Math Easy Solutions

- MES

In this video I go over further into Euclidean Division and this time look at the theorem and algorithm for univariate (i.e. single-variable) polynomials. The theorem is very similar to that for integers which I covered in my earlier video, in that both cases require proving that the associated Euclidean Division Algorithm is valid to ultimately prove that the theorem is valid. The main difference between the two algorithms is that for the integer case each step involved adding 1 to the quotient; whereas for the polynomial case we need to apply a special value, s, to ensure that the degree of the remainder keeps decreasing incrementally.

The Euclidean Theorem for Division of Polynomials is that for any two univariate polynomials a and b, where b is not equal to 0 (to avoid the case of a/0 or dividing by zero) there are associated polynomials q and r called the quotient and remainder, respectively, such that a = bq + r and the degree of r is less than the degree of b. Note also that in case the remainder is 0 and the degree of b is 0, deg(0) is defined as negative. The Division Algorithm involves starting from q = 0 and then applying the theorem formula to obtain r and checking if the degree of r is less than the degree of b. If it is, then we are done. If it is not, then we add a special polynomial s, which I show is such that involves the leading coefficients of both the remainder and the divisor, b, as well as their degrees (or number of their highest power). The selection of s is such that the remainder keeps subtracting by 1 its degree until ultimately it is less than the degree of b. Thus ultimately the algorithm is proved and thus so is the theorem!

The theorem and associated algorithm is the basis for polynomial long division where a/b = q + r/b and can be rearranged so that theorem holds true a = bq + r. This is a truly fascinating concept and make sure to walk-through both the algorithm as well as several examples to make Euclidean Division as clear as possible, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh4EItHHqj75MMNeJbQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/euclidean-division-of-polynomials-theorem-and-proof

Related Videos:

Euclidean Division of Integers: Theorem and Proof: https://youtu.be/66juubotzi0
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Long Division by Hand - An in depth look: http://youtu.be/giBZg5Vqryo
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y .

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In this video I go over Part 2 of the Laboratory Project: Bezier Curves. In this part I look at Question 2 of the project, which asks us to prove that the tangent line at the first control point, from the graph produced in Part 1, extends through to the second control point. Similarly I prove that this is the case for the tangent line at the 4th control point, which passes through to the 3rd control point. To prove this I show that we need to first calculate the derivative dy/dx by using it’s parametric form, i.e. (dy/dt)/(dx/dt), and then showing that it is equal to the slope of the line connecting the two control points. The good thing when calculating the derivative is that the Bezier curves are defined by identical x and y parametric equations, albeit the x/y’s are interchanged, so cuts the effort in calculating dy/dx by half. This is a very interesting video on the nature of Bezier curves, cubic Bezier curves to be exact, so make sure to watch this video, and stay tuned for later parts!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhucAFpvC1Wz3pa2I1g

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/laboratory-project-bezier-curves-part-2

Related Videos:

Laboratory Project: Bezier Curves Part 1: https://youtu.be/S7aApca_B04
Parametric Calculus: Surface Area Part 1: https://youtu.be/4bMEIf6WD8M
Parametric Calculus: Arc Length Part 1: https://youtu.be/AWvJDK-m6wQ
Parametric Calculus: Areas: https://youtu.be/XdplYV61xlM
Parametric Calculus: Tangents: https://youtu.be/deQwD2o0Sas
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE .

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In this video I go over a pretty extensive “formal” proof of what otherwise seems to be a straight forward theorem known as the Euclidean Division. When an integer, known as the dividend, is divided by another integer, known as the divisor, we get an answer broken up into two parts. The first part is an integer known as the quotient that represents how many times the divisor can divide evenly into the dividend. The second part is the remaining fraction of the divisor that doesn’t divide cleanly, and in which the numerator is known as the Remainder. When working this out by hand such as the division 9/2 = 4+1/2 we can clearly see the breakdown of this division.

The formulization of this process is known as Euclidean Division of Integers, and the theorem is as follows: The division a/b of two integers, where b is not equal to zero, involves the existence and uniqueness of two integers q and r, such that a = bq + r for 0 ≤ r less than |b|. Now the proof of this theorem is also in the algorithm in obtaining these integers q and r. First I show that whether a or b are positive or negative, the theorem gets reduced to just the positive case. Working in an incremental step by step method, I show the Division Algorithm needed to obtain these numbers, thus proving their existence.
The uniqueness proof involves some out of the box thinking to first assume that there are other values that q and r can take to fit the theorem, but then showing that this is impossible. This is a very interesting part of the overall derivation so I highly recommend you watch and understand the very unique reasoning applied in it!

