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A History Of Mathematics

Victor J. Katz

Chapter 1

Ancient Mathematics - all with Video Answers

Educators


Chapter Questions

01:43

Problem 1

Determine the words for the numbers 18 and 40 in all the languages you and your classmates know. Compare the construction of these words. Are there any forms essentially different from the ones in Sidebar 1.1?

Nidhi Singhi
Nidhi Singhi
Numerade Educator

Problem 2

Represent 125 in Egyptian hieroglyphics and Babylonian cuneiform.

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Problem 3

The Greeks used a ciphered system based on their alphabet to represent numbers, at least from about 450 B.C.E. The representation was as follows:
$$
\begin{array}{llllll}
\alpha & 1 & \iota & 10 & \rho & 100 \\
\beta & 2 & \kappa & 20 & \sigma & 200 \\
\gamma & 3 & \lambda & 30 & \tau & 300 \\
\delta & 4 & \mu & 40 & v & 400 \\
\epsilon & 5 & \nu & 50 & \phi & 500 \\
\zeta & 6 & \xi & 60 & \chi & 600 \\
\zeta & 7 & o & 70 & \psi & 700 \\
\eta & 8 & \pi & 80 & \omega & 800 \\
\theta & 9 & i & 90 & \lambda & 900
\end{array}
$$
where the letters $\zeta$ (digamma) for 6,9 (koppa) for 90 , and $D \lambda$ (sampi) for 900 are letters that by this time were no longer in use. Hence 754 was written $\psi v \delta$ and 293 was written $\sigma \varphi \gamma$. To represent thousands, a mark was made to the left of the letters $\alpha$ through $\theta$; for example, $\theta$ represented 9000 . Larger numbers still were written using the letter $M$ to represent myriads $(10,000)$ with the number of myriads written above: $M^\delta=40,000, M^{\text {thou }}=71,750,000$. Represent 125 , 62, 4821. and 23,855 in Greek alphabetic notation.

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Problem 4

The basic Chinese symbols for numbers from the Shang period are

$$
\begin{array}{cccccccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 100 & 1000 \\
- & = & \equiv & \vdots & \text { 《 } & + & \text { ) } & 5 & 1 & \text { (4) } & 7
\end{array}
$$

There were compound symbols for 20, 30, 40 (namely, $\cup \Psi U$ U), but in general notation followed the plan indicated in the text. Hence 88 is ) () (and 162 is (x) 1 ) $=$ Write the Chinese form of $56,554,63$, and 3282 .

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Problem 5

Use Egyptian techniques to divide 84 by 5 .

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00:19

Problem 6

Multiply $7 \overline{2} \overline{4} \overline{8}$ by $12 \overline{3}$ using the Egyptian multiplication technique. Note that it is necessary to multiply each term of the multiplicand by $\overline{\overline{3}}$ separately.

Tiffany Tran
Tiffany Tran
Numerade Educator
00:52

Problem 7

A part of the Rhind Mathematical Papyrus table of division by 2 follows: $2 \div 11=\overline{6} \overline{66}, 2 \div 13=\overline{8} 52 \overline{104}$.
$2 \div 23=12276$. The calculation of $2 \div 13$ is given as follows:
table cant copy
Perform similar calculations for the divisions of 2 by 11 and 23 to check the results.

Heather Zimmers
Heather Zimmers
Numerade Educator
00:27

Problem 8

Given the value $2 \div 13=852104$ above, the unit fraction values of $3,4,5 \ldots, 12$ divided by 13 can be casily found. For example, $3 \div 13=\overline{8} \overline{13} \overline{52} 104($ since $3=1+2)$ and $4 \div 13=\overline{4} \mathbf{2 6} 52$ (since $4=2 \times 2$ ). Similarly, calculate $5 \div 13.6 \div 13$. and $8 \div 13$.

James Kiss
James Kiss
Numerade Educator
00:54

Problem 9

The second part of problem 79 of the Rhind Papyrus reads:

It has heen surmised that this was a problem similar to the Old English children's rhyme "As 1 was going to St. Ives." Thus the complete problem may have read: "An estate has 7 houses, each house has 7 cats, each cat catches 7 mice. each mouse eats 7 spelt, each spelt was capable of producing 7 hekats of grain. How many things were there in the estate?" The first part of the problem shows that the product of 2801 by 7 is 19.607 . Show that this is the correct answer for the sum of the geometric series $7+49+343+2401+16.807$.

Willis James
Willis James
Numerade Educator
01:13

Problem 10

. Show that $1 \div 7$ gives the periodic sexagesimal fraction $0: 8,34,17,8,34,17 \ldots$ by dividing in base 60 .

