390 Quick Answers 10 February

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. 

It is common for artifacts to have a ruler for scale and a standard colour chart for colour comparison. 

I am happy to take more exam topics, but they do not count as lecture reactions for credit. 

Remember - the idea of broad ranging public education is _very_ new. 

Lecture Reactions

Why "Pythagorean Theorem"?  Tradition and western ethnocentrism.  I'm in favour of anyone pushing back on this in your classes.

I didn't mention this last time, but it's worth saying:  Chinese have commas every 4 places in their numerals which leads to an interesting change in pronunciation.  Chinese also avoid problems of Germanic languages by pronouncing 15 as ten-five and 50 as five-ten. 

Keep the timing in mind.  The very ancient different circle formulas (as they likely felt they were all good) are much much older than one solving quartic equations. 

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. 

The Chinese method for roots is very similar to what was used in the 50s in curriculum here.  Yes, it led to it.  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, or division, but how do you compute the cube root of 31 to 5 decimal places.  Synthetic division, which was also used but I omitted, is for dividing polynomials by (x-a).  These are the steps in the text for x^4=100.  Given that the Chinese had access to synthetic division, I would expect they could divide out one root to hunt for others.  That being said, while they did respect negatives far earlier than further West because they didn't think as geometrically, hence even being able to consider 4th powers, they definitely were not thinking about complex roots this long ago.  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 … If I wrote x^4 =100 as 100 0 0 0 100, then I was wrong, it should have been 1 0 0 0 100.  I was not joking or lying.  I make mistakes.  I was wrong. 

Oh, the Chinese remainder theorem is also not merely division.  People keep talking about remainders, and I'm not sure what they mean, but whichever one you mean, what the Chinese were doing is more than you think it is.  This is a modular equivalence problem. 

Both the Chinese and the Islamic calendars are Lunisolar, keeping track of both moon and sun.  This is far enough back now - do not include it in your reactions. 

The Arithmetic Triangle (seen from Yang Hui in China, will be reseen) contains both binomial coefficients (i.e. the coefficients when expanding a binomial) and combinations.  We will see it reappear.  The Chinese saw it as binomial coefficients.  The Indians will see it as combinatorics.  Combinatorics is counting.

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.

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. 

Reading Reactions

We'll see this again through history.  If you say "a number of people were there", would that include one?  Zero?  The idea that 2 is the first number seems … practical.  And … unimportant. 

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 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 Friday, 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).