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Geometry Proofs: Help & Tutorials5 chapters | 42 lessons

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Lesson Transcript

Instructor:
*Christopher Muscato*

Chris has a master's degree in history and teaches at the University of Northern Colorado.

In this lesson, you'll explore the life and achievements of the Greek mathematician Euclid, and test your understanding about Ancient Greece, early math, and the principles of Euclidean geometry.

**Euclid** was an ancient Greek mathematician who lived in the Greek city of Alexandria in Egypt during the 3rd century BCE. After Alexander the Great conquered Egypt, he set up Alexandria as the political and economic center, and many Greeks lived or worked there. **Euclid** is often referred to as the 'father of geometry' and his book *Elements* was used well into the 20th century as the standard textbook for teaching geometry.

There is a lot about Euclid's life that is a mystery, including the exact dates of his birth and death, and in many historical accounts he is simply referred to as 'the author of *Elements*'. This is not a reflection of his importance, just a testament of how hard it is to maintain good records over 2,300 years. Euclid seems to have known, worked with, or influenced other major Greek figures, including Plato and Archimedes. There are at least six major works attributed to Euclid. Most of them deal with mathematical formulas, but also delve into things like the math of mirrors and reflections, astronomy, and optical illusions.

The most famous work by Euclid is the 13-volume set called ** Elements**. This collection is a combination of Euclid's own work and the first compilation of important mathematical formulas by other mathematicians into a single, organized format. Thus, it made mathematical learning much more accessible.

Among these are Euclid's **theorems**, or statements proven by compounding different previously proven statements. Two of Euclid's theorems form foundational understandings about arithmetic and number theory. The first theorem is that every positive integer greater than 1 can be written as a product of prime numbers. For example, 21=3x7 or 31= 31x1. Euclid's second theorem states that there are an infinite number of prime numbers. These theorems may sound basic, but Euclid had to develop formulas to prove them. In fact, these are some of the fundamental concepts of arithmetic and had to be proven before more advanced theorems could be built upon them.

Euclid's *Elements* contains several **axioms**, or foundational premises so evident they must be true, about geometry. These include such basic principles as when two non-parallel lines will meet, that opposite angles of an isosceles triangle are equal, and how to find the area of a right triangle. *Elements* also contains geometric interpretations of algebra, such as ideas like a(b+c)=ab+ac. Most important among these is Euclid's algorithm, a formula for devising the greatest common factor of two integers.

Euclid also worked on the properties of shapes like triangles and circles, as well as their ratios and proportions. Together, these ideas make up **Euclidean geometry**, or the math of shapes that conforms to Euclid's axioms. Euclidean geometry is one of the foundational principles of modern math and was instrumental in developing more complex theories in math, art, and science. Intellectuals from Copernicus to Isaac Newton to Albert Einstein have reported the impact that *Elements* and its mathematical proofs, theorems, axioms and geometry had on their lives, and have used Euclidean geometry in their own work.

**Euclid** was an ancient Greek mathematician in Alexandria, Egypt. Due to his groundbreaking work in math, he is often referred to as the 'Father of Geometry'. Euclid's most well-known collection of works, called ** Elements**, outlines some of the most fundamental principles of geometry. It presents several axioms, or mathematical premises so evident they must be true, which formed the basis of Euclidean geometry.

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Geometry Proofs: Help & Tutorials5 chapters | 42 lessons

- Euclid's Axiomatic Geometry: Developments & Postulates 5:58
- Inductive & Deductive Reasoning in Geometry: Definition & Uses 4:59
- Properties and Postulates of Geometric Figures 4:53
- Algebraic Laws and Geometric Postulates 5:37
- Critical Thinking and Logic in Mathematics 4:27
- Logic Laws: Converse, Inverse, Contrapositive & Counterexample 7:09
- Converse of a Statement: Explanation and Example 5:09
- Direct Proofs: Definition and Applications 7:11
- Geometric Proofs: Definition and Format 8:35
- Who is Euclid? - Biography, Contribution & Theorems 4:05
- Go to Fundamentals of Geometry Proofs

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