390 Quick
Answers 10 March
We
have an exam on Friday. Bring computers; you will be typing
in class. Do *NOT* access any materials during class.
Please please have a plan. I am happy to discuss plans in
depth during office hours. I have an office hour every day
between now and then, although I wouldnât recommend coming on
Friday for this just because you wonât have time to make a new
plan. I will be grading exams on a long day during break,
and by the time you return, you will have current averages before
we meet after break.
April
7 is far sooner than it feels. Please please please be
diligent and dedicated in working on your drafts now. I am
always happy to talk with anyone about it. I will read paper
drafts in the order I receive them. I will talk about them
in person, but I will not be reading drafts online and giving
feedback before they are due. I strongly recommend visiting
either the writing learning centre or the history writing learning
centre. If you turn your paper draft in on 7 April, I may
return it to you as late as early May. Please submit them
early.
âDoes
the bibliography for the final paper (and draft) need to have the
annotations, or is a list of the sources cited in the proper
format sufficient?â No annotations when you write a paper.
One
more tiny point ⌠thereâs nothing stopping you from
completing the reactions due the Monday after break as early as
today.
I'm
glad that people are liking the portraits. I do try to
include them whenever I can. I expect each of them knew and
in some sense acknowledged the portraits that we are seeing.
I only didn't before because we don't have any from
contemporaries.
I feel that reactions were taken more seriously this time, after
my comment last. Thank you.
Lecture
Reactions
Bombelli
was the first in history to work with complex numbers. He
did so by trusting that â-1 would behave like other numbers,
including squaring to be -1. It
is very natural to say that â-1 does not exist. It was a
big step to think that it could be useful. Bombelli
trusted that complex numbers would work the way other numbers
do, and that led to the results and methods. The point of
Bombelliâs work was to find a way to work with the answer as del
Ferroâs work produced it. Bombelli tried a guess - it
worked out, that's nice. If it hadn't, we wouldn't be
talking about it, and he might have tried another.
The form of solutions produced by del Ferro's cubic method is
pretty intractable, even today.
Ferrari was _not_ thinking of the fourth dimension. That's
a noteworthy transition toward how you think of x^4. The
key to both del Ferro's and Ferrari's method is including new
variables which gives more flexibility, more control more way to
make it as you want.
Don't feel bad for not having heard about
before, it's not as common as haversine.
Remember, printing is a driving force to standardising
symbols.
I think it's very reasonable that you don't know of Riese.
He wrote a German arithmetic textbook for children. Remember
that education was surely not widespread yet.
Someone
asked when do we know that the sun moves. This was an idea
of Halley, Cassini and Herschel in the 18th century, verified by
Argelander in the 19th century. [I had to look this
up.]
Reading
Reactions
Re: Nostradamus and in general - be very careful to never confuse
Astrology and Astronomy.
Viete did not fix the calendar, in fact, he was complaining
about it being fixed. Pope Gregory didn't either, but the
Gregorian calendar bears his name (like the Julian calendar in
that way). I will finish our calendar story.
Oh this is a good time to say - yes, coding is definitely
mathematics. The NSA employs more mathematicians than
anyone in the world. They have positions at all
levels. If you are comfortable not telling anyone what you
do, I have heard good things from alumni and friends of mine who
work there. It is a curious compromise between academics
and industry, as they do spend a significant amount of time
learning new mathematics - taking and teaching classes - all
about the things that we donât know about.
Presumably van Roomen created the problem using powers
of sines as Viete solved it - by expanding sin(45ø) using
angle addition repeatedly. There is not another
reasonable way to imagine the problem being created.
Ok, I don't think I appreciated this enough before - yes, Roberval
is the first known to consider the sine curve, in it's role as the
companion of the cycloid. Did he connect it with the sine
trigonometry? It seems perhaps not yet.
Re:
Fermat in general - asking questions is valuable, making
conjectures is valuable, and proving things is also
valuable. They all have their place. We need all of
them.
It
is a tricky problem to ask who first considered integrals.
Integrals are area problems. And area problems are as old
as history. Some of them are done by cutting regions into
smaller pieces. Thatâs old too. When do they become
integrals? Perhaps when they connect to derivatives, or
when they become systematic (which happens the same time).
And that is what calculus is all about. We are hot on the
heels of calculus right now. We need one more piece at the
beginning of 7.2, and then weâll have full calculus story next
week. Well, at least that part of it. Also if youâre
looking for this, you might remember to look back at Stevin, who
had some early insights. 3 main problems of
calculus: derivatives (slopes), integrals (areas), and
optimisation.
I might not have mentioned this without someone asking, but yes,
it is very reasonable to expect that Pascal was familiar with
other's work on the arithmetic triangle.
Cardano
did work with probability as a gambler before Pascal.
For
anyone thinking about computing machines as topic for the final,
Pascal is a good place to start. .