390 Quick
Answers February 12
Lecture
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The
MCRTT is about combining two squares to make a larger
square. It’s about areas. I can’t say this
enough. Please don’t lose track of this.
In the square doubling picture, two squares are combined into one by
cutting one of them up into pieces and arranging them around the
first square to make a larger square. From the construction,
you would then look at the side length to get an approximation of
√2.
There's nothing about radians yet.
Varahamihira's triangle is identical to Yang Hui's triangle which is
identical to what I hope you called Pascal's triangle (i.e. I hope
that you have seen it). He (Varahamihira) used it for
computing combinations. Please recognise - just because
you sometimes used combinatorics in your probability classes doesn't
mean that when you answer "how many ways …" that it is a probability
question.
I am dismayed at so many being unfamiliar with different sizes of
infinity as it is expected by the department to be standard material
in 239.
What is in the Lilivati 8 operations on fractions chapter?
This is a reminder to check my book.
- finding common denominators
- multiplying fractions
- mixed numbers to improper fractions
- and examples of these things, perhaps they were counting
methods + examples
The
purpose of the pulveriser is to to find all natural number
solutions to an equation like 17x + 5 = 15y. Solving integer
equations is difficult, 61x^2+ 1 = y^2 is yet more difficult. I
don't know a shorter way to solve the pulveriser problem. I
would be surprised if there is a much shorter one. This is
similar to the Euclidean algorithm which is the most efficient known
method. The putting zero step is to make each of the steps in
the sequence the same afterwards.
Reading
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The single most important thing to remember about the rise of
Islamic culture is being open to ideas from other cultures.
Islamic mathematics combines input from the Grecoromans to the west,
the Indian and Chinese to the east, and even some from ancient
Babylonian in the same region.
We will talk about how and why we read numerals backwards.
Don't believe otherwise, we do.
ibn Labban shows a curiosity of early use of base ten numerals,
early on the whole number part was in base ten, but the fraction
part was in base 60. This is not so different from using
degrees, minutes, seconds.
Negatives are … complicated at this stage. Many times when
seeking an answer it is about seeking a length, but negatives could
show up part way toward a solution.
Oh
no, it’s time for the silly bible π story. I had so happily
banished this from my mind. There is _one_ value for
π in the bible. It is 3. That is the value that
the Indiana state legislature attempted to make law. Yes,
that is true. What isn’t true is that there is a much more
accurate value encoded in the bible. That is a fanciful tale
and please see that and neither believe that or any similarly
outlandish conspiracy that you read.
I
enjoy that “algebra” which sounds fancy to many students has such
a simple meaning. Please share it with students learning it
for the first time. Algebra at this level
(precalculus algebra) is mostly not something that is proven, but a
language to speak and a way to express and find answers. This
is consistent with the fact that in HS algebra you didn't spend much
time proving. Furthermore, remember this is still all verbal
algebra, no symbols, no equations.
Abu
Kamil’s pentagon isn’t so difficult. Put x on all the sides
of the pentagon. Remember the square sides are 10.
You’ll get some right triangles where the side lengths are all
known in terms of x. The MCRTT can be used to find x.
This is very likely his method, which is noteworthy for being
similar to what you would do. The angles are clearly not
congruent - one is a right angle, and the others clearly are
not.
Now that you know the peculiar history behind the term "sine",
please do know that there are excellent reasons for the co-functions
("of the complement") and tangent (to the circle) and secant (to the
circle). If
you want to give a good meaningful name for sine, call it
half-chord.
Surely memorisation in history, long throughout history, was much
more relied upon than it is now.
al-Sijzi’s
derivation of (a+b)^3 would be similar to what one might do now
with Algeblocks. I will show here.
We will discuss the solution to the cubic, and Khayyami in depth on
Friday. These are geometric solutions to equations. They
have nothing to do with equations for curves as you know them.
Ok,
here’s what’s happening … al-Khayyami is getting pushed to Friday,
and some of Friday is getting pushed to Monday for when you will
read about what’s happening in Europe all this time. Spoiler
alert - it’s not much. So, we’ll wrap up Islamic mathematics
next Monday.