Editing Chapter 2: An ancient theorem and a modern question
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=== Translation === | === Translation === | ||
In Euclidean geometry, a [https://www.mathwarehouse.com/transformations/translations-in-math.php | In Euclidean geometry, a [https://www.mathwarehouse.com/transformations/translations-in-math.php translatio] is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. | ||
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=== Exponents === | === Exponents === | ||
Exponents can be | Exponents can be though of as repeated multiplication, meaning: | ||
<math> 2^3 = 2 \cdot 2 \cdot 2 </math> | <math> 2^3 = 2 \cdot 2 \cdot 2 </math> | ||
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<math> 2^a \cdot 2^b = 2^{a+b} </math> | <math> 2^a \cdot 2^b = 2^{a+b} </math> | ||
Now, you may notice that this doesn't help if we are interested in numbers like <math>2^{\frac{1}{2}}</math> or <math>2^{-1}</math>. These cases are covered in the recommended section if you are interested but are not strictly necessary for understanding this chapter. | Now, you may notice that this doesn't help if we are interested in numbers like <math> 2^{\frac{1}{2}}</math> or <math>2^{-1}</math>. These cases are covered in the [[Recommended| recommended]] section if you are interested but are not strictly necessary for understanding this chapter. | ||
=== Pythagorean Theorem <math>a^2 + b^2 = c^2</math> === | Β | ||
=== Pythagorean Theorem <math> a^2 + b^2 = c^2 </math>=== | |||
''For any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.'' | ''For any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.'' | ||
Here are some [https://www.youtube.com/watch?v=COkhrDbNcuA animated proofs] as well as a [https://www.khanacademy.org/math/basic-geo/basic-geometry-pythagorean-theorem/geo-pythagorean-theorem/e/pythagorean_theorem_1 quiz] to test your understanding. | Here are some [https://www.youtube.com/watch?v=COkhrDbNcuA animated proofs] as well as a [https://www.khanacademy.org/math/basic-geo/basic-geometry-pythagorean-theorem/geo-pythagorean-theorem/e/pythagorean_theorem_1 quiz] to test your understanding. | ||
=== Euclidian Geometry === | === Euclidian Geometry === | ||
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This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows. | This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows. | ||
==== Euclidian Postualtes ==== | |||
==== Euclidian | |||
# A straight line segment can be drawn joining any two points | # A straight line segment can be drawn joining any two points | ||
# Any straight line segment can be extended indefinitely in a straight line. | # Any straight line segment can be extended indefinitely in a straight line. | ||
# Given any straight line segment, a circle can be drawn having that segment as its radius | # Given any straight line segment, a circle can be drawn having that segment as its radius | ||
# All right angles are congruent. | # All right angles are congruent. | ||
# If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. | # If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. | ||
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A good video explaining these postulates as well as what postulates ''are'' can be found [https://www.youtube.com/watch?v=gLMIFRLw9LU here]. | A good video explaining these postulates as well as what postulates ''are'' can be found [https://www.youtube.com/watch?v=gLMIFRLw9LU here]. | ||
Euclid's fifth postulate cannot be proven as a theorem | Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) | ||
=== Radians and <math>\pi</math> === | === Radians and <math> \pi </math> === | ||
<math>\pi</math> is introduced in the books as the sum of all angles of a triangles, which is <math>180^\circ</math>. This might be confusing to those who know that <math>\pi = 3.14 \cdots</math>. | <math> \pi </math> is introduced in the books as the sum of all angles of a triangles, which is <math> 180^\circ</math>. This might be confusing to those who know that <math> \pi = 3.14 \cdots </math>. | ||
The explanation for this is simple. <math>\pi</math> is simply used as a shorthand for <math>\pi R</math> where <math>R</math> stands for radian. An arc of a circle with the same length as the radius of that circle subtends an '''angle of 1 radian''' (roughly 57.29). Adding three radians together brings you almost '''180 degrees''' around. <math>\pi</math> radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of <math>2\pi</math>. To summarize: | The explanation for this is simple. <math> \pi </math> is simply used as a shorthand for <math> \pi R </math> where <math> R </math> stands for radian. An arc of a circle with the same length as the radius of that circle subtends an '''angle of 1 radian''' (roughly 57.29). Adding three radians together brings you almost '''180 degrees''' around. <math> \pi </math> radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of <math> 2\pi </math>. To summarize: | ||
<math> 1 Radian = 1R = 57.29^\circ </math>: | <math> 1 Radian = 1R = 57.29^\circ </math>: | ||
<math> \pi \cdot 57.29 = \pi r = 180^\circ </math> | <math> \pi \cdot 57.29 = \pi r = 180^\circ </math> | ||
