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Lessons
(Select One)
- 1. The Primes
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- The properties and patterns of prime numbers — whole numbers that are
divisible only by themselves and one — have been a source of wonder across
cultures for thousands of years, and the study of prime numbers is
fundamental to mathematics. This unit explores our fascination with primes,
culminating in the million-dollar puzzle of the Riemann Hypothesis, a
possible description of the pattern behind the primes, and the use of the
primes as the foundation of modern cryptography.
- 2. Combinatorics Counts
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- Counting is an act of organization, a listing of a collection of things
in an orderly fashion. Sometimes it's easy; for instance counting people in
a room. But listing all the possible seating arrangements of those people
around a circular table is more challenging. This unit looks at
combinatorics, the mathematics of counting complicated configurations. In an
age in which the organization of bits and bytes of data is of paramount
importance — as with the human genome — combinatorics is essential.
- 3. How Big Is Infinity?
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- Throughout the ages, the notion of infinity has been a source of mystery
and paradox, a philosophical question to ponder. As a mathematical concept,
infinity is at the heart of calculus, the notion of irrational numbers — and
even measurement. This unit explores how mathematics attempts to understand
infinity, including the creative and intriguing work of Georg Cantor, who
initiated the study of infinity as a number, and the role of infinity in
standardized measurement.
- 4. Topology's Twists and Turns
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- Topology, known as "rubber sheet math," is a field of mathematics that
concerns those properties of an object that remain the same even when the
object is stretched and squashed. In this unit we investigate topology's
seminal relationship to network theory, the study of connectedness, and its
critical function in understanding the shape of the universe in which we
live.
- 5. Other Dimensions
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- The conventional notion of dimension consists of three degrees of
freedom: length, width, and height, each of which is a quantity that can be
measured independently of the others. Many mathematical objects, however,
require more — potentially many more — than just three numbers to describe
them. This unit explores different aspects of the concept of dimension, what
it means to have higher dimensions, and how fractional or "fractal"
dimensions may be better for measuring real-world objects such as ferns,
mountains, and coastlines.
- 6. The Beauty of Symmetry
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- In mathematics, symmetry has more than just a visual or geometric
quality. Mathematicians comprehend symmetries as motions — motions whose
interactions and overall structure give rise to an important mathematical
concept called a "group." This unit explores Group Theory, the mathematical
quantification of symmetry, which is key to understanding how to remove
structure from (i.e., shuffle) a deck of cards or to fathom structure in a
crystal.
- 7. Making Sense of Randomness
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- Probability is the mathematical study of randomness, or events in which
the outcome is uncertain. This unit examines probability, tracing its
evolution from a way to improve chances at the gaming table to modern
applications of understanding traffic flow and financial markets.
- 8. Geometries Beyond Euclid
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- Our first exposure to geometry is that of Euclid, in which all triangles
have 180 degrees. As it turns out, triangles can have more or less than 180
degrees. This unit explores these curved spaces that are at once otherwordly
and firmly of this world — and present the key to understanding the human
brain.
- 9. Game Theory
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- Competition and cooperation can be studied mathematically, an idea that
first arose in the analysis of games like chess and checkers, but soon
showed its relevance to economics and geopolitical strategy. This unit shows
how conflict and strategies can be thought about mathematically, and how
doing so can reveal important insights about human and even animal
behaviors.
- 10. Harmonious Math
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- All sound is the product of airwaves crashing against our eardrums. The
mathematical technique for understanding this and other wave phenomena is
called Fourier analysis, which allows the disentangling of a complex wave
into basic waves called sinusoids, or sine waves. In this unit we discover
how Fourier analysis is used in creating electronic music and underpins all
digital technology.
- 11. Connecting with Networks
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- Connections can be physical, as with bridges, or immaterial, as with
friendships. Both types of connections can be understood using the same
mathematical framework called network theory, or graph theory, which is a
way to abstract and quantify the notion of connectivity. This unit looks at
how this branch of mathematics provides insights into extremely complicated
networks such as ecosystems.
- 12. In Sync
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- Systems of synchronization occur throughout the animate and inanimate
world. The regular beating of the human heart, the swaying and near collapse
of the Millennium Bridge, the simultaneous flashing of gangs of fireflies in
Southeast Asia: these varied phenomena all share the property of spontaneous
synchronization. This unit shows how synchronization can be analyzed,
studied, and modeled via the mathematics of differential equations, an
outgrowth of calculus, and the application of these ideas toward
understanding the workings of the heart.
- 13. The Concepts of Chaos
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- The flapping of a butterfly's wings over Bermuda causes a rainstorm in
Texas. Two sticks start side by side on the surface of a brook, only to
follow divergent paths downstream. Both are examples of the phenomenon of
chaos, characterized by a widely sensitive dependence of the future on
slight changes in a system's initial conditions. This unit explores the
mathematics of chaos, which involves the discovery of structure in what
initially appears to be randomness, and which imposes limits on
predictability.
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About This Course
This is a video course for high
school teachers and college-level learners. There are 13 half-hour video
programs.
Acknowledgements
Video for Mathematics
Illuminated and
the individual program descriptions are provided courtesy of Annenberg/CPB.
This site is not affiliated with nor endorsed by
Annenberg/CPB |
