Australian Politics Forum - Renaissamce mathematicians

John_Taverner

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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.

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

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Frank

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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)

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

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Bias_2012

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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...

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Our Lives Are Governed By The Feast & Famine Variable

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Jovial Monk Gold Member Offline Dogs not cats! Posts: 46675 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.
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Bobby.

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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.

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Jovial Monk Gold Member Offline Dogs not cats! Posts: 46675 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.
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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?? 😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳😳
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Frank Gold Member Offline Australian Politics Posts: 46154 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
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John_Taverner

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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.

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

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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?
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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.
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Jasin Gold Member Offline Posts: 48205 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: 48205 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

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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.

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

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John_Taverner

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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.

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

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