Hypatia held a teaching chair in Philosophy. She was particularly known for teaching (what we now call) Neo-platonism. She taught mathematics as a discipline for the mind that would allow students to better comprehend philosophy. Two good books on Hypatia are Hypatia of Alexandria by Maria Dzielska (Harvard University Press, 1995) and Hypatia of Alexandria: Mathematician and Martyr by Michael A. B. Deakin (Prometheus Books, 2007.) I've reviewed both books on my blog and have a number of other posts on Hypatia.
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Hypatia taught philosophy and astronomy in addition to mathematics. She was known for her lectures on Neoplatonism, a philosophical school that combined elements of Plato's teachings with other philosophical traditions. Hypatia's contributions to astronomy included observations of celestial bodies and the development of new mathematical techniques for calculating their properties.
Other than the fact that much of mathematics is about patterns, NO.
It is not particularly significant, other than being a prime.
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.
In mathematics, a googol is the number expressed by the digit one followed by a hundred zeroes. There is actually no mathematical significance to the number other than for comparison.
Numerical reasoning is only a part of maths. Other major components of maths are Geometry and Algebra. The word' maths'. is short for 'mathematics'. The word 'mathematics' comes from Classical Greece, and means to 'learn'.