Math 1230, Notes

Aug. 26, 2014

Math 1230, Notes Aug. 26, 2014 1 / 13

1 Euclid’s Elements and Hilbert’s update.

2 Platonic Solids. Definitions. How many are there? History. Existenceproofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

1 Euclid’s Elements and Hilbert’s update.2 Platonic Solids. Definitions. How many are there? History. Existence

proofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

1 Euclid’s Elements and Hilbert’s update.2 Platonic Solids. Definitions. How many are there? History. Existence

proofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

1 Euclid’s Elements and Hilbert’s update.2 Platonic Solids. Definitions. How many are there? History. Existence

proofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

1 Euclid’s Elements and Hilbert’s update.2 Platonic Solids. Definitions. How many are there? History. Existence

proofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

1 Euclid’s Elements and Hilbert’s update.2 Platonic Solids. Definitions. How many are there? History. Existence

proofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

1 Euclid’s Elements and Hilbert’s update.2 Platonic Solids. Definitions. How many are there? History. Existence

proofs. Aristotle’s error; time line: 430 BC to 2010 AD. Moderndevelopments, applications.

3 Symmetry; Symmetry groups of the Platonic solids. Application:Crystals. The crystallographic theorem. quasi-crystals and Penrosetiles

4 Prime numbers. proof that there are infinitely many. proof thatevery number can be decomposed into prime factors. Sieve ofErostothenes. relation to coding and internet security. algebraicgeometry approach (defer to after part 3).

5 history of numbers – irrationality of√

2. countability of rationals.Countability of algebraic numbers. Existence of transcendentalnumbers (Cantor diagonal argument). solving cubics, quartics.Complex numbers and the complex plane (Argand).

6 Fundamental Theorem of Algebra – Gauss’s first proof – why it iswrong – how is it proved today?

7 Radon transform, and application to CT scans

Math 1230, Notes Aug. 26, 2014 2 / 13

8. chaos and a crazy pendulum

9. quarternions and computer games

10 famous conjectures, unsolved problems – 4th grade problem, twinprimes

11. Archimedes (method of exhaustion, integrals– see Euclid) , Newton,Gauss.

Math 1230, Notes Aug. 26, 2014 3 / 13

8. chaos and a crazy pendulum

9. quarternions and computer games

10 famous conjectures, unsolved problems – 4th grade problem, twinprimes

11. Archimedes (method of exhaustion, integrals– see Euclid) , Newton,Gauss.

Math 1230, Notes Aug. 26, 2014 3 / 13

8. chaos and a crazy pendulum

9. quarternions and computer games

10 famous conjectures, unsolved problems – 4th grade problem, twinprimes

11. Archimedes (method of exhaustion, integrals– see Euclid) , Newton,Gauss.

Math 1230, Notes Aug. 26, 2014 3 / 13

8. chaos and a crazy pendulum

9. quarternions and computer games

10 famous conjectures, unsolved problems – 4th grade problem, twinprimes

11. Archimedes (method of exhaustion, integrals– see Euclid) , Newton,Gauss.

Math 1230, Notes Aug. 26, 2014 3 / 13

Mathematics before Plato: often ”mystical”, or ”philosophical”

Socrates: I cannot satisfy myself that, when one is added to one, theone to which the addition is made becomes two, or that the two unitsadded together make two by reason of addition. I cannot understandhow when separated from the other, each of them was one and not two,and now, when they are brought together, the mere juxtaposition ormeeting of them should be the cause of them becoming two.

This is not a course on philosophy. For that see references by Hardyand by Davis and Hersh. See also Part III of Gulliver’s travels. )

Math 1230, Notes Aug. 26, 2014 4 / 13

Mathematics before Plato: often ”mystical”, or ”philosophical”

Socrates: I cannot satisfy myself that, when one is added to one, theone to which the addition is made becomes two, or that the two unitsadded together make two by reason of addition. I cannot understandhow when separated from the other, each of them was one and not two,and now, when they are brought together, the mere juxtaposition ormeeting of them should be the cause of them becoming two.

