The Mechanics of Constructable Numbers
For Greek mathematicians, magnitudes were represented by the lengths of straight lines,
a fact which explains why 0 was not considered a number and why there were no
negative numbers! Starting with one or more magnitudes, any magnitude generated using
only straight-edge and compass is constructable. Therefore the five arithmetic operations
of addition, subtraction, multiplication, division and square roots are all constructable
operations in that is it possible to construct the sum, difference, product, quotient and
square roots of given magnitudes using only straight-edge and compass.
Addition: Let AB and AC be two line segments (magnitudes) sharing a common point. If
the two line segments which don’t share a common point, Euclid’s Proposition I.2
showed how to construct two equivalent line segments that did. Suppose we wish to
construct a line segment CD equal to their sum; i.e. CD = AB + AC. To do this construct
a circle with center A and radius AB and extend line segment CA to intersect the circle at
D. Then CD = AB + AC. Thus addition!
D
A
E
C
B
2.
Subtraction: EC = AC – AB is easily obtained from the above diagram.
3.
Multiplication: To construct AB × AC, assume AB is perpendicular to AC. On
the line determined by AC let AD be equal to length 1 and construct the right triangle
ADB. Construct a second right triangle ACE similar to ADB. Using similar triangles
AE
= AC
but since AD = 1 it follows that AE = AB × AC.
AB
AD
E
B
B
E
A
D
multiplication
C
A
D
C
division
4.
Division: Constructing AE = AB/AC by using a diagram similar to the one for
multiplication.
5.
Square roots: Given any line show it is possible to construct a second line whose
length is the square root of the first; that is given the length AB construct the
length AB . Begin by constructing a semi-circle with base AC = AB + BC where BC =
1. Let D be a point on the semi-circle such that BD is perpendicular to AC. Thus right
triangle ABD is similar to right triangle DBC so AB
= BD
. Since BC = 1 by
BD
BC
construction, AB = BD 2 so BD = AB .
D
A
B
C
Put this all together. Start with a single magnitude and call it length 1. Describe what
kinds of numbers are constructable.
A Challenge: Construct an isosceles trapezoid whose sides are in the ratio of 1:1:1: 3 .
Use the fact that
3 −1
is constructable!
2
OnConstructableNumbers.doc 09/04/2008