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By Steven G. Krantz

This publication is set the concept that of mathematical adulthood. Mathematical adulthood is critical to a arithmetic schooling. The objective of a arithmetic schooling is to rework the scholar from somebody who treats mathematical rules empirically and intuitively to anyone who treats mathematical principles analytically and will keep watch over and manage them effectively.

Put extra without delay, a mathematically mature individual is person who can learn, research, and evaluation proofs. And, most importantly, he/she is one that can create proofs. For this can be what smooth arithmetic is all approximately: bobbing up with new rules and validating them with proofs.

The publication offers heritage, information, and research for realizing the concept that of mathematical adulthood. It turns the belief of mathematical adulthood from a subject for coffee-room dialog to a subject for research and critical consideration.

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But modern marketing strategies today entail that each brand of pill have its own special “caplet” shape. And in fact, for practical reasons of drafting, the caplet shapes were specified for me as several arcs of circles (of different radii) pasted together. Calculating surface areas and volumes for such shapes is rather tricky. The people who asked me to do this job were not mathematically mature. They had only had a bit of high school mathematics. No calculus. The way they had been handling the task before I came along was a two-stage process.

In a slightly different vein, we might consider the experience of James Simons. In the late 1960s and early 1970s, Simons was a very accomplished differential geometer. He won the Veblen Prize, and his work with Chern is ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2011/11/9 — 15:21 — page 18 — #36 ✐ ✐ 18 1. Introductory Thoughts still cited today. Simons was also the founding Chairman of the mathematics department at SUNY Stony Brook, and deserves much of the credit for making Stony Brook the geometric powerhouse that it is today.

This was a moment of opportunity for Kepler. ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2011/11/9 — 15:21 — page 24 — #42 ✐ ✐ 24 1. Introductory Thoughts He was able to negotiate with Brahe’s family and obtain the much-needed data. As a result, Kepler was able to do his now-famous calculations of the planetary orbits and, as a result, formulate his time-honored laws of planetary motion. Particulary, using the data on planet Mars, Kepler published his first and second laws (in Latin) in Astronomia Nova in 1609. This paper was translated into English by Donahue.

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