This is not a course on philosophy. For that see references by Hardyand by Davis and Hersh. See also Part III of Gulliver’s travels. )

Math 1230, Notes Aug. 26, 2014 4 / 13

Mathematics before Plato: often ”mystical”, or ”philosophical”

Socrates: I cannot satisfy myself that, when one is added to one, theone to which the addition is made becomes two, or that the two unitsadded together make two by reason of addition. I cannot understandhow when separated from the other, each of them was one and not two,and now, when they are brought together, the mere juxtaposition ormeeting of them should be the cause of them becoming two.

This is not a course on philosophy. For that see references by Hardyand by Davis and Hersh. See also Part III of Gulliver’s travels. )

Math 1230, Notes Aug. 26, 2014 4 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Euclid’s Elements:

Summarized what was known about geometry around 300 BC. (little orno original work)

Based on 20 definitions, 5 postulates, and 5 ”common notions” – e.g.Things equal to the same thing are also equal to each other.

Some of the Definitions:

1. A point is that of which there is no part.

4. A straight line is one which lies evenly with the points on itself.

8. And a plane angle is the inclination of the lines to one another, whentwo lines in a plane meet each other and are not in a straight line.

15. A circle is a plane figure contained by a single line (which is calledthe circumference) such that all of the lines radiating from the one pointinside the figure are equal to each other.

16. And the point is called the center of the circle.

17. Parallel lines are lines in the same plane, and being produced(extended) to infinity in each direction, do not meet.

Math 1230, Notes Aug. 26, 2014 5 / 13

Postulates:

1. If p and q are points, then there is a line containing both p and q.

2. A line can be extended forever.

3. For any point p and length r there is a circle with center p andradius r .

4. Any two right angles are equal to each other.

5. If two different lines intersect a third line, making angles on the sameside whose sum is less than two right angles, then these two lines meeton that side of the third line.

Math 1230, Notes Aug. 26, 2014 6 / 13

Postulates:

1. If p and q are points, then there is a line containing both p and q.

2. A line can be extended forever.

3. For any point p and length r there is a circle with center p andradius r .

4. Any two right angles are equal to each other.

5. If two different lines intersect a third line, making angles on the sameside whose sum is less than two right angles, then these two lines meeton that side of the third line.

Math 1230, Notes Aug. 26, 2014 6 / 13

Postulates:

1. If p and q are points, then there is a line containing both p and q.

2. A line can be extended forever.

3. For any point p and length r there is a circle with center p andradius r .

4. Any two right angles are equal to each other.

5. If two different lines intersect a third line, making angles on the sameside whose sum is less than two right angles, then these two lines meeton that side of the third line.

Math 1230, Notes Aug. 26, 2014 6 / 13

Postulates:

1. If p and q are points, then there is a line containing both p and q.

2. A line can be extended forever.

3. For any point p and length r there is a circle with center p andradius r .

4. Any two right angles are equal to each other.

5. If two different lines intersect a third line, making angles on the sameside whose sum is less than two right angles, then these two lines meeton that side of the third line.

Math 1230, Notes Aug. 26, 2014 6 / 13

Postulates:

1. If p and q are points, then there is a line containing both p and q.

2. A line can be extended forever.

3. For any point p and length r there is a circle with center p andradius r .

4. Any two right angles are equal to each other.

5. If two different lines intersect a third line, making angles on the sameside whose sum is less than two right angles, then these two lines meeton that side of the third line.

Math 1230, Notes Aug. 26, 2014 6 / 13

Postulates:

1. If p and q are points, then there is a line containing both p and q.

2. A line can be extended forever.

3. For any point p and length r there is a circle with center p andradius r .

4. Any two right angles are equal to each other.

5. If two different lines intersect a third line, making angles on the sameside whose sum is less than two right angles, then these two lines meeton that side of the third line.

