Lapses in Mathematical Reasoning

, by Bradis, V. M.; Minkovskii, L.; Kharcheva, A. K.

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ISBN: 9780486409184 | 048640918X
Cover: Paperback
Copyright: 11/16/2016

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Unique method of teaching high school students mathematical reasoning skills leads students deliberately toward a false conclusion. Later, teacher and student review and and analyze the reasoning lapse to correct the problem. Covers arithmetic, algebra, geometry, trigonometry and approximate computations. 1963 edition. Includes 2 prefaces and 45 figures.

From the Preface to the First Edition

Preface to the Second Edition

xiii

Exercises in Refuting Erroneous Mathematical Arguments and Their Classification

Introduction

(1)

Mathematical Sophisms and their Pedagogical Value

(4)

Classification of Exercises in Refuting False Mathematical Arguments

(32)

Incorrectness of speech

(3)

Extension to exceptional cases

(1)

Ascribing properties of a particular form to the whole species

(1)

Incorrect applications of the principle of direct deductions by the converse

(4)

Replacement of precise definitions by geometric intuition

(3)

Errors of construction

(10)

Errors which are the consequence of a literal interpretation of the abbreviated conventional formulation of some geometric propositions

(2)

Violation of the sense of conventional notations

(1)

Deviation from the thesis

(4)

Arithmetic

Examples of False Arguments

(9)

The location of thirteen persons, one each, in twelve rooms

(1)

Eleven fingers on two normal hands

(1)

Square dollars

(1)

45 - 45 = 45

(1)

40 ÷ 8 = 41

(1)

Two times two is---five!

(1)

Is there a proportionality here?

(1)

100 per cent economy

(1)

How should a mean percentage be computed?

(1)

What will a yearly growth of 40 per cent give in five years?

(1)

A new rule for multiplying fractions

(1)

What happened to the dollar?

(1)

Where did the extra dime come from?

(1)

A father's will

(1)

2 X 3 = 4

(1)

Analysis of the Examples

(16)

Algebra

Examples of False Reasoning

(24)

Half a dollar is equal to five cents

(1)

6 = 2

(1)

12 = 6 = 0

(1)

Divisibility of polynomials and divisibility of numbers

(2)

An arbitrary number a is equal to zero

(1)

7 = 13

(1)

Positive unity is equal to negative unity

(1)

Another ``proof'' of the equality of the positive and negative unities

(1)

The imaginary unit and the negative real unit are equal

(1)

i2 = 1

(1)

Every negative number is greater than the positive number having the same absolute value

(1)

If a > b, then a > 2b

(1)

If a and b are positive numbers, then a > b and b > a

(1)

A positive number is less than zero

(1)

The sum of natural numbers, each of which exceeds unity, is greater than their product

(1)

What is the square root of the number a2?

(1)

Another ``proof'' that an arbitrary number is equal to zero

(1)

A number does not change if in it arbitrary digits are transposed

(1)

What is the meaning of the theorem on the existence of a root in the algebra of complex numbers?

(1)

On a method of obtaining correct results, the application of which requires considerable caution

(3)

About the sum 1 -- 1 + 1 -- 1 +

(1)

Is the whole always greater than its part?

(1)

Another ``proof'' of the equality of two arbitrary numbers

(1)

The sum of two arbitrary equal numbers is equal to zero

(1)

A number does not change if 1 is added to it

(1)

Achilles and the tortoise

(1)

On some student's errors

(4)

Analysis of the Examples

(36)

Geometry

Examples of False Arguments

123

(22)

Segments of parallel straight lines bounded by the sides of an angle are equal

123

(1)

Segment of a straight line is equal to a proper part of itself

124

(1)

All triangles are of equal area

125

(1)

The sum of the bases of an arbitrary trapezoid is equal to zero

126

(2)

Inscribed and circumscribed

128

(1)

More about proportionality

129

(1)

The circumferences of two circles of different radii have one and the same length

130

(1)

The sum of the arms is equal to the hypotenuse

131

(1)

The length of a semicircle is equal to its diameter

132

(1)

The lateral surface of a right circular cone with base radius r and altitude h is expressed by the formula P = πr(r + h)

133

(3)

At a given point on a straight line two perpendiculars can be constructed to that line

136

(1)

Through one point it is possible to draw two straight lines parallel to a given straight line

136

(1)

A circle has two centres

137

(1)

Two perpendiculars may be dropped from a point to a straight line

138

(1)

Two straight lines may be drawn through two given points

139

(1)

Every triangle is isosceles

140

(1)

The arm of a right-angled triangle is equal to its hypotenuse

141

(1)

A right-angle is equal to an obtuse angle (planar version)

142

(1)

64 cm2 = 65 cm2

143

(1)

A problem of patching

144

(1)

Analysis of the Examples

145

(17)

Stories, with Explanations, of Causes of Erroneous Reasoning

162

(15)

Similar triangles with equal sides

162

(2)

Trisection of an angle

164

(4)

More about the trisection of an angle

168

(2)

Quadrature of the circle

170

(2)

On a proof of the theorem on the sum of the interior angles of a triangle

172

(2)

How to compute the volume of a truncated pyramid

174

(3)

Trigonometry

Examples of False Arguments

177

(7)

π/ 4 = 0

177

(1)

The sine of an angle decreases if one adds 360° to that angle

178

(1)

The cosine of any acute angle is greater than unity

179

(1)

1/4 > 1/2

179

(1)

22 = 42

180

(1)

The area of a rectangle is equal to zero

181

(1)

There exist congruent triangles, not all of whose sides are respectively equal

182

(1)

Every triangle is a right-angled triangle

183

(1)

Analysis of the Examples

184

(5)

Approximate Computations

Accounts explaining Sources of Erroneous Reasoning

189

(1)

How old is the ancient statue?

189

(1)

All large numbers are approximately equal to each other

190

(2)

On the precision of the product of approximate numbers

192

(3)

Is the formula sin α / cos α = tan α true?

195

(1)

How many digits should be known in a number under a radical sign in order to obtain the root with a given accuracy?

196

(2)

Why does one get rid of irrationals in the denominator?

198

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Lapses in Mathematical Reasoning

Summary

Table of Contents

Supplemental Materials