This is a very interesting and extensive proof video of a seemingly basic division procedure but its applications are far-reaching, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh4Bp4CwnSx7ecaSmQA

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/euclidean-division-of-integers-theorem-and-proof

Related Videos:

Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Long Division by Hand - An in depth look: http://youtu.be/giBZg5Vqryo
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y .

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In this video I go over Part 3 of the Laboratory Project: Bezier Curves. In this part I look at Question 3 of the project, which looks at moving the second control point in hopes of obtaining a loop within the Bezier curve. I test out different values for the second control point and after experimenting, it appears that in general moving the right of the third control point creates a looped Bezier Curve. What’s also interesting is that the findings of Part 2 hold true for this looped curve, in that the tangent lines at the first control points still pass through the second control point; and like-wise the tangent line at the fourth control point passes through the third control point. This is a great video to understand how manipulating the control points changes the resulting curve in a controlled way, hence the name “control”. This behavior is why the Bezier Curves are used in many computer-aided design technologies, including simply printing letters. I will illustrate this even further in later parts, so stay tuned!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhucQPtmk4LWWlNrxzA

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/laboratory-project-bezier-curves-part-3

Related Videos:

Laboratory Project: Bezier Curves Part 2: https://youtu.be/HN-xNfncklU
Laboratory Project: Bezier Curves Part 1: https://youtu.be/S7aApca_B04
Parametric Calculus: Surface Area Part 1: https://youtu.be/4bMEIf6WD8M
Parametric Calculus: Arc Length Part 1: https://youtu.be/AWvJDK-m6wQ
Parametric Calculus: Areas: https://youtu.be/XdplYV61xlM
Parametric Calculus: Tangents: https://youtu.be/deQwD2o0Sas
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE .

In #PizzaGate Part 30 I revisit the infamous Madeleine McCann case and illustrate how it is an extremely sensitive issue for the current cowardly elites, and may just go right to the very top of the global political establishment. First off I recap the many pedophilia circumstances surrounding the case, such as the witness statement testimonies from the Gaspars’ and Yvonne Martin that suggest McCann family friend David Payne and Gerry McCann himself may in fact be pedophiles; and pedophile sir Clement Freud “befriending” the McCanns. Kate McCann’s book also makes bizarre references to pedophiles and Madeleine’s genitalia. The pedophilia theme itself is not the only disturbing issue surrounding the McCann case.

The efits of a potential suspect produced by the McCann team in fact look identical to two Washington DC power brokers and lobbyists in John and Tony Podesta. Now it is a stretch to say that these United States lobbyists “kidnapped” Madeleine, but that is mainly because according to Richard D. Hall’s brilliant Madeleine McCann documentaries, http://madeleinefilms.net, there was no kidnapping. Madeleine almost certainly died before the reported “disappearance”. Also the efits are almost certainly a forgery since the Irish Smith family said to have described the suspect had made initial statements to the police that they couldn’t identify the suspect from a photograph or in person. So how were the efits produced??

The efits were produced by ex-MI5 agents hired by the McCanns allegedly a year after Madeleine’s “disappearance” and were withheld by the McCanns and/or police until made public in 2013 by BBC Crimewatch yet they never mentioned the efits were produced by the McCanns! The company behind the efits was a shady organization Washington DC based company known as Oakley International and run an even shadier Kevin Richard Halligan. In this video I illustrate quite clearly that Halligan is almost certainly...

In this video I go over the graph of hyperbolic sine or sinh(x) in a step-by-step manual method. To graph the function, I first show how to find the domain, range, concavity, and critical points. The graph appears like a typical cubic, or x^3, function but I show that there are some stark differences, especially at larger values of y. One of the main differences is that the derivative, or slope, of sinh(x) is always positive and thus we never get a horizontal tangent as we do with x^3. Also, when we change the y-scale to very large values in the thousands to millions, we see that the sinh(x) becomes much more steep than the cubic function! This is actually quite amazing and is one of the main properties of hyperbolic functions that make it useful for many different real-world applications. This is a very good video in understanding the behavior of hyperbolic functions, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhv5kAoWHLnCYKJDIVQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-hyperbolic-functions-graphing-sinh-x

Related Videos:

Hyperbolic Functions: Graphing cosh(x) - (Revisited): https://youtu.be/iug2YP7hX1I
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Guidelines to Curve Sketching: http://youtu.be/tOn7ZSAntKs
Conic Sections: Parabolas: Definition and Formula: https://youtu.be/kCJjXuuIqbE .

