Chapter 7: Patterns in Numbers: The origins of number theory
“There is something fascinating about numbers. Plain, unadorned whole numbers, 1, 2, 3, 4, 5, … What could possibly be simpler? But that simple exterior conceals hidden depths, and many of the most baffling questions in mathematics are about apparently straightforward properties of whole numbers. This area is known as number theory.” (page 119)
Contributors to number theory: Euclid (proofs, prime factorization is unique), Diophantus (found all whole number solutions of a2 + b2 = c2), Fermat (primes, his last theorem), Leonhard Euler, Joseph-Louis Lagrange, Carl Friedrich Gauss (modular arithmetic. Disquisitiones Arithmeticae).
Fermat’s Last Theorem: “To resolve a cube into the sum of two cubes, a fourth power into two fourths or, in general, any power higher than the second into two of the same kind is impossible; of which fact I have found a remarkable proof. The margin is too small to contain it.”
“It often takes time for a good mathematical idea to acquire practical importance – sometimes hundreds of years – but eventually most topics that mathematicians find significant for their own sake turn out to be valuable in the real world too.”
Contributors to number theory: Euclid (proofs, prime factorization is unique), Diophantus (found all whole number solutions of a2 + b2 = c2), Fermat (primes, his last theorem), Leonhard Euler, Joseph-Louis Lagrange, Carl Friedrich Gauss (modular arithmetic. Disquisitiones Arithmeticae).
Fermat’s Last Theorem: “To resolve a cube into the sum of two cubes, a fourth power into two fourths or, in general, any power higher than the second into two of the same kind is impossible; of which fact I have found a remarkable proof. The margin is too small to contain it.”
“It often takes time for a good mathematical idea to acquire practical importance – sometimes hundreds of years – but eventually most topics that mathematicians find significant for their own sake turn out to be valuable in the real world too.”
Chapter 8: The System of the World: The invention of calculus
“The most significant single advance in the history of mathematics was calculus, invented independently around 1680 by Isaac Newton and Gottfried Leibniz.”
“Newton turned calculus into a central technique of the budding subject of mathematical physics, humanity’s most effective known route to the understanding of the natural world. Newton called his theory “The System of the World””. (in Principia Mathematica)
Calculus: mathematics of instantaneous rates of change – how rapidly is some particular quantity changing at this very instant. It divides into differential calculus and integral calculus.
The main motivation to the development of calculus came from physics and the patterns from nature and the universe.
There was a battle going on in Renaissance Europe between the existence of God (Church) and Science. Ptolemy, Copernicus, Kepler, Galileo, Leibniz, and Newton played an essential role in the development of science and the creation of the model of solar system as we know it. These lead to discoveries of patterns in nature.
“Newton’s great discovery was that nature’s patterns seem to manifest themselves not as regularities in certain quantities, but as relations among their derivatives. The laws of nature are written in the language of calculus; what matters are not the values of physical variables, but the rates at which they change. It was a profound insight, and it created a revolution, leading more or less directly to modern science, and changing our planet forever.” (page 162)
“Newton turned calculus into a central technique of the budding subject of mathematical physics, humanity’s most effective known route to the understanding of the natural world. Newton called his theory “The System of the World””. (in Principia Mathematica)
Calculus: mathematics of instantaneous rates of change – how rapidly is some particular quantity changing at this very instant. It divides into differential calculus and integral calculus.
The main motivation to the development of calculus came from physics and the patterns from nature and the universe.
There was a battle going on in Renaissance Europe between the existence of God (Church) and Science. Ptolemy, Copernicus, Kepler, Galileo, Leibniz, and Newton played an essential role in the development of science and the creation of the model of solar system as we know it. These lead to discoveries of patterns in nature.
“Newton’s great discovery was that nature’s patterns seem to manifest themselves not as regularities in certain quantities, but as relations among their derivatives. The laws of nature are written in the language of calculus; what matters are not the values of physical variables, but the rates at which they change. It was a profound insight, and it created a revolution, leading more or less directly to modern science, and changing our planet forever.” (page 162)