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Numbers & Geometry
Constructible Lengths
And
Irrational Numbers
The tools the ancient Greeks used to do mathematics was the compass and straightedge.
They were not only tools used to make new discoveries but it also served as their means
of doing computation (i.e. the calculator of the day).
Compass and Straightedge
The “tools” we use to copy parts of a triangle are a
compass and a straightedge.
A compass is a device used to draw circles or parts
of circles called arcs.
A straightedge is like a ruler but with no markings
on it. A ruler or yard stick is often used but you
must ignore the markings.
Copying a segment
The compass and straightedge can be used together to
transfer a segment of a given length onto a line. This is
done in two steps:
B
A
C
1. Put point on A and open till mark is on B
2. Lift off and put point on C and mark point D
D
Side-Side-Side (SSS) Triangle Congruence
If three sides of a triangle are congruent to the three corresponding sides of another
triangle, then the two triangles are congruent. We show this by showing how segments
from one triangle can be translated (copied by a compass and straightedge) to form the
other triangle.
A
B
D
C
E
Steps to copy a triangle by coping the sides:
1. Copy segment 𝐵𝐶 to locate point F
2. Make arc of length 𝐴𝐶 with point at F
3. Copy segment 𝐴𝐵 to locate point D on the arc from step (2)
4. Use straightedge to fill in segment 𝐷𝐹
Now we have, ABC  DEF
F
Copying an Angle
Copying an angle can be accomplished by copying a triangle that is included in that angle.
A
B
C
Steps to copy an angle:
1. Swing arc on the original angle (ABC) and without changing it make same arc on the
other ray you want to copy it onto.
2. Make arc from where the arc in step (1) passed through the original angle and
transfer it to the ray you want to copy it onto.
3. With your straightedge draw the line that connect the endpoint and where the arcs
cross.
Side-Angle-Side (SAS) Triangle Congruence
If two sides and the included angle of one triangle are congruent to two corresponding sides and the
included angle of another triangle with the corresponding sides being congruent, then the triangles
are congruent.
The included angle of two sides of a triangle is the angle that is formed by the two sides
of the triangle. It can not just be any two congruent sides and an angle, but the angle
that is between the two sides.
Below we show how to use a compass and straightedge to copy the side-angle-side of a
triangle.
A
Steps to copy a triangle by copying a side-angle-side:
1. Copy ABC with vertex at point E.
B
C
2. Use straightedge to draw in 𝐸𝐷
3. Copy 𝐴𝐵 onto 𝐸𝐷
D
4. Copy 𝐵𝐶 onto 𝐸𝐹
E
5. Draw segment 𝐷𝐹
F
6. ABC  DEF
A compass and straightedge
can be used to construct
both angle bisectors and
perpendicular bisectors of
segments.
Angle Bisector
Perpendicular Bisector
Base Angles of Isosceles Triangles are Congruent
A
In an isosceles triangle the angles made with the noncongruent side and one of the congruent sides are called
the base angles. In the triangle to the right ABD and
ACD are the base angles. The base angles are congruent.
The reason for this is as follows:
1. Construct angle bisector for CAB and call the point of
intersection with 𝐵𝐶 point D.
2. BAD  CAD (Side-Angle-Side)
3. ABD  ACD (They are the corresponding parts of the
congruent triangles.)
B
D
C
Number Representation as Lengths
At this point in history numbers where thought of as having two
components. These were referred to as the “whole” and the
“part”. It is sort of like how we think of mixed numbers today.
whole
1
The arithmetic of the day was carried out with a compass
and straightedge. Once a unit length was established you
could add, subtract and divide two whole values.
b
a
2
5
3
part
a
b
1
b
2
3
a+b
a
a-b
1
1
1
Eventually people worked out the idea of a “common unit”, today what we call a common
denominator so that mixed numbers could be added without needing the direct compass
construction.
A MATHEMATICAL CRISIS : WHAT IS 𝟐?
To the Greeks rational numbers (lengths) could be
understood and constructed using a compass and
straight edge. The thinking of the time was that any
length would have a rational representation. The
diagonal of a unit square can clearly be constructed.
Experimentation with the compass and straightedge
suggested that it was the ratio between the whole
numbers 7 and 5, but this would prove to be
incorrect as computation understanding of rational
numbers improved. This was well within the
experimental accuracy of the compass and
straightedge.