Math 1230, Notes Aug. 26, 2014 6 / 13

BOOK 1, PROPOSITION 1.

Math 1230, Notes Aug. 26, 2014 7 / 13

Book 1, Proposition 28 (part). If a straight line falling across twostraight lines makes the sum of the external angles equal to the internaland opposite angle on the same side then the two straight lines will beparallel to one another.

Book 1, Proposition 29 (part). A straight line falling across parallelstraight lines makes the alternate angles equal to each other.

Proof: (First proposition to require postulate 5)

Math 1230, Notes Aug. 26, 2014 8 / 13

Book 1, Proposition 28 (part). If a straight line falling across twostraight lines makes the sum of the external angles equal to the internaland opposite angle on the same side then the two straight lines will beparallel to one another.

Book 1, Proposition 29 (part). A straight line falling across parallelstraight lines makes the alternate angles equal to each other.

Proof: (First proposition to require postulate 5)

Math 1230, Notes Aug. 26, 2014 8 / 13

Book 1, Proposition 28 (part). If a straight line falling across twostraight lines makes the sum of the external angles equal to the internaland opposite angle on the same side then the two straight lines will beparallel to one another.

Book 1, Proposition 29 (part). A straight line falling across parallelstraight lines makes the alternate angles equal to each other.

Proof: (First proposition to require postulate 5)

Math 1230, Notes Aug. 26, 2014 8 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

More from the Elements:

book 2, prop.4 (121 of Boyer)

book 2, prop. 6 (122 of Boyer.)

book 2, Prop 11.

book 2, prop 12

book 7, props 1,2

book 9, prop 35.

book 11.

Math 1230, Notes Aug. 26, 2014 9 / 13

2.4: If a straight line be cut at random, the square on the whole isequal to the squares on the segments and twice the rectanglecontained by the segments

(a+ b)2 = a2 + b2 + 2ab

2.5: If a straight line be cut into equal and unequal segments, therectangle contained by the unequal segments of the whole, togetherwith the square on the straight line between the points is equal to thesquare on the half. (??)

For let any straight line AB have been cut - equally at C andunequally at D. I say that the rectangle contained by AD and DB plusthe square on CD is equal to the square on CB.

a2 − b2 = (a+ b)(a− b)

2.6 If a straight line be bisected and a straight line be added to it in astraight line, the rectangle contained by the whole (with the addedstraight line) and the added straight line together with the square onthe half is equal to the square on the straight line made up of the halfand the added straight line.

Math 1230, Notes Aug. 26, 2014 10 / 13

2.4: If a straight line be cut at random, the square on the whole isequal to the squares on the segments and twice the rectanglecontained by the segments

(a+ b)2 = a2 + b2 + 2ab

2.5: If a straight line be cut into equal and unequal segments, therectangle contained by the unequal segments of the whole, togetherwith the square on the straight line between the points is equal to thesquare on the half. (??)

For let any straight line AB have been cut - equally at C andunequally at D. I say that the rectangle contained by AD and DB plusthe square on CD is equal to the square on CB.

a2 − b2 = (a+ b)(a− b)

2.6 If a straight line be bisected and a straight line be added to it in astraight line, the rectangle contained by the whole (with the addedstraight line) and the added straight line together with the square onthe half is equal to the square on the straight line made up of the halfand the added straight line.

Math 1230, Notes Aug. 26, 2014 10 / 13

2.4: If a straight line be cut at random, the square on the whole isequal to the squares on the segments and twice the rectanglecontained by the segments

(a+ b)2 = a2 + b2 + 2ab

2.5: If a straight line be cut into equal and unequal segments, therectangle contained by the unequal segments of the whole, togetherwith the square on the straight line between the points is equal to thesquare on the half. (??)

