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The word base has several meaning in mathematics depending on the context in which it arises. Most of the uses are related to the common meaning of base as a starting point or building block.
In geometry, for example, a student may say that the area of a rectangle is equal to the length of its base times its height. In this case, base refers to one side of the rectangle, typically the side oriented most nearly horizontally from the perspective of the viewer.
In algebra, a student may wish to compute 32. In this case, the number 3 is the base and the number 2 is the power. More generally, when describing the expression ba we call b the base.
In arithmetic (or number theory), the word base is related to place value and how a number is represented. Students in elementary school learn to use the base 10 system. Here, each place value is a power of 10. In contrast, computers represent information as numbers written to base 2. Here, each place value is a power of 2 (and the only allowable digits are 0 and 1). Of course, numbers can be written to other bases as well.
The word base occurs when describing mathematical induction. Proofs by mathematical induction are often conceived in two steps: a base case followed by an inductive step.