A student asks what exactly Euclidean geometry is. How
do you answer?
Since this is not my area of expertise, I searched Google under the key words "Euclidean geometry" to get this:
http://www.google.com/search?client=safari&rls=en&q=Euclidean+geometry&ie=UTF-8&oe=UTF-8
In the future, you can find the information you desire more quickly, if you use appropriate key words to do your own search. Also see http://hanlib.sou.edu/searchtools/.
he had a huge one
To explain what Euclidean geometry is, you can start by breaking it down into its key components. Euclidean geometry is a branch of mathematics that focuses on the study of shapes, sizes, and properties of figures in two-dimensional and three-dimensional space. It was developed by the ancient Greek mathematician Euclid around 300 BCE.
Here's how you can provide a more detailed explanation to the student:
1. Begin with the historical background: Mention that Euclidean geometry is named after the mathematician Euclid, who wrote the book "Elements." This book is one of the oldest and most influential mathematical works, providing the foundation for the study of geometry.
2. Introduce the main concepts: Euclidean geometry primarily deals with points, lines, planes, angles, and solids. It explores the properties of these geometric objects and establishes logical relationships between them.
3. Explain the key axioms and postulates: Euclidean geometry is built upon a set of assumptions called axioms or postulates. These assumptions are self-evident truths that are accepted without proof. For example, the "parallel postulate" states that through a point not on a given line, there is only one parallel line to that given line.
4. Discuss the five fundamental Euclidean postulates: Euclid's "Elements" is organized into 13 books, but you can highlight the five main postulates that form the core of Euclidean geometry:
a. A straight line segment can be drawn between any two points.
b. A straight line can be extended infinitely in both directions.
c. A circle can be drawn with any point as the center and any distance as the radius.
d. All right angles are congruent.
e. If two lines are drawn that intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, the two lines will eventually meet on that side.
5. Emphasize the importance and applications: Euclidean geometry provided the foundation for geometric reasoning for thousands of years. It was used extensively in various fields such as architecture, engineering, and physics. Moreover, Euclid's work established a rigorous approach to logical deduction that is still influential in modern mathematics.
By following this approach, you'll be able to provide the student with a comprehensive understanding of what Euclidean geometry is and its significance in the field of mathematics.