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Q: How you define a line in math?

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There is no such thing as "Straght" in math: the word does not [yet] exist in the English language. It is not easy to define "staright". You start off with a set of primitive mathematical concepts and then define other concepts using them. A straight line is a primitive geometric concept: it is the shortest distance between two distinct points in a plane. [You will probably need to use the concept of straight to define shortest!]

They define one plane. A line is defined by two points, and it takes three points to define a plane, so two points on the line, and one more point not on the line equals one plane.

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gradient in math refers to the slope of a line

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