Numbers & Geometry

Constructible LengthsAnd

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:

1. Put point on A and open till mark is on B

2. Lift off and put point on C and mark point D

A

B

C

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.

B

A

C E

Steps to copy a triangle by coping the sides:

1. Copy segment to locate point F

2. Make arc of length with point at F

3. Copy segment to locate point D on the arc from step (2)

4. Use straightedge to fill in segment

Now we have, ABC DEF

D

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.

B

A

C

D

EF

Steps to copy a triangle by copying a side-angle-side:

1. Copy ABC with vertex at point E.

2. Use straightedge to draw in

3. Copy onto

4. Copy onto

5. Draw segment

6. ABC DEF

D

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

In an isosceles triangle the angles made with the non-congruent 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 point D.

2. BAD CAD (Side-Angle-Side)

3. ABD ACD (They are the corresponding parts of the congruent triangles.)

B C

A

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. 5

23

whole part

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.

a

b

1

ab

a + b

b

a a-b

1

1 1 1

23

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 ?

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.

√21

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.

A NEW TYPE OF NUMBER: IRRATIONAL NUMBERSThe 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.

Proof that is not the ratio of two whole numbers. (i.e. is irrational)

Proof: Assume for the purpose of contradiction it is in other words , whole numbers.Squaring both sides and multiplying by we get that : On the right side, the number of factors of the prime 2 in the factorization of is a multiple of 2.On the left side, the number of factors of the prime 2 in the factorization of 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 contradiction. Therefore, where are whole numbers.

CALCULATIONS WITH IRRATIONAL NUMBERSHow 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.

 

 

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𝑥Central angle congruence with a subtended cord (i.e. )

If three points are chosen on a circle. Central angles are constructed with the corresponding cords.1. and (isosceles Δ’s)2. (algebra)3. (divide by 2)4. (substitute & base angles)

This says that the measure of angle depends only on the angle made by the cord and the center of the circle regardless of where is chosen.

Since both and we get .

Angles that subtend the same cord of a circle are congruent.

 

 

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Consider two cords of a circle that intersect in the interior of the circle.1. The two angles labeled are congruent because they

subtend the same arc of the circle.2. The two angles labeled are congruent because they

form vertical angles.3. The two angles labeled are congruent because all

angles of a triangle sum to 180. Both angles will measure .

This means that the two triangles are similar.

The sides of the triangles are proportional:

Or equivalently,

c

a

b

d

The Greeks used this fact to multiply, divide and take the square root of any length.

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.

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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.