In this video I illustrate how complex numbers can be interpreted as vector rotations. This follows from my earlier video on equating the Rotation Matrix as a complex number. Rotating a vector by 0° is equivalent to multiplying the vector by the Identity Matrix [I]. Rotating a vector by 90° is equivalent to multiplying the vector by the Imaginary Matrix [i]. This is quite amazing, both in that complex numbers can be interpreted as vector rotations and also in that the imaginary unit i can be viewed as a 90° rotation! This video was taken from my earlier video listed below: - Complex Numbers as Rotation Matrices: https://youtu.be/Mgp2vrQeLEw - Video notes: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices - Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0E6x0dZEAx77Kqv7LLxYdbx Related Videos: Vectors and the Geometry of Space Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I do a debunk of Eric Dubay's debunk of my earlier Flat Earth debunk and show that the Mountain's shadow forms opposite triangular shapes depending on if it is on top or below the clouds. When the Sun goes below the horizon, it can form an epic shadow of large objects such as Mountain's or skyscrapers unto the underside of the clouds. In this case, the shadow has a triangular shape where the pointy part is near the peak of the Mountain and the wider base is on the far side away on the clouds. If the Mountain peak is above the clouds, then as the Sun approaches near the horizon, it can form a shadow above the clouds that is in reverse direction as to the one below the clouds. This is pretty straight forward stuff, but that doesn't stop professional liar Eric Dubay from lying about what our eyes clearly see. I debunk Eric's many lies in methodical fashion. Video links: - MES Links: https://mes.fm/links - Debunking Flat Earth Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUyOvHJiyX74GmJ79exCkh - Debunking Flat Earth: Mountain's Shadow on Clouds: https://youtu.be/j6Ey7zUpoos - Thumbnail: https://t.me/meslinks/22541 - Shadow on Clouds: https://flatearth.ws/shadow-on-clouds - Professional liar Eric Dubay pushing bogus disinfo: https://youtu.be/xDH6a3TJqpA - Lie 1: Alludes to the Sun reflecting to ground as a possible explanation. - Lie 2: Says ALL Mountain cloud shadow images are when mountain peak is above clouds. - Lie 3: Says shadow is from above the clouds. (Clouds are in REVERSE direction) - Lie 4: Says Sun never goes below the horizon. - Why is the top of the mountain illuminated but below it is not? What is causing that shadow in FRONT of the Mountain? Is the Sun below the clouds (answer is yes). - Lie 5: He shows the clouds in REVERSE direction to what it should be, confirming he is deliberately misleading his audience aka lying. - Lie 6: His graphic confirms that he is ignoring the shadow on the front of the Mountain. - Continues lying at end of video etc. - Good debunk: Mount Rainier's Shadow and Electroflerfspectivism. https://youtu.be/-MhWmD05dN0 - Many photos of Mountain peak CLEARLY below clouds: https://presearch.com/images?q=mountain%27s+shadow+on+clouds - Mountain Shadow Above and Below Clouds Experiments: https://youtu.be/sAhQN86ZBy4 - DRAFT Experiments: https://www.youtube.com/playlist?list=PLai3U8-WIK0FfZ_7hUuyO7xN8lp5oaDiu - Mount Rainier Epic photo: https://twitter.com/KClarkWx/status/1325837129851703298 Timestamps: - Intro and Links: 0:00 - MES Links: 1:40 - Previous video: Mountain's shadow on clouds: 2:10 - Debunking Eric Dubay: 3:07 - Lie 1: Eric suggesting shadow reflections as an explanation: 3:41 - Lie 2: Eric states ALL images have Mountain peak above clouds: 4:40 - Lie 3: Eric says shadow is above the clouds: 6:06 - Lie 4: Eric says Sun never goes below the horizon: 6:52 - Top of the mountain is illuminated but not below it: 7:18 - Lie 6: Eric confirms that he is ignoring the shadow IN FRONT of the Mountain: 8:23 - Lie 5: Eric shows clouds in REVERSE direction to what it should be: 9:16 - Good debunk by Mike the Engineer: 10:22 - MES Experiment: Mountain shadow above and below clouds: 11:43 - Draft MES Experiments and Outro: 15:17 . ------------------------------------------------------ Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over an in-depth derivation to determine an explicit formula for the n-th Fibonacci number. The Fibonacci sequence is the sequence of terms where the previous 2 numbers are added together. Prior to making this video, I had not known there was an explicit formula to determine any given number in the Fibonacci sequence. The derivation involves first determining the Maclaurin series for the given function by writing it out as a power series. Doing so yield the coefficients of the Maclaurin series to be just the Fibonacci numbers. The next part of the derivation is to solve for the series of the given function again but this time using a different method, by using partial fractions. This yields two partial fractions that are in the form of the sum of a convergent geometric series. Replacing the partial fractions with their corresponding Geometric series, simplifying the result, and comparing with our prior Maclaurin series, I note that we have in fact an explicit formula for the n-th Fibonacci number. Absolutely amazing and mind-boggling stuff! The timestamps of key parts of the video are listed below: - Problem 24: Series involving Fibonacci series: 0:00 - Solution to (a): The function as a Maclaurin series: 1:37 - Comparing coefficients of powers of x: 7:25 - Each coefficient is equal to the n-th Fibonacci number: 11:27 - Solution to (b): Explicit formula for the n-th Fibonacci number: 13:06 - Completing the square: 13:32 - Writing f(x) as partial fractions: 21:41 - Summary of f(x) as partial fractions: 29:13 - Partial fractions are in the form of the sum of a convergent geometric series: 35:15 - Simplifying the resulting series: 43:52 - Explicit formula for the n-th Fibonacci number: 49:24 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . ------------------------------------------------------ Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps ... https://www.youtube.com/watch?v=vRfeWzBNPF8

In this video I go over the distance formula between a point and a line once again, but this time take a look at a Geometric Proof. This proof, just line in my earlier Algebraic Proof, is only valid for slanted lines but nonetheless the final result is still applicable for horizontal and vertical lines. The proof uses the fact that we make a triangle between the point and the line, we get an angle that is also shared by a second triangle made by the slope of that line. This connection allows us to determine the distance using just the constants of the Line and coordinates of the point. This is a very unique derivation and is easier to go through than the algebraic proof. This is a great video to understand how there are often many different mathematical approaches that can be used to solve the same problem, so make sure to watch this video! Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhv9uc7-3FwaU7MwHaw View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-point-to-line-distance-formula-geometric-proof-1 Related Videos: Point-To-Line Distance Formula: Algebraic Proof: https://youtu.be/tngPrM2d-OM Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8 . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: Official Website: https://MES.fm Steemit: https://steemit.com/@mes Gab: https://gab.ai/matheasysolutions Minds: https://minds.com/matheasysolutions Twitter: https://twitter.com/MathEasySolns Facebook: https://fb.com/MathEasySolutions LinkedIn: https://mes.fm/linkedin Pinterest: https://pinterest.com/MathEasySolns Instagram: https://instagram.com/MathEasySolutions Email me: contact@mes.fm Try our Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Try our Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps ... https://www.youtube.com/watch?v=9ilcXZn9CfE