Babita Kumari
Babita Kumari
Numerade Educator
00:17

Problem 11

Find the reciprocals in base 60 of 18, 32, 54, and 64 $(=1,04)$. What is the condition on the integer $n$ that insures that it is a regular sexagesimal-that is. that its reciprocal is a finite sexagesimal fraction?

Ramzi Deek
Ramzi Deek
Numerade Educator

Problem 12

. In the Babylonian system, multiply 25 by 1,04 and 18 by 1.21. Divide 50 by 18 and 1,21 by 32 (using reciprocals). Use our standard multiplication al gorithm modified for base 60.

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01:21

Problem 13

Solve by the method of false position: A quantity and its $1 / 7$ added together become 19. What is the quantity?

Stephanie Carter
Stephanie Carter
Numerade Educator
00:52

Problem 14

Problem 72 of the Rhind Mathematical Papyrus reads " 100 loaves of pesu 10 are exchanged for loaves of pesu 45 . How many of these loaves are there?" The solution is given as, "Find the excess of 45 over 10 . It is 35 . Divide this 35 by 10. You get 3 2. Multiply $3 \overline{2}$ by 100 . Result 350 . Add 100 to this 350 . You get 450 . Say then that the exchange is 100 loaves of pesu 10 for 450 loaves of pesu 45 . ${ }^{-29}$ Translate this solution into modern terminology. Compare the method here to the solution in the text of problem 75 . How does this solution demonstrate "linearity"?

Heather Zimmers
Heather Zimmers
Numerade Educator
03:14

Problem 15

Solve problem 3 of Chapter 3 of the JiuFJang: Three people, who have 560,350 , and 180 coins respectively, are required to pay a total tax of 100 coins in proportion to their wealth. How much does each pay?

Zach Steedman
Zach Steedman
Numerade Educator
01:20

Problem 16

Find the solution to problem 3 of Chapter 8 of the Jiuzhang using the Chinese method: The yields of 2 bundles of the best grain. 3 bundles of ordinary grain, and 4 bundles of the wont grain are neither sufficient to make a whole measure. If we add to the good grain 1 bundle of the ordinary, to the ordinary I bundle of the worst, and to the worst I bundle of the best, then each yield is exactly one measure. How many measures does 1 bundle of each of the three types of grain contain? Show that the solution according to the Chinese method involves the use of negative numbers.

Heather Zimmers
Heather Zimmers
Numerade Educator

Problem 17

Solve problem 1 of Chapter 7 of the Jiuzhang using the method of surplus and deficiency: Several people purchased in common one item. If each person paid 8 coins, the surplus is 3 ; if each paid 7 , the deficiency is 4 . How many people were there and what is the price of the item?

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05:20

Problem 18

Solve problem 26 of Chapter 6 of the Jinzhang: There is a reservoir with five channels bringing in water. If only the first channel is open. the reservoir can be filled in $1 / 3$ of a day. The second channel by itself will fill the reservoir in 1 day, the third channel in 2 1/2 days, the fourth one in 3 days, and the fifth one in 5 days. If all the channels are open together. how long will it take to fill the reservoir? (This problem is the earliest known one of this type. Similar problems appear in later Greek. Indian, and Western mathematics texts.)

Gus Steppen
Gus Steppen
Numerade Educator

Problem 19

Solve problem 28 of Chapter 6 of the Jiustang: A man is carrying rice on a journey. He passes through three customs stations. At the first, he gives up $1 / 3$ of his rice, at the second $1 / 5$ of what was left. and at the third, $1 / 7$ of what remains. After passing through all three customs stations, he has left 5 pounds of rice. How much did he have when he started? (Versions of this problem occur in later sources in varrous civilizations.)

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04:41

Problem 20

Given a circle of radius 1 and a chord cutting off a central angle of $90^{\circ}$, show that $s$, the length of the chord, is $\sqrt{2}$ and that $p$, the length of the "arrow," is $\frac{2-\sqrt{2}}{2}$, Calculate the area of the segment determined by this chord using both the Chinese formula given in the text and modern methods. Compare the answers. Do the same for a segment in an angle of $60^{\circ}$ and one in an angle of $45^{\circ}$.

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
07:51

Problem 21

Show that the area of the Babylonian "barge" (see Fig. 1.14) is given by $A=(2 / 9) a^2$, where $a$ is the length of the are (one-quarter of the circumference). Also show that the length of the long transversal of the barge is $(17 / 18) a$ and the length of the short transversal is $(7 / 18) a$. (Use the Babylonian values of $C^2 / 12$ for the area of a circle and 17/12 for $\sqrt{2}$.)

