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An axiom is something used in logic. It is a statement which has no proof. This means it can not be proved within the discussion of a problem. So inside some discussion, it is thought to be true. There are many reasons why it has no proof. For example,
- The statement might be obvious. This means most people think it is clearly true. An example of an obvious axiom is the principle of contradiction. It says that a statement and its opposite can not both be true at the same time and in the same place.
- The statement is based on physical laws and can easily be observed. An example is Newton's laws of motion. They are easily observed in the physical world.
- The statement is a proposition. This means we care more about what happens if the axiom is true. We do not care so much if it is actually true. The proposition is given at the start of an argument. This is a more modern definition of an axiom.
Logic can be used to find theorems from the axioms. Then those theorems can be used to make more theorems. This is often how math works. Axioms are important because logical arguments start with them.
Euclid of Alexandria was a Greek mathematician. Around the year 300BC, he made a list of axioms:
- Two numbers that are both the same as a third number are the same number.
- If A and B are two numbers that are the same, and C and D are also the same, A+C is the same as B+D.
- If A and B are two numbers that are the same, and C and D are also the same, A-C is the same as B-D.
- Two shapes that fill exactly the same space are the same shape.
- If you divide a number by anything more than 1, the quotient (result) will be less than the original number.