In case I gave the wrong impression, I do not mean to say that you are not permitted to include academic journals in your research.  I only mean to say that you probably don't need to be reading research mathematics - which is typically sometimes difficult.  

Someone should make a timeline of these things - putting together our different cultures.  If someone does - I will happily post it if "someone" is willing to share.  

§3.2 Quick Answers

Please remember cross-cultural communication is very slow, and even in-cultural communication isn't easy.  

Much of ancient and medieval Indian mathematics is, in fact, written in verse, and is quite poetic.  

Some of the mythology of large numbers is clearly just to make numbers that are large, but doesn't intend to be scientific with them.  It reminds me of Archimedes sand-reckoning, but I do think the Indians were widely considering exceptionally large numbersmore than other cultures.  

My references only refer to the Indians using the arithmetic triangle for combinatorics, unlike the Chinese, who seemed to only use it for expanding powers.  

With regard to expontents, 1 isn't very interesting 1^p = 1 and p^1 = p.  This may have something to with the sentiment that 2 is the first "actual" number.  They didn't begin counting with 2 or avoid using 1 in any way.  

Rules for operating with zero aren't as practically necessary as others.  And with regard to multiplication and division, they as significantly different than other numbers.  These are probably part of why they were difficult for humans to discern.

Remember there was less an idea of ownership of ideas in these past cultures than there is today.    

I don't know how Aryabhata deduced his approximation to pi.  I hope you can easily see that his text does give the same value indicated.  

We read that the Chinese weren't bothered by irrationals (didn't even make the distinction).  I don't know that the Indians did either, but I'm only speculating.  

Indeterminate equations are an area of study that is pursued extensively by Indians.  

Amicable numbers related to perfect numbers (a number whose factors add to itself).  Two numbers are amicable if their factors add to each other.  220 and 284 are the smallest example.  Try it.  

There are about 4.2 million Jains in the world today.  It is the 14th largest religion in the world.  

A fun quote from your classmate:  "Pythagoras is starting to seem like he was one of the last people to "discover" his theorem."

Varahamihira's work on the orbit of the earth did not involve differential equations.  Perhaps difference equations, but definitely nothing continuous.  Mostly this was done at the time by making adjustments, because the circular approximations are naturally inaccurate for an elliptical orbit.  

class notes


Math is a part of religion - to study mathematics and refine it is more religious, not in contrast to it.

history of numeration

The brahmi numerals were even preceded by the kharosthi numerals - which look a lot like roman.  Babylonians place value 60 used in Greece but didn't influence much else.  Chinese had base 10 multiplicative system -- not place value, so had 3 hundred 4 ten 7 rather than 347.  India had much the same.  Around 600 CE they started dropping the "hundred" "ten".  Severus Sebokht, in Syria, 662, refers to them.  Dot first used for zero.  System does seem heavily influenced by Chinese counting boards.  Definitely fully developed by 800, and even already transmitted to Islamic translators.  

1. (Baudhayana, Manava, Apastamba) Vedas 700 BCE
        "pythagorean" theorem, constructing altars and converse, square root of two

2.  Bakshali (Son of Chajaka) between 200 CE and 1200 CE ??
        linear indeterminant equations, quadratic formula, "rule of three"
(for solving proportions)
     Sphujidhvaja 300 CE - verbal base 10 with placeholder.

3.  Aryabhata 476 - ? CE
        more verbal base ten
        sum of whole numbers, squares, cubes
        display trig table from Calinger (both he and Varahamihira) [built trigonometry on ideas from Greeks, but before Ptolemy]
        solves linear indeterminants via continued fractions.

4.  Varahamihira 505-587 CE
        sin^2ø = 1/2(1-cosø)
        Arithmetic triangle - about 500 years before China - used for
combinations

5.  Brahmagupta 598-670
        better planar and spherical trigonometry
        according to Bhaskara, the pulveriser is used for ax + c = by.
        method for ax^2 + 1 = y^2
        integer arithmetic rules and some for zero
        area of cyclic quadrilaterals (which reduces to Heron's formula)

6.  Mahavira (Sridhara) 800s
         formula for combinations   

     Bhaskara 1114 - 1185
       Contents of Lilavati
        more theoretical trigonometry
        sum and squares