Let s=1 + x + x^2 + ... say academic morons.
Then, 1/(1-x)=1+ sx
That's almost like the Banash-Tarski paradox which states that one can form two identical balls equal to the same size as the original ball using only the original ball. It's just that in this case, one can form as many sums as one likes, all equal to the original "series":
1+ x(1 + x + x^2 + ...)= 1 + x(1 +x(1+ x + x^2 + ...))
=1 +x+sx^2=1 +x+x^2(1+ x(1+x + x^2 + ...))
=1 +x+x^2+ x^3s=1 +x+x^2+ x^3 + x^4s
Now only a moron can believe that
1+sx= 1+x + sx^2 = 1 +x+x^2+ sx^3 = 1 +x+x^2+ x^3 + sx^4
Try finding a value of s that is not 1/1-x, that will satisfy all of those! Yet, highly stupid academics continue in the this path.
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https://www.youtube.com/watch?v=sAdtI4MIotg
This is the first of three videos on arithmetic with magnitudes in geometry.
There are two main ways of representing ratios.
1. As line segments (magnitudes of any kind will do) separated by a colon
2. As sides of a triangle (proportional ratios are found from similar triangles)
A ratio is literally nothing more than a comparison of two magnitudes. For example, if we write ____ : _________ , this means we are comparing the line lengths that appear on either side of the colon. The comparison is a means of identifying the line that appears BEFORE the colon (called the antecedent) using the line that appears AFTER the colon (called the consequent). This is purely a matter of convention. The antecedent and consequent are called aliquot parts of the ratio. We cannot talk about the properties or anything else with respect to ____, unless we have something to compare it against.
Euclid defined ratio as follows (Book V, Def.3):
Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητά ποια σχέσις.
Thomas Heath translated it incorrectly as:
A ratio is a certain type of condition with respect to size of two magnitudes of the same kind.
Heath's translation is just a bunch of baloney. Heath did not understand what was being said which is not surprising because like most mainstream math academics, Heath had no clue what the number concept meant.
Here is the correct translation:
A ratio is about the quotientness (πηλικότητά) of two homogenous magnitudes.
There is no English equivalent for the Greek word πηλικότητά, so I made up the word "quotientness".
Euclid writes "ποια σχέσις" meaning "what relationship" or "quotientness".
This means that given any ratio, the most important aspect is whether or not the antecedent and consequent parts (called aliquot parts) have a common measure or divisor. Aliquot means part of another whole.
This property of having a common measure is called quotientness. There are two possibilities - either the aliquot parts have a common measure or they do not. Can you think of examples?
1:2 has the property of quotientness.
circumference : diameter does NOT have the property of quotientness.
In geometry it does not matter if the ratios have quotientness or not, because every ratio has the property of proportion with innumerably many other proportional ratios.
Using the scheme described above, we can find any proportional ratio using a parallel line which adjusts the aliquot parts as it moves between the vertical and horizontal lines.
The concept of measure arises once we choose a "unit" ratio, that is, any ratio whose aliquot parts are equal and assumed as the basis of identifying the antecedent of any other ratio.
Since any such ratio
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https://www.youtube.com/watch?v=sznfB7A8fEc
The Historic Geometric Theorem (HGT) was discovered from my New Calculus. The HGT is not New Calculus, but is realised from it:
https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus
A link to the original article here:
https://www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020
A conversation with a ChatBot:
https://www.academia.edu/105028579/Most_advanced_AI_Claude_admits_my_historic_geometric_theorem_of_January_2020_is_profound_discovery
Almost 200 articles here:
https://independent.academia.edu/JohnGabriel30
The first attempt to discredit me and steal my work:
https://www.academia.edu/44928764/How_stupid_are_mainstream_math_professors
Want to get instant updates for the newest math around? Join our discord server! https://discord.gg/CJ9Ks3WerR
Merchandise Store:
https://new-calculus.printify.me/products
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https://www.youtube.com/watch?v=kk99l6E_BDU
The theorem is an interesting and very useful result realised in the New Calculus. You can verify it using your bogus mainstream calculus.
The link to the applet here:
https://drive.google.com/open?id=1rgU-OrtZ48arNLe5rt15Xa-M7qM3B530
What this theorem tells you in simple words is this:
You can pick any point on a curve and calculate the area using equal horizontal distances from the point. IF the arithmetic mean (or average value as known in mainstream nose picking circles) remains unchanged and equal to the y ordinate at the same point, then that point is an inflection point!
This theorem was not realised in your bogus mainstream calculus because you've never understood what it means to be a definite integral which is well defined in the New Calculus. It's also an example of the importance of the arithmetic mean with respect to the core of calculus. You can't understand area, volume, etc without understanding the arithmetic mean. Some cranks like Dennis Muller of logical phalluses seems to think the arithmetic mean is entirely unremarkable.
