Difference between revisions of "Geometry"
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| − | '''Geometry''' is the field of [[mathematics]] dealing with figures in a given [[space]]. It is one of the two oldest branches of mathematics, along with [[arithmetic]] (which eventually branched into number theory and algebra). The geometry usually studied is | + | '''Geometry''' is the field of [[mathematics]] dealing with figures in a given [[space]]. It is one of the two oldest branches of mathematics, along with [[arithmetic]] (which eventually branched into [[number theory]] and [[algebra]]). The geometry usually studied is |
| + | |||
| + | ==Introductory Videos== | ||
| + | https://youtu.be/51K3uCzntWs?t=842 \\ | ||
| + | https://youtu.be/j3QSD5eDpzU | ||
== Euclidean Geometry == | == Euclidean Geometry == | ||
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===Parallel Postulate=== | ===Parallel Postulate=== | ||
{{main|Parallel Postulate}} | {{main|Parallel Postulate}} | ||
| − | The fifth [[postulate]] stated in the book, | + | The fifth [[postulate]] stated in the book, equivalent to the following statement, |
:''“Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”'' | :''“Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”'' | ||
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*A [[point]] is a geometric structure with no area, length, width, or dimension. Its only property is space. It is said to be zero-dimensional. | *A [[point]] is a geometric structure with no area, length, width, or dimension. Its only property is space. It is said to be zero-dimensional. | ||
*A [[line]] is generally taken to mean a straight line, which is the locus of points on the [[Cartesian plane]] satisfying a [[linear]] [[function]]. It has length and position, but no other properties. It is one-dimensional. A [[line segment]] means a finite segment of a line, while a [[ray]] is a line infinitely extending in only one direction. | *A [[line]] is generally taken to mean a straight line, which is the locus of points on the [[Cartesian plane]] satisfying a [[linear]] [[function]]. It has length and position, but no other properties. It is one-dimensional. A [[line segment]] means a finite segment of a line, while a [[ray]] is a line infinitely extending in only one direction. | ||
| − | *A [[plane]] is a line | + | *A [[plane]] is a line but in a Cartesian space. It as length, width, and position. It is two-dimensional. The point/line/plane sequence can be extended to spaces and higher dimensions. |
*An [[angle]] is a structure formed by the intersection two [[ray]]s at their endpoints. It is measure in either [[degree]]s or [[radian]]s, though the less-common [[Système international|metric]] unit [[gradian]] is also used. | *An [[angle]] is a structure formed by the intersection two [[ray]]s at their endpoints. It is measure in either [[degree]]s or [[radian]]s, though the less-common [[Système international|metric]] unit [[gradian]] is also used. | ||
Revision as of 06:59, 25 September 2020
Geometry is the field of mathematics dealing with figures in a given space. It is one of the two oldest branches of mathematics, along with arithmetic (which eventually branched into number theory and algebra). The geometry usually studied is
Contents
Introductory Videos
https://youtu.be/51K3uCzntWs?t=842 \\ https://youtu.be/j3QSD5eDpzU
Euclidean Geometry
- Main article: Euclidean geometry
The most common type of geometry used in pre-collegiate mathematics competitions is Euclidean geometry. This type of geometry was first formally outlined by the Greek mathematician Euclid in his book The Elements.
Parallel Postulate
- Main article: Parallel Postulate
The fifth postulate stated in the book, equivalent to the following statement,
- “Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”
was the subject of a controversy for many centuries, with many attempted proofs. It is much less simple than the other postulates, and more wordy. This postulate is the basis of Euclidean geometry.
Non-Euclidean Geometry
Non-Euclidean geometry are geometries in which the fifth postulate is altered. Types of non-Euclidean geometry include:
Student Guides to Geometry
Main Concepts
- The notion of dimensions is fundamental to geometry. N-space is a term related to this concept.
- A point is a geometric structure with no area, length, width, or dimension. Its only property is space. It is said to be zero-dimensional.
- A line is generally taken to mean a straight line, which is the locus of points on the Cartesian plane satisfying a linear function. It has length and position, but no other properties. It is one-dimensional. A line segment means a finite segment of a line, while a ray is a line infinitely extending in only one direction.
- A plane is a line but in a Cartesian space. It as length, width, and position. It is two-dimensional. The point/line/plane sequence can be extended to spaces and higher dimensions.
- An angle is a structure formed by the intersection two rays at their endpoints. It is measure in either degrees or radians, though the less-common metric unit gradian is also used.
