Dueling with the Abacist to find cube roots

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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.

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First express which root to find in the form MATH. Apply the binomial theorem.

The first few terms for the $p^{\text{th}}$ power of the value $1+t$ are:

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$\vspace{1pt}$If $p$ is an integer, this series terminates; otherwise it is an infinite series. A much more compact form is
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The first order approximation has just one extra term.


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So, for more general quantities
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NOW the original number was 1729.03. Thus in our notation $a=1728$ and $t=1.03$, and $p=\dfrac{1}{3}$ (i.e. cube root) we have


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How good is this approximation? To twenty places MATH

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Advanced. Using more terms of the series improves accuracy. Using the next term of the series, we have


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Here the formula we used was MATH


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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 (1492-1559). As you can see one of the competitors is using numbers, the other is using a counting board - also referred to as an abacus.
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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 suan-pan, 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.
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The Japanese soroban can be traced to the 16$^{\text{th}}$ 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 country-side market place rather the industrialized city.

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A little more calculus

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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.

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"Send me a postcard."
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This problem is a classic. The mistake he makes is a classic; we've all made it at least once. We repeat it here.
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MATH $=$ MATH
$u=\sin x$ $du=\cos x~dx$ $= $ MATH
$dv=x^{2}dx$ MATH $=$ goes no where

However, by reversing the "parts" we get the correct answer.

MATH $=$ MATH
$u=x^{2}$ $du=~2xdx$ $=$ MATH
$dv=\sin x~dx$ $v=-\cos x$ $=$ MATH

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.

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Calculus is generally regarded to have been invented in the 17$^{\text{th}}$ century by Issac Newton and Gottfried Leibniz. However, calculus-like 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.
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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 (MATH). Above is the differential triangle diagram from his book Lectiones Geometricae which he used in the proof.


Still more calculus...

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Max Fisher

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."

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We know the area of the circle of radius $r$ is $A=\pi r^{2}.$ It is also known that the area of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is given by
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What you see Max doing in the film is solving this problem. Is his solution correct?

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Yes, it's right on the mark. Here is the solution derived:
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You can see in the images above and below that Max is essentially performing the same calculation.
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calculus__83.pngNow its your turn. Use the rates equation above.

  1. Follow the calculation of Feynman to approximate $\root{3} \of{28}.$(What should $a$ be?)

  2. Now find $\sqrt{28}.$ Here you need to take $p=\dfrac{1}{2}.$ You will need another $a$ as well.

  3. Here's a calculation problem that you can do in your head - should take about one minute. Calculate the value of this expression
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  4. (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?

  5. Determine MATH

  6. Determine MATH

  7. Find the area inside the ellipse MATH and outside the circle $x^{2}+y^{2}=a^{2}$. (See picture below. Find the area of the "blue" or darker region.)
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