A History of Mathematics: An Introduction

Victor J.Katz

Chapter 19 Algebra and Number Theory in the Eighteenth Century - all with Video Answers

Educators

Chapter Questions

Problem 1

Of three workmen, $A$ can finish a given job once in three weeks, $B$ can finish it three times in eight weeks, while $C$ can finish it five times in twelve weeks. How long will it take for the three workmen to complete the job together? (This exercise and the next two are from Newton's Universal Arithmetic.)

Nick Johnson

Numerade Educator

01:07

Problem 2

If 12 cattle eat up $31 / 3$ acres of meadow in 4 weeks and 21 cattle eat up 10 acres of exactly similar meadow in 9 weeks, how many cattle shall eat up 36 acres in 18 weeks? (Hint: The grass continues to grow.)

Patrick Burns

Numerade Educator

01:12

Problem 3

Given the perimeter $a$ and the area $b^{2}$ of a right triangle, find its hypotenuse.

Jonathon Brumley

Numerade Educator

02:31

Problem 4

Suppose that the distance between London and Edinburgh is 360 miles and that a courier for London sets out from the Scottish city running at $10 \mathrm{mph}$ at the same time that one sets out from the English capital for Edinburgh at $8 \mathrm{mph}$. Where will the couriers meet? (This exercise and the next two are from Maclaurin's Treatise of Algebra.)

Stephen Zaffke

Numerade Educator

00:53

Problem 5

Derive Cramer's rule for three equations in three unknowns from the rule for two equations in two unknowns: Given the system
$$
\begin{aligned}
&a x+b y+c z=m \\
&d x+e y+f z=n \\
&g x+h y+k z=p
\end{aligned}
$$
solve each equation for $x$ in terms of $y$ and $z$, then form two equations in those variables and solve for $z$. Finally, determine $y$ and $x$ by substitution.

Wendi Zhao

Numerade Educator

04:36

Problem 6

A company dining together find that the bill amounts to \$175. Two were not allowed to pay. The rest found that their shares amounted to $\$ 10$ per person more than if all had paid. How many were in the company?

Kelly Hughes

Numerade Educator

01:25

Problem 7

How can one reconcile the algebraic rule $\sqrt{a} \sqrt{b}=\sqrt{a b}$ with the computation $\sqrt{-1} \sqrt{-4}=-2 \neq \sqrt{(-1)(-4)}$ ? Why do you think Euler erred in his discussion of this matter?

Amy Jiang

Numerade Educator

02:36

Problem 8

Twenty persons, men and women, dine at a tavern. The share of the bill for one man is $\$ 8$, for one woman $\$ 7$, and the entire bill amounts to $\$ 145$. Required, the number of men and women separately. (This exercise and the next two are from Euler's Introduction to Algebra.)

Jake Zanazzi

Numerade Educator

04:38

Problem 9

A horse dealer bought a horse for a certain number of crowns and sold it again for 119 crowns, by which means his profit was as much per cent as the horse cost him. What was the purchase price?

Ruby P

Numerade Educator

02:01

Problem 10

Three brothers bought a vineyard for $\$ 100$. The youngest says that he could pay for it alone if the second gave him half the money which he had; the second says that if the eldest would give him only a third of his money, he could pay for the vineyard singly; lastly, the eldest asks only a fourth part of the money of the youngest, to pay for the vineyard himself. How much money did each have?

Nick Johnson

Numerade Educator

00:10

Problem 11

Factor Leibniz's polynomial $x^{4}+a^{4}$ into two real quadratic polynomials. ( Hint: Add and subtract $2 a^{2} x^{2}$.)

Joseph Liao

Numerade Educator

01:05

Problem 12

Factor $x^{5}-1$ into linear and real quadratic factors.

John Irizar

Numerade Educator

02:37

Problem 13

Nicolaus Bernoulli claimed that the polynoma1 $x^{2}-4 x=$ $2 x^{2}+4 x+4$ could not be factored into a product of two quadratic polynomials. Show that in fact the factors are
$$
\left(x^{2}-(2+\sqrt{4+2 \sqrt{7}}) x+(1+\sqrt{4+2 \sqrt{7}}+\sqrt{7})\right)
$$
and
$$
\left(x^{2}-(2-\sqrt{4+2 \sqrt{7}}) x+(1-\sqrt{4+2 \sqrt{7}}+\sqrt{7})\right)
$$.

Abhijith V

Numerade Educator

01:44

Problem 14

Use Euler's procedure from his proof that all real quartics factor to determine the factorization of $x^{4}-2 x^{2}+8 x-3$ as a product of two quadratic polynomials.

Joshua Kinney

Numerade Educator

03:57

Problem 15

Find a cubic curve and a quadratic curve that intersect in six real points.

Clarissa Noh

Numerade Educator

01:37

Problem 16

Consider the following system of linear equations given by Euler:
$$
\begin{array}{r}
5 x+7 y-4 z+3 v-24=0 \\
2 x-3 y+5 z-6 v-20=0 \\
x+13 y-14 z+15 v+16=0 \\
3 x+10 y-9 z+9 v-4=0
\end{array}
$$
Show that these four equations are "worth only two," so that they do not determine a unique 4 -tuple as a solution.

Elisa Ma

Numerade Educator

07:15

Problem 17

Show that if $n$ is prime, then the roots of $x^{n}-1=0$ can all be expressed as powers of any such root $\alpha \neq 1$.

