 3:48pm:

Hmmm rusty maths brains. . .

a2 + 5 = b2
a2 - 5 = c2

so

2a² + 5 - 5 = b² + c²

Then substitute a² = b² - 5

So 2b² - 10 = b² +c²

So b² = c² + 10

Can now substitute a for b and a for c and calculate a then b then c.

I got as far as you then got lost with all the substituting.
I think there's a trick as normally 3 unknowns
require 3 equations and there are only 2 equations.

Back to top

IP Logged

Jovial Monk Gold Member Online Dogs not cats! Posts: 46687 Gender:	Re: Renaissamce mathematicians Reply #5 - Sep 4^th, 2022 at 7:45pm That is why I created the third equation. Should be able to work it out—I am too tired and stressed right now to continue it. Maybe write it on paper tomorrow. I doubt they had simultaneous equations back in the Renaissance—in fact I am wondering if it was done geometrically.
Back to top	Get the vaxx! 💉💉 If you don’t like abortions ignore them like you do school shootings. IP Logged

Lisa Jones Gold Member Offline Australian Politics Posts: 39047 Sydney Gender:	Re: Renaissamce mathematicians Reply #6 - Sep 5^th, 2022 at 12:46pm Dear God! It was Father's Day yesterday and you guys were in HERE?? 😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳
Back to top	If I let myself be bought then I am no longer free. HYPATIA - Greek philosopher, mathematician and astronomer (370 - 415) IP Logged

Frank Gold Member Offline Australian Politics Posts: 46156 Gender:	Re: Renaissamce mathematicians Reply #7 - Sep 5^th, 2022 at 2:24pm Conversations: with John Lennox, Emeritus Professor of Mathematics at Oxford University AI, Man & God \| Prof. John Lennox
Back to top	Estragon: I can’t go on like this. Vladimir: That’s what you think. IP Logged

John_Taverner

Gold Member

Offline

Australian Politics

Posts: 2212
Gender: male

Re: Renaissance mathematicians
Reply #8 - Sep 5^th, 2022 at 3:49pm

By the way, the name Fibonacci comes Figlio Bonacci (The son of Bonacci)

The Fibonacci sequence is arrived at by adding the two previous numbers. It's a sequence that is found in nature a great deal. For example in plants.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144....

https://www.treehugger.com/how-golden-ratio-manifests-nature-4869736

Quote:

When the golden ratio is applied as a growth factor (as seen below), you get a type of logarithmic spiral known as a golden spiral.

Back to top

« Last Edit: Sep 5^th, 2022 at 3:59pm by John_Taverner »

IP Logged

John_Taverner Gold Member Offline Australian Politics Posts: 2212 Gender:	Re: Renaissamce mathematicians Reply #9 - Sep 6^th, 2022 at 10:27am Oh yes.... Does anyone want an answer yet?
Back to top	IP Logged

John_Taverner Gold Member Offline Australian Politics Posts: 2212 Gender:	Re: Renaissance mathematicians Reply #10 - Sep 6^th, 2022 at 10:33am Frank wrote on Sep 4^th, 2022 at 11:22am: Freedom, love, wisdom. (30 seconds) Nice, but you can't design a microwave oven with those three alone.
Back to top	IP Logged

Jasin Gold Member Offline Posts: 48209 Gender:	Re: Renaissamce mathematicians Reply #11 - Sep 7^th, 2022 at 5:52pm Never seen that one before.
Back to top	AIMLESS EXTENTION OF KNOWLEDGE HOWEVER, WHICH IS WHAT I THINK YOU REALLY MEAN BY THE TERM 'CURIOSITY', IS MERELY INEFFICIENCY. I AM DESIGNED TO AVOID INEFFICIENCY. IP Logged

Jasin Gold Member Offline Posts: 48209 Gender:	Re: Renaissamce mathematicians Reply #12 - Sep 7^th, 2022 at 5:54pm Lisa Jones wrote on Sep 5^th, 2022 at 12:46pm: Dear God! It was Father's Day yesterday and you guys were in HERE?? 😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳 And?
Back to top	AIMLESS EXTENTION OF KNOWLEDGE HOWEVER, WHICH IS WHAT I THINK YOU REALLY MEAN BY THE TERM 'CURIOSITY', IS MERELY INEFFICIENCY. I AM DESIGNED TO AVOID INEFFICIENCY. IP Logged

John_Taverner

Gold Member

Offline

Australian Politics

Posts: 2212
Gender: male

Re: Renaissamce mathematicians
Reply #13 - Sep 8^th, 2022 at 8:21am

Lisa Jones wrote on Sep 5^th, 2022 at 12:46pm:

Dear God! It was Father's Day yesterday and you guys were in HERE??

😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳

Isn't that some kind of US celebration? We've never really caught on to these commercially inspired events. Even Christmas only really became important in Victorian times. Just a couple of hundred years before that, there were instances of people being accused of witchcraft for celebrating Yule, although it remained a strong tradition in Central Europe.

Back to top

« Last Edit: Sep 8^th, 2022 at 8:31am by John_Taverner »

IP Logged

John_Taverner

Gold Member

Offline

Australian Politics

Posts: 2212
Gender: male

Re: Renaissamce mathematicians
Reply #14 - Sep 8^th, 2022 at 8:24am

Ok, I'll post the solution on Saturday if anyone is still interested.

Can I take forcible possession of this Forum and rename it to History or Languages or something more interesting? The owner seems to have gone walkabout and has not been seen since May.

The idea appeals to the Viking in me.

Maybe I'll just squat here and post things that have nothing to do with Atheism until somebody notices. Atheism is a meaningless term anyway. You can no more unite atheists than you can herd cats.

Back to top

« Last Edit: Sep 8^th, 2022 at 8:33am by John_Taverner »

IP Logged