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Mathematics became very analytical around the time of Riemann (1826-1866). The mathematics that followed from this is known as modern mathematics. Applied mathematicians may consider more recent mathematics in the second half of the 1900's to be modern mathematics, when computers, economics, and finance (etc) all became huge fields in mathematics.
"René Descartes
Modern society would be impossible to run without mathematics. Even fairly primitive societies unconsciously depend on mathematics.
Some ancient Babylonian inventions are mathematics and astronomy. The Babylonians invented the accurate calendar and knew place value. they also traded with other neighboring cities for economy.
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.