Constructible Lengths And Irrational Numbers 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: 1. Put point on A and open till mark is on B 2. Lift off and put point on C and mark point D B A 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. A D F B 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

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. 2. Use straightedge to draw in 𝐸𝐷 3. Copy 𝐴𝐵 onto 𝐸𝐷 4. Copy 𝐵𝐶 onto 𝐸𝐹 5. Draw segment 𝐷𝐹 6. ABC  DEF B C D E F

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 𝐵𝐶 point D. 2. BAD  CAD (Side-Angle-Side) 3. ABD  ACD (They are the corresponding parts of the congruent triangles.) A 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 part 5 2 3 1 a 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 1 b a b 2 3 a a-b 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. 2 1 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 5 =1.4 2 = 7 5 2= 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. 𝑛= 𝑝 1 𝑒 1 ∙ 𝑝 2 ∙⋯∙ 𝑝 𝑘 𝑒 𝑘 Proof that 2 is not the ratio of two whole numbers. (i.e. 2 is irrational) Proof: Assume for the purpose of contradiction it is in other words 2 = 𝑎 𝑏 , 𝑎,𝑏 whole numbers. Squaring both sides and multiplying by 𝑏 2 we get that : 2 𝑏 2 = 𝑎 2 On the right side, the number of factors of the prime 2 in the factorization of 𝑎 2 is a multiple of 2. On the left side, the number of factors of the prime 2 in the factorization of 2 𝑏 2 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, 2 ≠ 𝑎 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. Central angle congruence with a subtended cord (i.e. 𝜃 2 =𝜙) If three points are chosen on a circle. Central angles are constructed with the corresponding cords. 𝜃=180−2𝑥 and 𝜏=180−2𝛼 (isosceles Δ’s) 𝛾+𝛿=2𝛿=180− 𝜃+𝜏 =2𝑥+2𝛼−180 (algebra) 𝛿=𝑥+𝛼−90 (divide by 2) 𝜙+𝛿=𝜙+ 𝑥+𝛼−90 =𝛼 (substitute & base angles) 𝜙=90−𝑥= 𝜃 2   𝛼 𝛽 𝛾 𝛿 𝜃 𝜏 𝜙 O 𝑥   𝜙 1 O 𝜃 𝜙 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. Since both 𝜙 1 = 𝜃 2 and 𝜙 2 = 𝜃 2 we get 𝜙 1 = 𝜙 2 . Angles that subtend the same cord of a circle are congruent.

This means that the two triangles are similar. Consider two cords of a circle that intersect in the interior of the circle. The two angles labeled 𝛼 are congruent because they subtend the same arc of the circle. The two angles labeled 𝛽 are congruent because they form vertical angles. The two angles labeled 𝛾 are congruent because all angles of a triangle sum to 180. Both angles will measure 180−𝛼−𝛽. 𝛼 𝛼 𝛽 𝛾 𝛽 𝛾 c a b d This means that the two triangles are similar. The sides of the triangles are proportional: 𝑎 𝑑 = 𝑐 𝑏 Or equivalently, 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 a 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. b 1 𝑎 𝑏 b a a 1 𝑎 a b 1 𝑎𝑏 𝑎