Problem Set 4 Solutions
Math 4033, History of Mathematics
Spring 2007
Problems 4.5: 1, 4, 5; 5.3: 2, 3, 13, 14, 19; 5.4: 1, 2, 5
4.5.1 Verify the following statements of Archimedes. (We assume that modern volume and
surface area formulas are known.)
(b) T
Problem Set 1 Solutions
Math 4033, History of Mathematics
Spring 2007
Problems 2.3: 4, 5, 20; 2.4: 1c, 2, 9; 2.5: 3, 11; 2.6: 7, 8
1
1
2.3.4 (a) Show that the product of 14 by 1 + 1 + 4 is equal to 1 .
2
8
Solution: We form an Egyptian-style multiplicatio
Math 4033, History of Mathematics
Midterm Exam Solutions
March 8, 2007
1. (a) (20 points) Fill in the following table with the equivalent numbers from the various
number systems weve studied.
Egyptian
Babylonian
Mayan
Hindu-Arabic
LXXV
75
MCCLXXXI
(i)
Rom
Math 303 Final Exam Solutions
May 12, 2008
1. (10 pts) Give conditions under which the following equation would be true a + (b c) = (a + b) (a + c). This is just a matter of solving the equation. a + (b c) = (a + b) (a + c) a+bc=a+bac a+bc=bc a=0 Since al
Math 303 Exam 2 Solutions
Apr 9, 2008
1. (10 pts) Eight times any triangular number, plus 1, is a square number. Show that this is true for the rst four triangular numbers. 1(1 + 1) 2 2(2 + 1) 8T2 + 1 = 8 2 3(3 + 1) 8T3 + 1 = 8 2 4(4 + 1) 8T4 + 1 = 8 2 8T
Math 303 Exam 1 Solutions
Feb 27, 2008
1. (10 pts) Identify the largest number expressible in a simple grouping system which uses a base of 5 and 10 distinct symbols. Give your answer as a number in the Hindu-Arabic decimal system. Explain carefully why t
Math 105 History of Mathematics
Second Test Answers
Prof. D. Joyce, November, 2010
Scale.
85104 A. 7484 B. 5973 C. Median 79.
Problem 1.
B.
Here is the rst stage of the computation.
x3
1
Essay. [25] Select one of the two topics A or
20
1
Topic A. One qual
Math 105 History of Mathematics
First Test Answers
Prof. D. Joyce, October, 2010
Scale. 89101 A. 7888 B. 5877 C. Median 84.
arguments, and even in previously accepted statements. It can be used to nd hidden assumptions.
More importantly, adhering to a str
HISTORY OF MATH
HOMEWORK 7 SOLUTIONS
11.4,6,11
Exercise from Class: Prove the formula used to make Wallis rst table:
1
(1 x
q
) dx =
p+q
1
(1 x1/p )q1 dx,
1/p q
0
0
where p and q are positive integers.
Call the left hand integral Bq . Use integration by p
HISTORY OF MATH
HOMEWORK 6 SOLUTIONS
11.4,6,11
11.4. Find the equation of the locus of points P = (x, y ) whose squared distances to two given
points (a, 0) have a given sum m.
We have
[(x + a)2 + y 2 ] + [(x a)2 + y 2 ] = m,
which simplies to
1
x 2 + y 2
HISTORY OF MATH
HOMEWORK 5 SOLUTIONS
9.30, 32, 35, 37, 38
9.30. Use Cardanos formula to solve x3 = 6x + 6.
We have p = 2, q = 3, so q 2 + p3 = 1. The positive real solution is
x = [3 + 1]1/3 + [3 1]1/3 = 41/3 + 21/3 .
The general solution is
w+
2
,
w
wher
HISTORY OF MATH
HOMEWORK 4 SOLUTIONS
7.3,6,13,15,18
7.3. For the equation x2 = bx + c: Halve the number of roots (b/2), multiply this by itself
(b/2)2 ). Add this to the number (b/2)2 + c), extract the square root ( (b/2)2 + c) and and this
to half the nu
HISTORY OF MATH
HOMEWORK 3 SOLUTIONS
A. Using modern methods, prove Prop. 3 of Quadrature of the Parabola:
From two points Q, Q on the parabola, draw lines parallel to the tangent line to the vertex P
and let V, V be the intersections of these lines with
HISTORY OF MATH
HOMEWORK 2 SOLUTIONS
2.12,13,14,18
2.12 To apply to [construct on] a given straight line AB a rectangle equal to a given rectangle.
Lets also construct the diagram in the book, just for practice.
