Complex numbers. This is an introduction to complex numbers. It includes the mathematics and a little bit of history as well. It is intended for a general audience. The necessary background in a familiarity with ordinary real numbers (all positive and negative numbers and zero) and algebra. In one section some background in trigonometry is needed as indicated with the symbol. That section goes further into complex numbers and is optional in an introduction. I. 1. Solution of quadratics, solution of cubics 2. 3. The Fundamental Theorem of Algebra proved! II. 4. Notation, arithmetic operations on C, parallelogram rule, addition as translation, negation and subtraction 5. The unit circle, the triangle inequality 6. Multiplication done algebraically, multiplying a complex number by a real number, multiplication and absolute value, powers of i, roots of unity, multiplying a complex number by i, a geometric interpretation of multiplication 7. 8.
Reciprocals done geometrically, complex conjugates, division 9. Plug Flow Reactor - Free PPT downloads. Plug Flow Model - Chmltech - Biodiesel Projects Multiphase Chemical Reactor Engineering Quak Foo Lee Ph.D. Candidate Chemical and Biological Engineering The University of British Columbia Different Types of Reactor Fixed Bed Rector Fluidized Bed Reactor Batch Reactor Straight Through Transport Reactor Slurry Phase Distillate Reactor Packed ... Gas-Solid Reactor Models - Chmltech - Biodiesel Projects ... fluidized beds The Packed Bed Reactor The flow and contacting can be simply represented by the plug flow model. ... (fast) fluidized beds The Packed Bed Reactor The flow and contacting can be simply represented by the plug flow model.
Reactor Design ... Reactor Design - Chemical Engineering Department, University ... Reactor Design S,S&L Chapter 7 Terry A. Slide 1 * The chemical industry has some of the most easily-controllable processes in existence. Chapter 5a - CE 351 Chemical Reaction Engineering: Reactor Design Project PowerPoint Presentation Bioreactor Operation Modes-2. ... ... ... Uwe Alfer Privatseiten Fibonaccifolge und Goldener Schnitt in der Phyllotaxis.
Maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/pinecone3.gif. Time and Quantum Physics relationships to Phi. The Golden Ratio seems to be appearing in several places in the Quantum Physics Model. Because an electron has an electric charge and an intrinsic rotational motion, or spin, it behaves in some respects like a small bar magnet and is said to have a magnetic moment. Because the electron also has mass, it behaves in some respects like a spinning top, and is said to have spin angular momentum. The g factor of the electron is defined as the ratio of its magnetic moment to its spin angular momentum. The electron g-factor is due to the stretching of space-time as the electron spins at the speed of light. Mathematically, the electron g-factor is approximately: gfactore = -2 / sin (Ø) and the proton g-factor is approximately: gfactorp = 2Ø / sin (1/Ø) Thus it appears that the Golden Ratio, or Phi, is a constant produced by time.
The National Institute of Standards and Technology (NIST) states these gfactor constants as per the table below. Insights on this page were contributed by David W. Spirals illustrating Phi. Fibonacci numbers and Phi are related to spiral growth in nature If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number. This property results in the Fibonacci spiral, based on the following progression and properties of the Fibonacci series: 12 + 12 + 22 + 32 + 52 = 5 x 8 12 + 12 + . . . + F(n)2 = F(n) x F(n+1) A Golden spiral is very similar to the Fibonacci spiral but is based on a series of identically proportioned golden rectangles, each having a golden ratio of 1.618 of the length of the long side to that of the short side of the rectangle: The Fibonacci spiral gets closer and closer to a Golden Spiral as it increases in size because of the ratio of each number in the Fibonacci series to the one before it converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively) Golden spiral in human ear Be Sociable.
Mineral Growth. Math Mysticism: is Hurricane Shape a Fibonaci Spiral? Please beware of the golden ratio math mysticism spreading online. Many people are sharing this image online. It is true that the hurricanes have the general shape of a spiral. In particular, it's a equiangular spiral (aka logarithmic spiral), which is the shape that many (but not all) natural growth takes (⁖ Seashells), but saying that hurricane has golden-spiral is completely baseless, it's like seeing Jesus face on Mars.
The so-called “Fibonacci spiral” (aka golden-spiral) you see in the pic is made up of parts of circles, stacked on blocks of squares. (equiangular spiral is a family of spirals. Also, hurricane does not really have a definite shape, we can only say it's roughly that of equiangular spiral based partly on physics and appearance. If you don't have a degree in math, you might be all confused about this Fibonacci number , the Golden ratio, and the various spirals. For highschool-level explanation of the equiangular spiral, see: Equiangular Spiral.