[Today’s class: Maya arithmetic, especially subtraction (use toothpicks and small candies---or pencil and paper),

review of Pythagorean Theorem and its converse,

Puzzle Proofs of Pythagorean Theorem activity,

Proofs of Pythagorean Theorem via area and algebra (see Pythagorean Theorem activity),

Historical applications of Pythagorean Theorem]

** **

**Mathematics 115 **

**Homework Assignment #4**

*Due **Monday, January 14, 2002*

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__Prof.__ __Beery's__ __office__ __hours__ __this__ __week__: ** **Thursday
1/10 10:30 a.m.-12:30
p.m., 4-5 p.m.

Friday 1/11 1:30 - 3:30 p.m.

Monday 1/14 10:30 a.m.-12:30 p.m., 4-5 p.m.

and by appointment, Hentschke 203D, x3118

__Tutorial__** session: Sunday, Jan. 13, **

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* *

__Read__: "No Stone Unturned (Early
Southern California math artifacts?)"

"Kernel revealing history of humans in the New World"

"Mayan Arithmetic" (you may skip Section 4, Division)

"Mayan Head Variant Numerals"

"Ancient Mayan Symbols Yield New Meaning"

All of these selections are in the Mathematics of the Americas section.

The following articles are in the Mathematics of Ancient Greece section.

“Pythagoras of Samos” (pages 399-401 in “The Greeks”)

“Pythagoras’ Trousers”

“The Pythagorean Problem” (pages 103 and 104 and top of page 105)

__ __

__Do__: Exercises 1 - 8 from the "Pythagorean
Theorem Problems" handout

Answer the following questions about the "Mayan Arithmetic" essay.

1. What's the difference between the two Mayan number systems? List at least

four (4) differences.

2. Write the numbers 12 bak 16 kal 9 hun and 8 bak 6 kal 16 hun using Mayan

symbols,
then add them together. Show __all__ steps.

3. Write 12 kal, 6 hun and 5 kal, 18 hun using Mayan symbols, then

subtract
the second number from the first number. Show __all__ steps.

4. Write 12 bak 16 kal 9 hun and 8 bak 6 kal 16 hun using Mayan symbols,

then
subtract the second number from the first number. Show __all__ steps.

** **

**Monday's
class will begin at ****1:15 p.m.** and
will end at the usual time.

**Monday's
quiz** (50
points) will cover all class work, reading, and homework since

the start of Interim!

**Read
ahead:** "Hydraulic
Societies" and "Prime Numbers" (way up front)

**1.** Use the
Pythagorean Theorem to find the length of the third side of the right triangle.

**1a.**
**1b.**

**1c.**

**2.** The converse
of the Pythagorean Theorem says: * If a triangle has sides of lengths a, b,
and c and a ^{2} + b^{2} = c^{2}, then the triangle is a
right triangle with legs of lengths a and b and hypotenuse of length c.* Use
the Pythagorean Theorem and its converse to determine whether or not each
triangle is a right triangle.

**2a.**
**2b.**

**2c.**

Exercise 3 is from a medieval European military handbook.

**3. **(Johnson and Mowry,
364) An army wishes to scale a 24-foot high castle turret. If the turret lies
across an 18-foot wide moat, how long must the army’s ladder be?

** **

Exercise 4 is from the *Cairo Papyrus,* an Egyptian
mathematical papyrus dating from about 300 BCE. Of its 40 mathematical
problems, nine require use of the Pythagorean Theorem.

** **

**4.** (Burton, 76) A ladder of
length 10 cubits leans against a wall. If the foot of the ladder stands 6
cubits from the wall, how high up the wall does the ladder reach?

Exercise 5 is from an ancient Mesopotamian clay tablet from the Old Babylonian period (1900-1600 BCE).

**5. **(Bunt, *et al,* 59)
A beam of length 30 units stands vertically against a wall. As the top of the
beam slides down the wall, the bottom of the beam slides away from the wall. If
the top of the beam slides 6 units down the wall, how far does the bottom of
the beam slide away from the wall? (Hint: The beam becomes the hypotenuse of
a right triangle.)

** **

Exercise
6 is from the *Lilavati,* written by the Indian mathematician and
astronomer, Bhaskara II, in about 1150 CE.

** **

**6. **(Plofker, 23) In a
lake with flocks of geese and cranes, the tip of a lotus bud stands 1/2 *hasta*
above the surface. The wind slowly pushes the bud until it is submerged at a
distance of exactly 2 *hastas* from its original position. What is the
depth of the water? What is the height of the lotus plant?

** **

Exercises
7 and 8 are from the** **Chinese text, *Nine Chapters on the Mathematical
Art,* which originally was composed in about 200 BCE, and rewritten and
expanded in about 250 CE by Liu Hui. Like Euclid's *Elements,* the *Nine Chapters*
was a compilation of the mathematics known to that day (Swetz and Kao, 17,
26-27). The subject of Chapter 9 of the *Nine Chapters* is the
Pythagorean Theorem.

**7.** (Katz, 42)**
**A pole leans against a 10 *chi* high wall so that the top of the pole
is even with the top of the wall. If the bottom of the pole is moved just 1 *chi*
further from the wall, the pole will fall flat on the ground without sliding
any further from the wall. What is the length of the pole? (Hint: Begin by
writing the length of the hypotenuse in terms of x.)

**8. **(Swetz and Kao,
42-44) A measuring rod has the same length as the diagonal of a rectangular
door. If the measuring rod is 4 *chi* longer than the width of the door
and 2 *chi* longer than the height of the door, find the dimensions of the
door. (Hint: What are the lengths of the legs of the triangle in terms of x?)**
**

References:

** **

Bunt, Lucas N.H., Phillip S.
Jones, and Jack D. Bedient, *The Historical Roots of Elementary Mathematics,*

Dover Publications, New York, 1988.

Burton, David, *The History of Mathematics: An
Introduction *(3^{rd} edition), McGraw-Hill, New York, 1997.

Johnson, David, and Thomas Mowry,
*Mathematics: A Practical Odyssey,* PWS Publishing, Boston, 1995.

Katz, Victor J., *A History
of Mathematics: An Introduction,* Addison-Wesley, Reading, Massachusetts,
1998.

Plofker, Kim
(translator), *The Lilavati of Bhaskara II* (unpublished manuscript), Brown University,

Providence, R.I., 2000.

Swetz, Frank J., and T.I.
Kao, *Was Pythagoras Chinese?: An Examination of Right Triangle Theory in *

* Ancient **China**,* National Council of Teachers of Mathematics, Reston, Virginia,
1977.