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    Euclidian axioms

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6th class > > Euclidian axioms

Euclidian axioms

Euclidean axioms are general principles in geometry that are accepted as true without proof, similar to postulates but more broad and fundamental. They serve as the foundation for logical reasoning in mathematics, especially in Euclidean geometry. Here's a simplified explanation of some key Euclidean axioms that help us understand how space and shapes work:

AXIOM 1:Things that are equal to the same thing are also equal to each other. Imagine you have two apples, and both are the same as another apple in size and weight. This means your two apples are also the same size and weight as each other.

AXIOM 2:If you add equal things to equal things, the wholes are equal. If you and your friend both have two marbles, and you each get one more, then you both still have the same number of marbles.

AXIOM 3:If you subtract equal things from equal things, the remainders are equal. If you and your friend have the same number of candies, and you both eat one, you'll still have the same number of candies left.

AXIOM 4:Things that coincide with one another are equal. This means if you can place one shape on top of another and they fit perfectly, those two shapes are exactly the same in size and shape.

AXIOM 5:The whole is greater than the part. If you have a whole pizza and you take a slice away, what's left of the pizza is less than the whole pizza you started with.

These axioms are very basic ideas that seem obvious, but they're really important because they help us make sure that the way we think about space and shapes is consistent and makes sense. Just like building blocks, these axioms let mathematicians build up more complicated ideas on a sturdy base.

One of the people who studied Euclid’s work was the American President Thomas Jefferson. When writing the Declaration of Independence in 1776, he wanted to follow a similar approach. He begins by stating a few, simple “axioms” and then “proves” more complex results:

“We hold these truths to be self-evident: that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.”

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We, therefore … declare, that these United Colonies are, and of right ought to be, free and independent states.”

This is just one example where Euclid’s ideas in mathematics have inspired completely different subjects.