During the duel with the abacist, how did Richard Feynman (Matthew Broderick) compute the cube root of 1729.03 so fast? In general, how can we compute roots?
Answer. We use the same formula that is used for powers, the same formula Feynman used.
First express which root to find in the form . Apply the binomial theorem.
The first few terms for the power of the value are:
If
is an integer, this series terminates; otherwise it is an infinite series. A
much more compact form
is
The first order approximation has just one extra term.
So, for more general quantities
NOW the original number was 1729.03. Thus in our notation and , and (i.e. cube root) we have
How good is this approximation? To twenty places
Advanced. Using more terms of the series improves accuracy. Using the next term of the series, we have
Here
the formula we used was
The contest between the abacist and the algorist has taken place for
centuries. In the picture below we see an example of the duel in an early
textbook on arithmetic by Adam Riese (14921559). As you can see one of the
competitors is using numbers, the other is using a counting board  also
referred to as an
abacus.
During
this period the arabic numerals (1,2,3,4,5,6,7,8,9,0) that we use today were
competing with the ancient Roman numerals (I,V,X,L,C,D,M), as they had for
centuries. It would be another century before our current numerals would
dominate in the world of numerical transactions.
The abacus, as we know it today appears in China in about 1200 CE. In
Chinese, it is called the suanpan, the Chinese word
'suan' meaning 'calculate.' Each rod has five beads below the bar and two
beads above the bar that move along bamboo rods. Such an abacus is pictured on
page 15.
The
Japanese soroban can be traced to the
16
century. It was probably borrowed and adapted from the Chinese. Its counters
are sharp edged and each rod has only one bead above the bar. and only four
beads below the bar. With an abacus the basic operations of arithmetic can be
carried out with remarkable speed and accuracy. Computing square and cube
roots is possible, but not very easy. The abacus is still used today in many
countries, but more in the countryside market place rather the industrialized
city.
In the movie Stand and Deliver, Jaime Escalante takes a group of East Los Angeles high school students to the AP Calculus exam. They must prepare for two years to get ready. The movie is about some of the personal stories of the young men and women who stick it out, sometimes against enormous odds. In the scene of the film clip, we see one of the boys struggling with a calculus problem.

This problem is a classic. The mistake he makes is a classic; we've all made
it at least once. We repeat it
here.
goes no where 
However, by reversing the "parts" we get the correct answer.
This film is about the remarkable spirit, perseverance and mental toughness of a teacher and his inspiration for a whole class of students to achieve what they thought was for them impossible. The Advanced Placement, AP, calculus exam is certainly no "walk in the park." Sometimes these situations make the very best stories.
Calculus is generally regarded to have been invented in the
17
century by Issac Newton and Gottfried Leibniz. However, calculuslike
calculations have been used for more than two thousand years. The earliest
such calculations were performed by Archimedes who for example proved when a
specially shaped boat would capsize. This is a tricky calculus problem, but
the methods of Archimedes are a tour de force of
brilliant computations.
What
is remarkable is the Sir Issac Newton's first teacher, Issac Barrow, actually
proved a version of the fundamental theorem of calculus somewhat earlier
().
Above is the differential triangle diagram from his book
Lectiones Geometricae which he used in the proof.

Rushmore is sort of a silly movie about a 15 year old, Max Fisher, who want to be an adult a little too soon. Placing his priorities on extracurricular activities, his grades plummet to the point of expulsion. In the film clip at the very beginning of the movie, he is fantacizing about solving what his teacher calls the " just about the hardest geometry problem in the world."
We know the area of the circle of radius
is
It is also known that the area of an ellipse with semimajor axis
and semiminor axis
is given
by
What
you see Max doing in the film is solving this problem. Is his solution
correct?
Yes, it's right on the mark. Here is the solution
derived:
You can see in the images above and below that Max is essentially performing
the same
calculation.
Now its your turn. Use the rates equation above.
Follow the calculation of Feynman to approximate (What should be?)
Now find Here you need to take You will need another as well.
Here's a calculation problem that you can do in your head  should take about
one minute. Calculate the value of this
expression
(Hotel Infinity) A hotel has an infinite number of rooms, and each is occupied. A new guest arrives at the front desk and asks for a room. The manager says, "Of course, you may have a room." Without expelling any current guest, how does the manager find a room for the new guest? Suppose the guest arrived with his infinite number of cousins. How could the manager accomodate each of them in an individual room?
Determine
Determine
Find the area inside the ellipse
and outside the circle
.
(See picture below. Find the area of the "blue" or darker
region.)