Chris Trentman
Chris Trentman
Numerade Educator
03:13

Problem 22

Show that the area of the Babylonian "bull's eye" (see Fig. 1.14) is given by $A=(9 / 32) a^2$, where $a$ is the length of the arc (one-third of the circumference). Also show that the length of the long transversal of the bull's eye is $(7 / 8) a$ while the length of the short transversal is $(1 / 2) a$. (Use the Babylonian values of $C^2 / 12$ for the area of a circle and $7 / 4$ for $\sqrt{3}$.)

Jay Patel
Jay Patel
Numerade Educator
00:25

Problem 23

Various conjectures have been made for the derivation of the Egyptian formula $A=\left(\frac{8}{9} d\right)^2$ for the area $A$ of a circle of diameter $d$. One of these uses circular counters, known to have been used in ancient Egypt. Show by experiment using pennies, for example, whose diameter can be taken as 1. that a circle of diameter 9 can essentially be filled by 64 circles of diameter I. (Begin with one penny in the center: surround it with a circle of 6 pennies, and so on.) Use the obvious fact that 64 circles of diameter 1 also fill a square of side 8 to show how the Egyptians may have derived their formula. ${ }^{30}$

Ashley High
Ashley High
Numerade Educator
04:57

Problem 24

For the truncated pyramid from the Moscow Papyrus, compare the correct volume given in the text with the volume calculated by means of the incorrect Babylonian formula $V=\frac{1}{2}\left(a^2+b^2\right) h$. Find the percentage error. Do the same for a truncated pyramid of lower base 10 , upper base 8 . and height 2 .

Jennifer Stoner
Jennifer Stoner
Numerade Educator
00:38

Problem 25

In the Indian Sulvasutras, the priests gave the following procedure for finding a circle whose area was equal to a given square. In square $A B C D$, let $M$ be the intersection of the diagonals (Fig. 1.30). Draw the circle with $M$ as center and MA as radius: let ME be the radius of the circle perpendicular to the side $A D$ and cutting $A D$ in $G$. Let $G N=\frac{1}{3} G E$. Then $M N$ is the radius of the desired circle. Show that if $A B=s$ and $2 M N=d$, then $\frac{d}{4}=\frac{2+\sqrt{2}}{3}$. Show that this implies a value for $\pi$ equal to 3.0883 .

Anurag Kumar
Anurag Kumar
Numerade Educator
01:11

Problem 26

. Look up the 18.6-year cycle of mornrise positions in an astronomy text. Discuss its astronomical basis and how it can be used in predicting eclipses. Do you think it reasonable that the priests at Stonehenge could have made such predictions?

Mayukh Banik
Mayukh Banik
Numerade Educator

Problem 27

Look up details on the ancient Chinese calendar. How did it reconcile the lunar and solar cycles?

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01:44

Problem 28

Consult a reference on the details of the Jewish calendar. Make a short report on how the length of each year is determined.

Carson Merrill
Carson Merrill
Numerade Educator
01:27

Problem 29

Convert the Babylonian approximation 1:24.51.10 to $\sqrt{2}$ to decimals and determine the accuracy of the approximation.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:41

Problem 30

. Use the assumed Babylonian square root algorithm of the text to show that $\sqrt{3}=1: 45$ by beginning with the value 2. Find a three-sexagesimal-place approximation to the reciprocal of 1:45 and use it to calculate a three-sexagesimalplace approximation to $\sqrt{3}$.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
07:38

Problem 31

Show that 123152432 is a good approximation to $\sqrt{164}$. (This value appears in a late Greek-Egyptian papyrus,)

Andrew Sum
Andrew Sum
Numerade Educator
03:16

Problem 32

Use the Chinese square root algorithm (which is the same as the one taught currently) to derive a sequence of decimal approximations to $\sqrt{2}$ to five-place accuracy. Use $\sqrt{a^2 \pm b}=a \pm \frac{1}{2} b\left(\frac{1}{a}\right)$ to do the same. Which algorithm works best: that is, which produces five-place accuracy with the least amount of calculation?

Julie Silva
Julie Silva
Numerade Educator
01:11

Problem 33

Show that taking $1+u=1: 48(=14 / 5)$ leads to line 15 of Plimpton 322 and that taking $\mathrm{v}+u=2: 05(=21 / 12)$ leads to line 9. Find the values for $y+w$ that lead to lines 6 and 13 of that tablet.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
00:35

Problem 34

The scribe of Plimpton 322 did not use the value $v+$ $u=2: 18,14,24$ with its associated reciprocal $v-u=$ 0:26.02, 30 in his work on the tablet. Find the smallest Pythagorean triple associated with those values.

