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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.