Algebra is a form (or subset) of mathematics.
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there is no difference between Mathematics and Arithmetic because Arithmetic is a branch of mathematics. there is no difference between Mathematics and Arithmetic because Arithmetic is a branch of mathematics.
They are both branches of mathematics.
The difference of this two kind of mathematics is that Non-Science Mathematics is used in cases that only logic is used to solve any problem, in the other hand; Scientific Mathematics is something related to Algebra, Arithmetic, and Geometry, because is used formulas, ways to solve big and tricky problems.
algebra 1a is the first part of algebra 1 and algebra 1b is the second part. :)
pure mathematics is mathematics studied and used reasons other than application. So some algebra certainly might be, but not all algebra. Modern or abstract algebra would fit better in that category.