F
G
H
E
D
B
M
L
A
Let BEF G be the given re
Library Assignment #2: Periodical Literature
Provide research summaries of ten papers on the history of mathematics (both words are crucial)
that you have looked up and read. One purpose for doing this is that it will provide everyone with
lots of ideas f
Library Assignment #1: Your Mathematician
Your Mathematician is
0. Look up YM in our text. Read the relevant pages and take notes.
1. Look up YM in the Dictionary of Scientic Biography or the Biographical Dictionary of Mathematicians. Give the volume, pag
History of Mathematics
Homework 5 Solutions
1. Multiply 8743 by 5692 prosthaphaeretically.
We use the formula
1
cos a cos b = (cos(a + b) + cos(a b),
2
where cos a = 0.8743 and cos b = 0.5692.
find then that
a 0.506804540161645,
From the cosine tables we
History of Mathematics
Homework 4 Solutions
1. Solve the following problem of Abu Kamil: Suppose 10 is divided into two parts and the
product of one part by itself equals the product of the other part by the square root of 10.
Find the parts.
Let one of t
History of Mathematics
Homework 3
Due Wednesday, October 6
1. Beginning, as did Archimedes, with a regular hexagon inscribed in a circle, use an
Archimedean recursion algorithm to nd either p12 and P12 or a12 and A12 . What value
of would be implied by th
History of Mathematics
Homework 2 Solutions
1. Write the number 0.0862 in Babylonian notation.
Note that
0.0862 = 5 601 + 10 602 + 19 603 + 12 604 ,
so that 0.0862 = 0; 5, 10, 19, 12.
2. Verify that if (c/a)2 is 1; 33, 45 and b = 45 and c = 1, 15, then a,
History of Mathematics
Homework 1 Solutions
1. Express 2/103 as a sum of two unequal unit fractions, and write these in Egyptian
hieroglyphic notation.
Using the formula from the book, we have
2
1
1
=
+
103
52 5356
1
2. Solve by false position the equatio
Problem Set 7
Math 4033, History of Mathematics
Solutions
Problems: 8.1: 6, 9, 10; 8.2: 1, 7, 8, 10; 8.3: 1, 3; 8.4: 2, 8, 11
8.1.6 Use Napiers rods to multiply 458 by 79.
Solution: The relevant boxes in Napiers rods would look like this:
4
.
.
.
5
.
.
.
Problem Set 6
Math 4033, History of Mathematics
Solutions
Problems: 7.3: 1, 2, 3, 14; 7.4: 1, 4, either 5 or 6
7.3.1 Find all three roots of the following cubic equations. (Note: For these you are intended
to do a substitution to cancel the quadratic term
Problem Set 5
Math 4033, History of Mathematics
Solutions
Problems 5.5: 1, 4, 5, 6, 7, 14, 15; 6.2: 5, 7; 6.3: 2, 5
Plus do the following problems from the OSU High School Math Contest.
5.5.1 Solve the following quadratic equations by the Arabic technique
Problem Set 3 Solutions
Math 4033, History of Mathematics
Spring 2007
Problems 4.2: 7, 9, 12; 4.3: 1b and d, 4, 9, 10b, 16c-d, 20, 24, 26; 4.4: 1, 7
4.2.7 Prove that if two opposite sides of a parallelogram are
equal and parallel, then the other two sides
Problem Set 2 Solutions
Math 4033, History of Mathematics
Spring 2007
Problems 3.2: 7, 11, 12; 3.3: 2, 7, 8, 13, 17, 21; 3.4: 3, 7; 3.5: 5, 6
3.2.7 (4 points, (b) and (d) only) Prove the following identities about oblong numbers algebraically and geometri