For let any straight line AB have been cut - equally at C andunequally at D. I say that the rectangle contained by AD and DB plusthe square on CD is equal to the square on CB.

a2 − b2 = (a+ b)(a− b)

2.6 If a straight line be bisected and a straight line be added to it in astraight line, the rectangle contained by the whole (with the addedstraight line) and the added straight line together with the square onthe half is equal to the square on the straight line made up of the halfand the added straight line.

Math 1230, Notes Aug. 26, 2014 10 / 13

2.4: If a straight line be cut at random, the square on the whole isequal to the squares on the segments and twice the rectanglecontained by the segments

(a+ b)2 = a2 + b2 + 2ab

2.5: If a straight line be cut into equal and unequal segments, therectangle contained by the unequal segments of the whole, togetherwith the square on the straight line between the points is equal to thesquare on the half. (??)

For let any straight line AB have been cut - equally at C andunequally at D. I say that the rectangle contained by AD and DB plusthe square on CD is equal to the square on CB.

a2 − b2 = (a+ b)(a− b)

2.6 If a straight line be bisected and a straight line be added to it in astraight line, the rectangle contained by the whole (with the addedstraight line) and the added straight line together with the square onthe half is equal to the square on the straight line made up of the halfand the added straight line.

Math 1230, Notes Aug. 26, 2014 10 / 13

2.4: If a straight line be cut at random, the square on the whole isequal to the squares on the segments and twice the rectanglecontained by the segments

(a+ b)2 = a2 + b2 + 2ab

2.5: If a straight line be cut into equal and unequal segments, therectangle contained by the unequal segments of the whole, togetherwith the square on the straight line between the points is equal to thesquare on the half. (??)

For let any straight line AB have been cut - equally at C andunequally at D. I say that the rectangle contained by AD and DB plusthe square on CD is equal to the square on CB.

a2 − b2 = (a+ b)(a− b)

2.6 If a straight line be bisected and a straight line be added to it in astraight line, the rectangle contained by the whole (with the addedstraight line) and the added straight line together with the square onthe half is equal to the square on the straight line made up of the halfand the added straight line.

Math 1230, Notes Aug. 26, 2014 10 / 13

2.4: If a straight line be cut at random, the square on the whole isequal to the squares on the segments and twice the rectanglecontained by the segments

(a+ b)2 = a2 + b2 + 2ab

2.5: If a straight line be cut into equal and unequal segments, therectangle contained by the unequal segments of the whole, togetherwith the square on the straight line between the points is equal to thesquare on the half. (??)

For let any straight line AB have been cut - equally at C andunequally at D. I say that the rectangle contained by AD and DB plusthe square on CD is equal to the square on CB.

a2 − b2 = (a+ b)(a− b)

2.6 If a straight line be bisected and a straight line be added to it in astraight line, the rectangle contained by the whole (with the addedstraight line) and the added straight line together with the square onthe half is equal to the square on the straight line made up of the halfand the added straight line.

Math 1230, Notes Aug. 26, 2014 10 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Goals of an Axiom System:

1. Consistency. No contradictions result.

2. No redundancy; none of the axioms can be proved from the otheraxioms.

Not known for Euclid’s axioms until 19th century

How can we verify consistency?

Model of Euclidean Geometry?

Definition: A point is an ordered pair (a, b) of real numbers.

Definition: If (a, b) and (c , d) are two different points, then a line(segment) is the set of all points (a+ t (c − a) , b+ t (d − b)) suchthat 0 ≤ t ≤ 1.

Then this satisfies all of the axioms, and so the axioms are consistent.

Second model: require that a and b are rational. (Does proposition1.1 hold?)

How can we verify that there is no redundancy?

Axioms of David Hilbert, 1862-1943:

Math 1230, Notes Aug. 26, 2014 11 / 13

Math 1230, Notes Aug. 26, 2014 12 / 13

Math 1230, Notes Aug. 26, 2014 13 / 13