390 Quick Answers 9 February

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Interesting question - was new mathematics as daunting "then" as it is now?  I will talk about this.  Are there modern parallel developments of mathematics?  Yes.  And yes, history of mathematics does push us to be philosophical about mathematics.  I think that's a good thing. 

Lecture Reactions

Having commas every 4 in Chinese numerals does not make anything more difficult.  Chinese also avoid problems of Germanic languages by pronouncing 15 as ten-five and 50 as five-ten. 

In the Zhoubi Suanjing proof of the MCRTT, we computed the area of the square in two different ways, set them equal and did minimal algebra to get the desired result. 

I just had this idea from one of your comments - The Nine Chapters was a collection of known mathematics, as more was known, more was added … I have never thought this before, but in a way it is not unlike Wikipedia. 

Rule of three is nothing at all more than "if you know 3 pieces in a proportion, you can find the fourth."  We'll see it mentioned by this name again. 

You do know about black Friday, right?  And "in the red"?  If not, talk to an accounting major.  [You may not use this as a reaction for next time.]

Combinatorics is counting.  We will see more clearly from the Indian perspective why the arithmetic triangle is combinatorial. 

Indeterminate problems have infinitely many solutions.  You learned about them in linear algebra.  Our example was an example of exactly one such problem. 

For Sun Zi’s Chinese remainder theorem - if we have a multiple of 3 and 5 which is 1 mod 7, then it contributes nothing to the 3 or 5 remainders and exactly 1 to the 7 remainders.  We then can multiply that to get something that contributes what we want to the 7 remainder.  We repeat this for the others.   The lcm of 3,5,7 is 105 so changing by 105 doesn’t change any of the multiples.

The power of the "Celestial Element" method we discussed for approximating polynomial roots is not to find exact values, but to approximate them.  This is not a replacement for the quadratic formula, or for factoring, but how do you compute the cube root of 31 to 5 decimal places. 

Solving quartics is kinda a big deal.  Others don't think about it because algebra is too geometric - so x^2 is a square, and x^3 is a cube so …

Why did we go through Chinese and Indian mathematics so quickly?  Good question.  Quick answers:  we have all of human history to cover, so everything is rushed.  Furthermore, we are focusing on tracing the mathematics you know.  There is also plenty of mathematics that doesn't lead to the mathematics you know.  We're mostly not following those paths. 

China and India have a curious interaction.  They are border countries, but have the world's most imposing boundary between them.  There is some transfer between, but it's also limited. 


Reading Reactions

I think the Indian proof of the MCRTT may be the oldest that I know. 

We will see this show up from time to time, If you have "a number of books" would that include 1?  Perhaps not.  So, maybe 1 is not a number.  It's not important, but this will recur. 

Zero being used in arithmetic and zero being a placeholder (both are important) naturally come hand in hand, because now you need to add or multiply 0 in multi-digit computations.  Without it being a placeholder, there was also little need to think much about the arithmetic. 

I'm out of my area here, but I don't think Jains would think of end of time, rather of cycles and starting over.  That is very more consistent with what I know of Indian philosophy. 

"Aryabhata expressed 57,753,336 where the syllables represent 6, 30, 300. What does this mean, how do they correlate?"

I do think Sridhara's Trisitaka is a poem about arithmetic, but I have not seen it. 

Primes wouldn't be amicable because their only proper factors are 1.  220 has proper factors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
284 has proper factors 1, 2, 4, 71, 142

I did not intend to talk about Brahmagupta's formula for cyclic quadrilaterals, but s = semiperimeter, like in Heron's formula, and the formula does not work for general quadrilaterals. 

"It makes sense to me to have the lowest magnitude first then increase from there. I wonder why we don't do it that way." - come back on Monday, this will be our starting point. Watch for it in the reading for next class

I like that there's eagerness for Islamic mathematics.  Jeff sets it up nicely, and it deserves this showcase treatment.  Stay tuned for next week (and a little more, we'll see).