Math is used to quantify scientific observations and predictions. Ie: how high will the ball bounce when dropped from 10 feet?
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Scientific notation is useful in mathematics because it makes very large or very small numbers easier to compute.
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It is the usage of the zero symbol that is still essential in mathematics even today.
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.