In this video I go over deriving the formula for the shortest distance between a point and a line. There are several different ways of deriving this, and in this video I use an algebraic derivation. This method involves using the fact that the shortest distance between a point and a line is the line that is perpendicular to the other line. Thus both lines are negative reciprocals of each other. Combined with the Pythagorean theorem to obtain the square of the distance in determines of the squares of the differences in x and y, we can then play around with some algebra to obtain our final formulation.

Note the general proof used in this video involves a derivation which is not valid for vertical or horizontal lines BUT the final result still holds true nonetheless! This is actually a very interesting result and illustrates how we must always use mathematical rigor regardless of whether the final formula is valid for cases that weren't valid in the proof methodology; so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhv8AcV6RCgPi8zuO4g

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-point-to-line-distance-formula-algebraic-proof

Related Videos:

Negative Reciprocals and Perpendicular Lines: http://youtu.be/Ue7FmrfmuX4
Foil Method - Simple Proof and Quick Alternative Method: http://youtu.be/tmj_r94D6wQ
Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8 .

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 .

In this video I go over part 4 of the Laboratory Project: Bezier Curves, and show how we can use Bezier curves to represent letters, specifically the letter C. We can do this by modifying the control points in such a way that the curve within forms a shape that looks like the letter C. From the early parts to this laboratory project, I showed that the tangent lines at the endpoints of a cubic Bezier curve extend through the middle control points. Using this fact, it makes it easy to guess how the shape of a Bezier curve would look like without actually calculating it using a graphing calculator. I also go over some even more AMAZING features of the Desmos calculator (https://www.desmos.com/calculator/yazgguimuo), and show how can manipulate and drag the control points in real-time to see how the curve gets affect. This is a very important and interesting video because it shows some of the mathematics behind the stuff we take for granted, such as the letters on a screen, or the process in which a printer prints out different characters and symbols. Also you don’t want to miss me playing around with the Desmos calculator so this video is a MUST WATCH!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhucYuMHZlv6EvhrhCQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-laboratory-project-bezier-curves-part-4-amazing-calculator

Related Videos:

Laboratory Project: Bezier Curves Part 3: https://youtu.be/MzsT3FzrkU0
Laboratory Project: Bezier Curves Part 2: https://youtu.be/HN-xNfncklU
Laboratory Project: Bezier Curves Part 1: https://youtu.be/S7aApca_B04
Parametric Calculus: Surface Area Part 1: https://youtu.be/4bMEIf6WD8M
Parametric Calculus: Arc Length Part 1: https://youtu.be/AWvJDK-m6wQ
Parametric Calculus: Areas: https://youtu.be/XdplYV61xlM
Parametric Calculus: Tangents: https://youtu.be/deQwD2o0Sas
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE

In this video I go over yet another video on deriving the distance formula between a line and a point. This time I solve it using a second Geometric Proof which is very similar to the first method but this time I use an ingenious strategy of identifying two triangles with identical areas but one of which contains the distance from the point to the line. To do this I move the triangle formed by the slope of the line directly on top of the other triangle which contains the distance. Thus we can develop two formulas for the overall area by identifying two different amplitudes or heights, and bases of the triangles. And as in my other derivations, the proof is only valid for slanted lines but nonetheless the final result is applicable for horizontal and vertical lines. This is a great video to understand the many different methods, as well as some ingenious thinking outside the box methods, that can be used to solve the same exact problem, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh4AMlboW9kw17Ko9IQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-point-to-line-distance-formula-geometric-proof-2

Related Videos:

Point-To-Line Distance Formula: Alternative Formula: https://youtu.be/z56Gx6m9uqs
Point-To-Line Distance Formula: Geometric Proof #1: https://youtu.be/9ilcXZn9CfE
Point-To-Line Distance Formula: Algebraic Proof: https://youtu.be/tngPrM2d-OM
Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8
Negative Reciprocals and Perpendicular Lines: http://youtu.be/Ue7FmrfmuX4 .

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Created 2 weeks, 5 days ago.

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I mainly teach math, but also do controversial videos that are censored on YouTube, as well as creating #FreeEnergy tech!