Construct √an-b" using the ruler and compass, where a > b> 0 in an ordered field F (assuming you have two points (0, 0) and (1,0), as always.)
Construct √an-b" using the ruler and compass, where a > b> 0 in an ordered field F (assuming you have two points (0, 0) and (1,0), as always.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
[Classical Geometry] How do you solve #1? The second picture is a hint (you don't need to solve the bullet points in the hint, just the asked question in the list of seven)
![1. Construct √an – bn using the ruler and compass, where a > b> 0 in an ordered field F
(assuming you have two points (0, 0) and (1,0), as always.)
2. Show that Q√3 = {a +b√3|a, b € Q} is a field by verifying the field axioms one by one.
3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a ¹.
4. Let II be an ordered field.
(a). Explain what does 3 mean in F.
(b). Give an example of ordered field F where 3 does not have a square root in F
(c). Show that there is an equilateral triangle in II if and only if √3 € F.
5. Show the congruence axiom C3 for IIF (don't assume the field is Pythagorean)
6. Can we discuss incidence in IIF31? Can we discuss betweenness in IIF3₁1? Explain your
answer.
7. Consider the vector space F³ = {(x, y, z)|x, y, z € F}, where F is field.
(a). let ~ be a relation on F³ \ {0, 0, 0}, defined by
a = (x1, y₁, 21) ~ B = (x2, Y2, 22)
if there exists some > € K s.t. Aa= B. Show that A is an equivalence relation.
(b). Describe the equivalent classes [a] in F3.
(c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We
interpret the primitive term as
• point: [a], a € F³.
● line: {[a] a = (x, y, z), ax + by + cz = 0} for some fixed a, b, c = F.
(d). show that the line is well defined: if ax+by+cz = 0 holds for some a = = (x, y, z),
then it holds for any B = (x1, 9₁, 2₁) s.t. B € [a].
(e). How many points are there in F₂P²? List them.
(f). How many lines are there in F₂P2? List them.
(g). Show that F₂P2 is isomorphic to the Fano plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbffe82f9-2d9a-4748-a8e9-659533c98536%2F7a9b993b-7aec-4f55-a5f4-551e2d675a37%2F9ovd8xn_processed.png&w=3840&q=75)
Transcribed Image Text:1. Construct √an – bn using the ruler and compass, where a > b> 0 in an ordered field F
(assuming you have two points (0, 0) and (1,0), as always.)
2. Show that Q√3 = {a +b√3|a, b € Q} is a field by verifying the field axioms one by one.
3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a ¹.
4. Let II be an ordered field.
(a). Explain what does 3 mean in F.
(b). Give an example of ordered field F where 3 does not have a square root in F
(c). Show that there is an equilateral triangle in II if and only if √3 € F.
5. Show the congruence axiom C3 for IIF (don't assume the field is Pythagorean)
6. Can we discuss incidence in IIF31? Can we discuss betweenness in IIF3₁1? Explain your
answer.
7. Consider the vector space F³ = {(x, y, z)|x, y, z € F}, where F is field.
(a). let ~ be a relation on F³ \ {0, 0, 0}, defined by
a = (x1, y₁, 21) ~ B = (x2, Y2, 22)
if there exists some > € K s.t. Aa= B. Show that A is an equivalence relation.
(b). Describe the equivalent classes [a] in F3.
(c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We
interpret the primitive term as
• point: [a], a € F³.
● line: {[a] a = (x, y, z), ax + by + cz = 0} for some fixed a, b, c = F.
(d). show that the line is well defined: if ax+by+cz = 0 holds for some a = = (x, y, z),
then it holds for any B = (x1, 9₁, 2₁) s.t. B € [a].
(e). How many points are there in F₂P²? List them.
(f). How many lines are there in F₂P2? List them.
(g). Show that F₂P2 is isomorphic to the Fano plane.
Transcribed Image Text:Q1: First, you should show that a > b>0 implies that a" - b" > 0, so that the sqrt makes sense. Remember that you are dealing with the field R so all elementary inequalities and
ruler-compass constructions work here. Now given a and b
• How do you construct a"?
• How do you construct b"?
with this in hand how do you construct a" - b?
• Recall for such a number you can construct the square root as well, following the Descartes theorem.
Theorem 3.1. Descartes
Suppose we are given points P; = (a¿, b;), 1 ≤ i ≤ n in the real Cartesian plane, where
Po = (0,0) and P₁ = (1,0). Then it is possible to construct a point Q = (a, ß) € R² via
the ruler and compass iffa and 3 can be obtained from a1, a2, an and b₁,b2, ..., bn by
operations of the following type
1. +, -, X, ÷.
2. Solving linear/quadratic equations.
3. Taking square root of a positive number.
...
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