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.docxThe mathematics we learn in school is only a glimpse of the vast and diverse field of mathematics. In this video, I aim to show you all the amazing things you don't know about it.
History of Mathematics
The origin of mathematics lies in counting, a skill found in other animals. Evidence for human counting dates back to prehistoric times, with check marks made in bones.
Throughout history, there have been numerous innovations: the first equation by the Egyptians, advancements in geometry and numerology by the Greeks, negative numbers in China, and zero in India. Persian mathematicians wrote the first book on algebra in the Golden Age of Islam. Mathematics experienced a boom during the Renaissance, along with the sciences, leading to new discoveries.
Modern Mathematics
Modern mathematics can be divided into pure mathematics and applied mathematics, both studying mathematics for its own sake and for solving real-world problems. Mathematicians often create theories that may not have immediate applications, but later they discover that these theories are exactly what they need.
Pure Mathematics
Pure mathematics can be divided into several sections:
Numbers
The study of numbers begins with natural numbers and arithmetic operations, moves on to integers, rational numbers, real numbers, and complex numbers. Some numbers have interesting properties, such as prime numbers and pi.
Structures
Algebra contains the rules for manipulating equations, including vectors and matrices. Linear algebra captures the rules for multi-dimensional numbers.
Group Theory
Number theory studies the properties of numbers, combinatorics looks at discrete structures like trees and graphs, group theory at objects related in groups, and order theory at arrangements of objects.
Geometry
Geometry includes Pythagoras, trigonometry, and fractal geometry. Topology looks at properties of spaces where deformation is allowed. Measure theory ties numbers and spaces together. Differential geometry looks at shapes on curved surfaces.
Changes
Calculus involves integrals and differentials. Vector calculus looks at changes for vectors. Dynamical systems study systems that evolve in time. Chaos theory studies sensitive dynamical systems. Complex analysis looks at properties of functions with complex numbers.
Applied Mathematics
Applied mathematics uses mathematics in physics, chemistry, engineering, and other sciences. Control theory is used in complex electrical systems. Numerical analysis uses approximations to solve complex problems. Game theory is used in economics, psychology, and biology. Probability and statistics study random events and data, related to mathematical finance. Optimisation tries to find the best choice amongst many options.
Computer Science
Computer science derives its rules from pure mathematics. Machine learning uses many areas in mathematics. Cryptography uses combinatorics and number theory.
Foundations of Mathematics
The foundations of mathematics ask about the rules of mathematics and their consistency. Mathematical logic, set theory, and category theory try to answer these questions. Gödel's incompleteness theorems suggest that mathematics does not have a complete and consistent set of axioms.
five questions on the text:
What were some of the earliest mathematical innovations, and which civilizations contributed to them?
How do pure mathematics and applied mathematics differ, and why do they often intersect?
What are some key areas of pure mathematics, and how do they relate to each other?
How is mathematics applied in fields like physics, engineering, and computer science?
What fundamental questions do the foundations of mathematics seek to answer, and what is the significance of Gödel's incompleteness theorems?