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This video introduces complex numbers, explains how they are represented mathematically, and shows how to perform operations with them. It also discusses the Mandelbrot set and how it is related to complex numbers.

**00:00:00**In this video, Professor Gilbert Strang from MIT discusses complex numbers, which are simply real numbers together with an imaginary number. Complex numbers can be thought of as imaginary numbers that have a negative square root. This video gives an example of how to solve an equation in complex form, and also discusses how complex numbers can be used to solve polynomial equations. Gauss, one of the greatest mathematicians of all time, proved that all polynomials can be factored into x squared minus one times x squared plus one.**00:05:00**The complex plane is a two-dimensional plane that can be viewed as the mirror image of the real world. It has real numbers (X) and imaginary numbers (I). Each point in the complex plane has a real and imaginary part, which can be represented by a number called Z. Z Bar is the complex number that corresponds to the mirror image of the number 1.**00:10:00**The video explains complex numbers, showing how to add and multiply them using conjugates. It also explains how to find the real and imaginary parts of a complex number.**00:15:00**This YouTube video explains complex numbers in an easy-to-understand way. It shows how multiplication works and how to calculate the imaginary part of a complex number.**00:20:00**This video explains how to calculate complex numbers in polar coordinates, which is simpler than calculating them in real or Cartesian coordinates. The video also provides an example of how to multiply a complex number a plus bi.**00:25:00**This video introduces complex numbers, which are a combination of real and imaginary numbers. The video explains that complex numbers can be represented in terms of real and imaginary parts, and that the real part is always good. It also explains that the angle part of a complex number is the same as the exponential of a real number raised to an imaginary power.**00:30:00**In this video, mathematician and YouTube creator John Horton explains how complex numbers work and how to multiply them. He starts by explaining how complex numbers are represented mathematically, and shows how to multiply two complex numbers using the Euler's Formula. He then goes on to explain how to find the distance and angle between two points using complex numbers. Finally, he demonstrates how to solve problems involving complex numbers using derivatives and the factorial function.**00:35:00**In this video, mathematician and YouTube personality John Green explains complex numbers, which are numbers that can be expressed as a combination of real and imaginary numbers. Complex numbers are important in calculus, and Green provides a simple explanation of how they work.**00:40:00**This video discusses complex numbers and their properties. It explains that, just like real numbers, complex numbers can be expressed in terms of their square roots, cubes, and exponents. It also covers the concept of the Mandelbrot set, which is a set of complex numbers that share some common properties.**00:45:00**The Mandelbrot set is a complex set with pieces that blow up if W is too large, and contains all the W's. The video explains how to calculate W, and shows an example of how it affects the set.**00:50:00**This YouTube video is a presentation about complex numbers, which are a type of number. Complex numbers are made up of real and imaginary parts, and can be used to solve quadratic equations. The key to understanding complex numbers is to see them in the complex plane, and to understand the unit circle in that plane.

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