Wasim Sher

Numerade Educator

01:45

Problem 18

Let $x_{1}, x_{2}$ be the two roots of the quadratic equation $x^{2}+$ $b x+c=0$. Since $t=x_{1}+x_{2}$ is invariant under the two permutations of the two roots, while $v=x_{1}-x_{2}$ takes on two distinct values, $v$ must satisfy an equation of degree 2 in $t$. Find the equation. Similarly, $x_{1}$ is invariant under the same permutations as $x_{1}-x_{2}$. Thus, $x_{1}$ can be expressed rationally in terms of $x_{1}-x_{2}$. Find such a rational expression. Use the rational expression and equation you found to "solve" the original quadratic equation.

Cory Kuzinski

Numerade Educator

04:35

Problem 19

Determine the three roots $x_{1}, x_{2}, x_{3}$ of $x^{3}-6 x-9=0$ Use Lagrange's procedure to find the sixth-degree equation satisfied by $y$, where $x=y+2 / y$. Determine all six solutions of this equation and express each explicitly as $\frac{1}{3}\left(x^{\prime}+\omega x^{\prime \prime}+\omega^{2} x^{\prime \prime \prime}\right)$, where $\left(x^{\prime}, x^{\prime \prime}, x^{\prime \prime \prime}\right)$ is a permutation of $\left(x_{1}, x_{2}, x_{3}\right)$ and $\omega$ is a complex root of $x^{3}-1=0$.

Elisa Ma

Numerade Educator

08:45

Problem 20

Show that the expression $x_{1} x_{2}+x_{3} x_{4}$ takes on only three distinct values under the 24 permutations of four elements.

Anas Venkitta

Numerade Educator

02:10

Problem 21

Let $x_{1}, x_{2}, x_{3}, x_{4}$ denote the four roots of the quartic equation $x^{4}+a x^{3}+b x^{2}+c x+d=0$. Set $\alpha=x_{1} x_{2}+x_{3} x_{4}$, $\beta=x_{1} x_{3}+x_{2} x_{4}, \gamma=x_{1} x_{4}+x_{2} x_{3} .$ Show that $\alpha+\beta+$ $\gamma=b$, that $\alpha \beta+\alpha \gamma+\beta \gamma=a c-4 d$, and that $\alpha \beta \gamma=$ $a^{2} d+c^{2}-4 b d$. Show that this implies that $\alpha, \beta$, and $\gamma$ are the roots of the cubic equation $y^{3}-b y^{2}+(a c-4 d) y-$ $\left(a^{2} d+c^{2}-4 b d\right)=0$.

Varsha Aggarwal

Numerade Educator

06:23

Problem 22

Use the results of Exercise 21 to determine the reduced equation for the quartic equation $x^{4}-12 x+3=0$. Solve this reduced equation, a cubic. Use the values you obtain for $\alpha, \beta$, and $\gamma$ to solve the original quartic equation.

Morgan Cheatham

Numerade Educator

04:29

Problem 23

In Euler's proof of the case $n=3$ of Fermat's Last Theorem, show that $\frac{1}{4} p$ and $p^{2}+3 q^{2}$ are relatively prime if $p$ is not divisible by 3 . (Recall that $x=p+q, y=p-q$ are odd and relatively prime.)

Trang Hoang

Numerade Educator

09:17

Problem 24

In Euler's proof of the case $n=3$ of Fermat's Last Theorem, we now consider the situation where $p=3 r$. We then know that $\frac{3}{4} r\left(9 r^{2}+3 q^{2}\right)=\frac{9}{4} r\left(3 r^{2}+q^{2}\right)$ must be a cube. Show that the two factors in this expression are relatively prime. It follows that each must be a cube. In particular, $q^{2}+3 r^{2}$ must be a cube. Factor this expression as in the text, using complex numbers of the form $a+b \sqrt{-3}$, and conclude that $q=t\left(t^{2}-9 u^{2}\right), r=3 u\left(t^{2}-u^{2}\right)$, where $t$ is odd and $u$ is even. Also, since $\frac{9}{4} r$ is a cube, show that $\frac{2}{3} r=2 u(t+$ u) $(t-u)$ is a cube where the factors are relatively prime. Conclude as in the case detailed in the text that we can now
find three integers smaller than the original set for which the sum of their cubes is a cube.

P Krishnamurthy

Numerade Educator

01:00

Problem 25

Calculate the distinct residues $1, \alpha, \beta, \ldots$ of $1,5,5^{2}, \ldots$ modulo 13 . Then pick a nonresidue $x$ of the sequence of powers and determine the coset $x, x \alpha, x \beta, \ldots$ Continue to pick nonresidues and determine the cosets until you have divided the group of all 12 nonzero residues modulo 13 into nonoverlapping subsets, the cosets of the group of powers of $5 .$

Raj Bala

Numerade Educator

01:46

Problem 26

Determine the quadratic residues modulo 13 .

James Chok

Numerade Educator

01:45

Problem 27

Prove that $-1$ is a quadratic residue with respect to a prime $q$ if and only if $q \equiv 1(\bmod 4)$.

James Chok

Numerade Educator

03:16

Problem 28

Benjamin Banneker was fond of solving mathematical puzzles and recorded many in his notebook, including his own version of the old hundred fowls problem: A gentleman sent his servant with $£ 100$ to buy 100 cattle, with orders to give $£ 5$ for each bullock, 20 shillings for each cow, and 1 shilling for each sheep. (Recall that 20 shillings equals $£ 1$.) What number of each sort of animal did he bring back to his master? $?^{23}$

Anas Venkitta

Numerade Educator

01:00

Problem 29

Divide 60 into four parts such that the first increased by 4, the second decreased by 4, the third multiplied by 4, and the fourth divided by 4 shall each equal the same number (Banneker).

Brandon Fox

Numerade Educator

04:06

Problem 30

Suppose a ladder 60 feet long is placed in a street so as to reach a window on one side 37 feet high, and without moving it at the bottom, to reach a window on the other side 23 feet high. How wide is the street? (Banneker))