Honors Geometry Unit 2 Notes

Definitions, Axioms, and Theorems

This is the beginning of our study of geometry. Geometry is one of the oldest branches of mathematics and is still studied for its historical significance as well as its applicability. Approximately 2000 years ago, a man named Euclid compiled the worlds knowledge of geometry and number theory into a series of thirteen books called the Elements that organized this knowledge into a new format. Each idea was either a definition, an word that we assigned a meaning to, a postulate (or axiom) or an idea that is considered simple enough that it can be accepted without proof and as a premise for building arguments, or theorems, statements that must be proven in order to be accepted.

Examples

• We can define a line as that which has length but no breadth, so that we all have the same understanding what we mean when we say "line."
• We can postulate that between any two points a line can be drawn.
• We can prove the Pythagorean Theorem.

Importance

This process provides formality to mathematics and allows us to show that the procedures we are following are true in all cases. It removed doubt of the validity of solutions.

Day 0

We will begin by looking at some algebraic properties to which you are probably more familiar.

• Reflexive Property
• Symmetric Property
• Transitive Property of Equality
• Associative Property of Addition/Multiplication
• Commutative Property of Addition/Multiplication
• Distributive Property
• Additive Property of Equality
• Multiplicative Property of Equality
• Additive Identity
• Multiplicative Identity
• Additive Inverse
• Multiplicative Inverse
  • What number does not have a multiplicative inverse?

These handful of properties allow us to solve a large number of algebraic problems and guarantee correctness.

$$y=x+3$$