Ali Soave
Ali Soave
Numerade Educator
04:10

Problem 35

Solve problem 8 of Chapter 9 of the Jiuchang: The height of a wall is 10 chith. A pole of unknown length leans against the wall so that its top is even with the top of the wall. If the bottom of the pole is moved 1 chith farther from the wall. the pole will fall to the ground. What is the length of the pole?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:55

Problem 36

Prove that the construction given in the Sidvasutras for constructing a square equal to the difference of two squares is correct (Fig. 1.31): Let $A B C D$ be the larger square with side equal to $a$, and let $P Q R S$ be the smaller square with side equal to $b$. Cut off $A K=b$ from $A B$ and draw $K L$ perpendicular to $A K$ internecting $D C$ in $L$. With $K$ as center and radius $K L$. draw an are mecting $A D$ at $M$. Then the square on $A: M$ is the required square.

Grace Muhihu
Grace Muhihu
Numerade Educator
01:15

Problem 37

Solve the problem from the Old Babylonian tablet BM 13901: The sum of the areas of two squares is 1525. The side of the second square is $2 / 3$ of that of the first plus 5 . Find the sides of each square.

Julie Silva
Julie Silva
Numerade Educator
05:30

Problem 38

Solve the problem from the Berlin Papyrus. If the area of a square of 100 square cubits is equal to the sum of the areas of two smaller squares, and if the side of one is $\overline{2} \overline{4}(=3 / 4)$ times the side of the other. then find the sides of the two unknown squares.

Sushmit Acharya
Sushmit Acharya
Numerade Educator
01:52

Problem 39

. Solve problem 20 of Chapter 9 of the Jiu-/hang: A square walled city of unknown dimensions has four gates, one at the center of each side. A tree stands 20 pu from the north gate. One must walk 14 pu southward from the south gate and then turn west and walk 1775 pu before one can see the tree. What are the dimensions of the city?

- -
- -
Numerade Educator
01:57

Problem 40

Give a geometric argument to justify the Babylonian "quadratic formula" that solves the equation $x^2-b x=c$.

Dushyant Barot
Dushyant Barot
Numerade Educator
05:47

Problem 41

Consider the system of equations taken from an Old BabyIonian text:

$$
x=30 \quad x y-(x-y)^2=500
$$

Show that the substitution of the first equation into the second leads to a quadratic equation in $y$ that has two positive roots, a type the Babylonians did not address. Show that subtraction of the second equation from the square of the first gives the equation $(x-y)^2+30(x-y)=400$, a quadratic in $x-y$ that has only one positive root.

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
04:14

Problem 42

Solve the following Babylonian problem:

$$
x+y=5 \frac{5}{6} \quad \frac{x}{7}+\frac{y}{7}+\frac{x y}{7}=2
$$

by first multiplying the second equation by 7 and then subtracting off the first equation, thus reducing the system to a standard form.

M Hassan Anwar
M Hassan Anwar
Numerade Educator

Problem 43

Discuss the pros and cons of using a grouping system, a ciphered system, and a place-value system (all base 10) in terms of brevity of expression, amount of memorization required. and ease of arithmetic calculation.

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Problem 44

Compare a base 10 and a base 60 place-value system in terms of the criteria of problem 43 . Is there a base that is better than either of these two? Explain.

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Problem 45

Devise a lesson on place value using the Babylonian system and. in particular. using the multiplication table of Fig. 1.8.

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Problem 46

Devise a lesson teaching the quadratic formula using geometric arguments similar to the (assumed) Babylonian ones.

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02:08

Problem 47

Compare the standard division algorithm with the Babylonian method of using reciprocals in terms of ease of use. Determine which algorithm your calculator uses to perform division.

R M
R M
Numerade Educator

Problem 48

Devise a lesson that explains the basic idea of linearity using examples from Egyptian and Chinese sources.

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00:56

Problem 49

Devise a lesson to convince students of the correctness of the formula $A=\pi r^2$ for the area of a circle, where $\pi$ is defined as $\frac{C}{d}$. How would a value for $\pi$ be determined? How would you convince students that there is a constant of proportionality between the circumference and the diameter of every circle?

Rishi Kavikondala
Rishi Kavikondala
Numerade Educator
01:32

Problem 50

Why is it useful to have a calendar whose months are determined by the cycles of the moon? Do we lose anything today by not having such a calendar?

Rodger Claar
Rodger Claar
Numerade Educator
03:21

Problem 51

Devise a lesson teaching the Pythagorean theorem using material from Chinese sources.

Charles Carter
Charles Carter
Numerade Educator