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Definitions and proof both play essential roles in mathematics. Definition is crucial, because you need to know what something is (and what it's not) before you can make conjectures about it and prove these conjectures.
When mathematicians create a definition, they strive to be concise -- to communicate a lot of information in a few words. This can make reading and understanding mathematical definitions difficult. This is further complicated by the fact that more than one definition may work for a given object. You may find yourself coming across an unfamiliar definition for a familiar object, and somehow you have to make sense of it.
A benefit of mathematical definitions is that you'll never find the circularity associated with dictionary definitions. For example:
Webster's attempt to define "dimension"
Dimension: | Any measurable extent, such as length, width, and depth. |
Extent: | The space, amount, or degree to which a thing extends; size; length; breadth. |
Measurement: | Extent, quantity, or size as determined by measuring. |
Size: | That quality of a thing that determines how much space it occupies; dimensions; extent. |
Length: | The measure of how long a thing is; the greatest of the two or three dimensions of anything; extent in space. |
Here's a mathematical definition of a geometric property called convexity. The word may be familiar to you, but try to focus just on the definition provided for this activity.
A figure is convex if, for every pair of points within the figure, the segment connecting the two points lies entirely within the figure.
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