Lots and lots and lots. Just off the top of my head, anything that uses quantum physics is based off of Hilbert space theory and operator analysis, which are both (older) fields of math.
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Contemporary mathematics refers to mathematics in general application and can be used anytime, anywhere and regardless of who uses it. In general, business mathematics and mathematics of investment may be well classified under this category.
It enables students to interact with numbers.
When trying to deliver a complex topic to a large audience creating a power point presentation is useful. You can highlight the key topics of the application of mathematics in the engineering field using such a tool.
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.
No, but incorrect application of mathematics can make it look like it does.