So just remember, <math>\pi = 180^\circ</math>. Further explanations are given in the [[Preliminaries| preliminaries]] | So just remember, <math> \pi = 180^\circ </math>. Further explanations are given in the [[Preliminaries| preliminaries]] section. | ||
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=== Hyperbolic Geometry === | === Hyperbolic Geometry === | ||
A type of geometry which can emerge when the fifth postulate is no longer taken to be true. Objects like triangles obey different rules in this type of geometry. For instance, [https://en.wikipedia.org/wiki/Hyperbolic_triangle hyperbolic triangles] have angles which sum to '''less''' than <math>\pi</math> radians. In fact, we have we have a triangle with an area represented by <math>\triangle</math> and three angles represented by <math>\alpha, \beta, \gamma</math> then by the ''Johann Heinrich Lambert formula'': | A type of geometry which can emerge when the fifth postulate is no longer taken to be true. Objects like triangles obey different rules in this type of geometry. For instance, [https://en.wikipedia.org/wiki/Hyperbolic_triangle hyperbolic triangles] have angles which sum to '''less''' than <math> \pi </math> radians. In fact, we have we have a triangle with an area represented by <math> \triangle </math> and three angles represented by <math> \alpha, \beta, \gamma </math> then by the ''Johann Heinrich Lambert formula'': | ||
<math> \pi - (\alpha + \beta + \gamma) = C \triangle </math> | <math> \pi - (\alpha + \beta + \gamma) = C \triangle </math> | ||
where <math>C</math> is | where <math> C </math> is jsut some constant. | ||
In contrast to euclidean geometry where the angels of a triangle alone donβt tell you anything about its size - in hyperbolic geometry if you know the sum of the angels of a triangle, you can calculate its area using the formula above. | In contrast to euclidean geometry where the angels of a triangle alone donβt tell you anything about its size - in hyperbolic geometry if you know the sum of the angels of a triangle, you can calculate its area using the formula above. | ||
== Preliminaries == | == Preliminaries == | ||
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== Essential == | == Essential == | ||
* An | * An additcting puzzle game where you do Euclidian constructions | ||
** [https://www.euclidea.xyz/en/game/packs Euclidia] | ** [https://www.euclidea.xyz/en/game/packs Euclidia] | ||
* | * An animated version of a proof of the Pythagorean Theorem | ||
** [https://timalex.github.io/royal-road/squareangle/ | ** [https://timalex.github.io/royal-road/squareangle/ Pythagorean Theorem Proof] by Community Contributor @TimAlex | ||
* A video game set on a hyperbolic plane | * A video game set on a hyperbolic plane | ||
** [https://play.google.com/store/apps/details?id=com.roguetemple.hyperroid&hl=en_US HyperRogue] for Android | ** [https://play.google.com/store/apps/details?id=com.roguetemple.hyperroid&hl=en_US HyperRogue] for Android | ||
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* Understanding fractional and negative powers | * Understanding fractional and negative powers | ||
** [https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ Understanding Exponents (Why does <math>0^0 | ** [https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ Understanding Exponents (Why does <math>0^0</math>=1)?] | ||
** [https://medium.com/i-math/what-do-fractional-exponents-mean-1bb9bd2fa9a8 What Do Fractional Exponents Mean?] | ** [https://medium.com/i-math/what-do-fractional-exponents-mean-1bb9bd2fa9a8 What Do Fractional Exponents Mean?] | ||
** [https://medium.com/i-math/negative-exponents-reciprocals-and-the-decimal-system-revisited-f4f08894e285 Netaive Exponents and the Decimal System] | ** [https://medium.com/i-math/negative-exponents-reciprocals-and-the-decimal-system-revisited-f4f08894e285 Netaive Exponents and the Decimal System] | ||
* A more in-depth description of the logarithms and exponents with applications | * A more in-depth description of the logarithms and exponents with applications | ||
** [https://www.youtube.com/watch?v=cEvgcoyZvB4&t=1620s Logarithm Fundamentals] | ** [https://www.youtube.com/watch?v=cEvgcoyZvB4&t=1620s Logarithm Fundamentals] | ||
* For those who want an additional explanation of radians | * For those who want an additional explanation of radians | ||
** | ** https://www.youtube.com/watch?v=tSsihw-xPHc | ||
* For those who want an additional explanation of radians and are mad about it | * For those who want an additional explanation of radians and are mad about it | ||
** [https://www.youtube.com/watch?v=jG7vhMMXagQ Pi Is (still) | ** [https://www.youtube.com/watch?v=jG7vhMMXagQ Pi Is (still Wrong).] | ||
* A spot of linear algebra | * A spot of linear algebra | ||
** [https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab The Essence of Linear Algebra] | ** [https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab The Essence of Linear Algebra] | ||
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* A more in depth introduction to linear algebra | * A more in depth introduction to linear algebra | ||
** [https://mathblog.com/linear-algebra Linear Algebra Done Right] by Sheldon Axler | ** [https://mathblog.com/linear-algebra Linear Algebra Done Right] by Sheldon Axler | ||