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Page 1: Introduction
Mathematics in the Modern World.
Page 2: Growth of Mathematics in the 18th Century
By the 18th century, mathematics had evolved into a modern science.
The rapid development of mathematics was facilitated by its introduction into schools.
This educational reform allowed a larger number of individuals the opportunity to learn basic mathematical principles.
Page 3: Emergence of New Mathematicians
The growth in mathematical education led to a surge of new mathematicians entering the field.
Their contributions included innovative ideas, solutions to longstanding problems, and new areas of mathematical research.
Existing fields in mathematics also experienced significant expansion due to these developments.
Page 4: Key Figures in Modern Mathematics
This section highlights some of the most influential mathematicians of modern times.
Page 5: Leonhard Euler
Euler famously resolved the equation e^{iπ} + 1 = 0.
Page 6: Contributions and Life of Leonhard Euler
Euler was a Swiss mathematician greatly influenced by Johann Bernoulli.
He began his career in St. Petersburg in 1727, becoming the head of the mathematics department in 1731.
In 1741, he moved to Berlin where he worked at the Berlin Academy for 25 years before returning to St. Petersburg to complete his life.
Page 7: Areas of Work and Innovations
Euler contributed to multiple mathematics areas, including geometry, calculus, trigonometry, algebra, applied mathematics, graph theory, and number theory, among others.
He introduced the concept of a function in its modern usage and popularized notations such as f(x) for functions, e for the natural logarithm base, Σ for summations, and i for the imaginary unit.
Page 8: Sir Isaac Newton
Page 9: Impact and Innovations of Newton
Newton is pivotal in numerous scientific fields: he was a co-inventor of calculus, designed the first reflecting telescope, and laid the foundations of classical mechanics with his work "Philosophiæ Naturalis Principia Mathematica."
He was also the first to analyze white light and identify its component colors and established the three laws of motion known as Newton's laws.
Page 10: Legacy of Sir Isaac Newton
Had Newton not existed, the technological landscape would be markedly different.
Other scientists might have eventually uncovered his ideas, but the timeline for such advancements is uncertain, potentially delaying progress significantly.
Page 11: Introduction to Carl Gauss
Page 12: Overview of Carl Gauss
Gauss is considered one of the greatest mathematicians surpassing even Newton in the mathematical domain.
Born in 1777 in Germany, Gauss displayed early brilliance in mathematics and has extensively influenced algebra, statistics, geometry, optics, astronomy, and various other fields.
Page 13: Contributions to Number Theory
Gauss published "Arithmetical Investigations," crucial in establishing number theory, essential for modern computing.
Number theory is foundational since computers rely fundamentally on binary digits - 1 and 0.
Page 14: Introduction to John von Neumann
Page 15: Life and Work of John von Neumann
A pivotal figure in 20th-century mathematics, von Neumann invented the architecture for virtually all modern computers.
Born in Budapest early in the 20th century, he earned a Ph.D. in mathematics by 22 and also completed a degree in chemical engineering.
He began working at Princeton with Albert Einstein in 1930 at the Institute of Advanced Study.
Page 16: Neumann's Lasting Influence
The devices we use today continuously execute routine processes, initially devised by von Neumann.
Throughout his career, he contributed significantly to set theory, geometry, quantum mechanics, game theory, statistics, and computer science.
He was also involved in the Manhattan Project.
Page 17: Introduction to Alan Turing
Page 18: Turing's Contributions and Legacy
Turing, regarded as the father of computer science, was instrumental during WWII in decoding Nazi codes, including the Enigma machine's messages.
Turing's life ended tragically when he was prosecuted for his sexuality; his death by apparent suicide occurred in 1954.
Page 19: Turing's Impact on Computing
He significantly contributed to the modern computer's designs, and his "Turing machine" concept remains vital today.
The "Turing test" measures artificial intelligence by assessing a program's ability to engage in human-like conversations without detection.
Page 20: Introduction to Benoit Mandelbrot
Page 21: Background of Benoit Mandelbrot
Born in Poland in 1924, Mandelbrot fled to France to escape Nazi persecution and later moved to the U.S. as an IBM Fellow.
His role at IBM allowed him to harness cutting-edge technology for his research, and he passed away from pancreatic cancer in 2010.
Page 22: Mandelbrot's Fractal Geometry
Mandelbrot was renowned for discovering fractal geometry, which involves complex shapes created by simple, self-replicating formulas.
Fractals are crucial to computer animations and graphics, influencing designs in cellphone antennas and microchips due to their efficient space utilization.
Page 23: Mathematics' Role in the Modern World
Mathematics is foundational; it allows for predictions and life-saving strategies, and is interwoven with art and music.
The subject remains enigmatic, full of surprises and wonder.
Page 24: Daily Applications of Mathematics
Mathematics and mathematicians have underpinned modern life, with many concepts applied in everyday scenarios.
Page 25: Examples of Mathematical Applications
Google relies on linear algebra, graph theory, and SVD.
Error correcting codes utilize Galois theory; the Internet depends on network theory, while security involves Fermat's and RSA theories.
Medical imaging and statistics employ techniques like the Radon Transform and contributions from figures like Nightingale.
Page 26: Nature of Mathematics
Mathematics is multifaceted: it is practical, theoretical, a thinking process, an art, and a universal language.
Page 27: Final Thoughts on Mathematics
Mathematics pervades our daily lives; respect for the discipline and its contributors is essential.
As mathematics continues to evolve, it promises to unlock new advancements in our universe.