2 = 1.414213562 ⋯
7
= 1.4
5
2
1
2=
2=
7
5
49
25
2 ∙ 25 = 49
50 = 49
A NEW TYPE OF NUMBER: IRRATIONAL NUMBERS
The idea that a number (length) was not
the ratio of two whole number lengths
was very outlandish to the Greek
understanding of numbers. In fact people
were put to death first over suggesting
this then later after it was established
revealing the “proof” of it.
Fundamental Theorem of Arithmetic.
If n is a whole number then n can be factored
uniquely into a product of prime numbers.
𝑒
𝑒
𝑛 = 𝑝11 ∙ 𝑝2 ∙ ⋯ ∙ 𝑝𝑘𝑘
Proof that 2 is not the ratio of two whole numbers. (i.e. 2 is irrational)
Proof:
𝑎
Assume for the purpose of contradiction it is in other words 2 = 𝑏, 𝑎, 𝑏 whole numbers.
Squaring both sides and multiplying by 𝑏2 we get that : 2𝑏2 = 𝑎2
On the right side, the number of factors of the prime 2 in the factorization of 𝑎2 is a
multiple of 2.
On the left side, the number of factors of the prime 2 in the factorization of 2𝑏2 is one more
than a multiple of 2.
Since the number of factors of the prime 2 in the factorization must be the same on each
side must be the same because the number factors into primes in only one way this is a
𝑎
contradiction. Therefore, 2 ≠ 𝑏 where 𝑎, 𝑏 are whole numbers.
∎
CALCULATIONS WITH IRRATIONAL NUMBERS
How do you accomplish the basic arithmetic operations of adding subtracting multiplying
and dividing if a number (length) can not be represented as a “whole” and a “part”? The
answer is the same as before when it came to addition and subtraction. Something new was
needed for multiplication and division.
𝜃
Central angle congruence with a subtended cord (i.e. 2 = 𝜙)
If three points are chosen on a circle. Central angles are
constructed with the corresponding cords.
1. 𝜃 = 180 − 2𝑥 and 𝜏 = 180 − 2𝛼 (isosceles Δ’s)
2. 𝛾 + 𝛿 = 2𝛿 = 180 − 𝜃 + 𝜏 = 2𝑥 + 2𝛼 − 180 (algebra)
3. 𝛿 = 𝑥 + 𝛼 − 90 (divide by 2)
4. 𝜙 + 𝛿 = 𝜙 + 𝑥 + 𝛼 − 90 = 𝛼 (substitute & base angles)
𝑥 𝛼
𝜙
𝛿
𝛽
𝛾
𝜃
𝜏
O
𝜃
5. 𝜙 = 90 − 𝑥 = 2
This says that the measure of angle 𝜙 depends only on the
angle made by the cord and the center of the circle regardless
of where 𝜙 is chosen.
𝜃
2
𝜃
2
Since both 𝜙1 = and 𝜙2 = we get 𝜙1 = 𝜙2 .
Angles that subtend the same cord of a circle are congruent.
𝜙1
𝜙2
𝜃
O
Consider two cords of a circle that intersect in the interior
of the circle.
1. The two angles labeled 𝛼 are congruent because they
subtend the same arc of the circle.
2. The two angles labeled 𝛽 are congruent because they
form vertical angles.
3. The two angles labeled 𝛾 are congruent because all
angles of a triangle sum to 180. Both angles will
measure 180 − 𝛼 − 𝛽.
𝛼
𝛾
𝛽
𝛾
This means that the two triangles are similar.
The sides of the triangles are proportional:
𝑎 𝑐
=
𝑑 𝑏
Or equivalently,
𝛼
𝛽
c
b
a
d
𝑎𝑑 = 𝑏𝑐
The Greeks used this fact to multiply, divide and take the square root of any length.
If you know two cords of a circle the center of the
circle can be found by constructing the perpendicular
bisectors of two cords. The point of intersection will be
the center. The distance from the center to the end
point of one of the cords will be the radius.
1
Again if we are given the unit length and two other we can
arrange them as cords of a circle to multiply, divide and take
the square root.
a
𝑎𝑏
b
𝑎
𝑏
1
b
a
1
a
𝑎
a
b
𝑎
1