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https://www.youtube.com/watch?v=bgFTB66SlnE
Some of you might be sick of hearing about the number concept. However, the most important concept in mathematics is the number and how it is realised from geometry together with all its arithmetic properties and operations. Mathematics is the abstract science of measure and number.
Although I have talked about this topic often in previous videos, its importance is paramount and in this video I summarise the concept of number as briefly as I can.
In a previous article I talked about "rates" and how we got rates:
https://www.academia.edu/89274499/What_exactly_is_a_rate
Thank me for enlightening you here:
https://gofund.me/af8a5312
In geometry we can perform all the operations of arithmetic on any two ratios regardless if they have quotientness or not. For example, we can determine pi/sqrt(2) exactly. This is not possible in algebra where we use the abstract unit for all measure.
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https://www.youtube.com/watch?v=tCCuLo00tJs
When the cult of mainstream mathematics academia breaks common sense, they need to issue new decrees (rules or executive orders) in order to keep their bullshit afloat.
Link to article used in presentation:
https://www.academia.edu/76786923/The_idiocy_behind_the_teaching_of_limit_theory_in_mainstream_mathematics
The hilarious act of massaging an expression (through multiplication by 1) which the prima donnas of mainstream mathematics don't understand in order so they can actually replace a symbol with the bullshit concept of "infinity number" doesn't even occur to the baboons of mainstream mathematics academia or the trash heap I call the BIG STUPID.
Ironically, the morons call 1 the "multiplication identity element". This kind of thinking becomes the norm when people totally inept at mathematics (such as MIT math professors and the like) are awarded doctorates for producing exquisite bowel movements on topology and set theory, both anti-mathematical bullshit.
The typical monkey business one finds in mainstream mathematics is demonstrated well by this video:
https://youtu.be/NeZmqpL4aiA
In the video, the presenter imagines that the function f(x)=sqrt(x^2+ax) - sqrt(x^2+bx) does not have a limit when x becomes indefinitely large since ∞ - ∞ =/= 0. He reaches this conclusion by trying f(∞) and quickly realises that the function limit cannot be determined this way, so then he "rationalises" (multiplies by 1 and inadvertently changes the expression which is stated in terms of *symbols* and not *variables* as is commonly believed) the numerator to arrive at:
f(x)=((a-b))/( sqrt(1+a/x) + sqrt(1+b/x)
in which he UNDEFINES f(0) but in his dysfunctional brain somehow defines f( ∞). MAGIC! ?
Algebra is not as sound as geometry. For example, when he divides through by x, isn't he in the least concerned that he is "dividing by 0 (a non-number)" or does he choose to forget? Chuckle.
lim (x → ∞) ( ((a-b))/( sqrt(1+a/x) + sqrt(1+b/x) ) = (a-b)/2
But this is not the same as the lim (x → ∞) (sqrt(x^2+ax) - sqrt(x^2+bx)).
Multiplying by 1 should never change the value of sqrt(x^2+ax) - sqrt(x^2+bx). However, in his example, what he does is equivalent to x = ∞ once the new expression is obtained.
This is the same monkey business in the example: lim (x→1) (x^2-1)/(x-1) which simplifies to lim (x→1) x+1. Now if f(1)=2 when f(x)=x+1, then f(1) must still equal to 2 after f(x) is multiplied by 1.
In his example f(∞) is undefined and suddenly defined after he multiplies by 1. While it is correct to say that f(x) has a limit of (a-b)/2, the arithmetic he used suggests that it's fine to think of the limit as f(∞)=(a-b)/2 which works for one form but not the other, meaning that the two forms cannot be the same.
This nons
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https://www.youtube.com/watch?v=0zh8IzChCJY
Mathematical induction is over 2000 years old. It has NOTHING to do with the bullshit of set theory, Boolean logic, first order logic or any of the rot of the last 150 years in mainstream academia. In fact, most academics deny this theorem when it does not suit them. For example, they claim that 1/3 = 0.333... when induction proves that whatever the evaluation of 0.333... might be, it will always be less than 1/3. Of course the wily reptiles have only Euler's decree in terms of their refutations, that is, S = Lim S:
https://www.linkedin.com/pulse/eulers-worst-definition-lim-john-gabriel
Link to article in induction:
https://drive.google.com/open?id=1Ws8xcHAlnueZsTlQhq-ouOUGB-1yhi1M
Link to proof of most important number theorem:
https://drive.google.com/open?id=1o5kcWvU35tdt_SY83UFnXZAlsKBAsSYH
Link to my free eBook:
https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view
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https://www.youtube.com/watch?v=y0HPJoZhqrI
The arithmetic mean is an incorrect name given to a very important concept in mathematics - the level magnitude or number or term.
The level magnitude is the reason calculus works. Apparently mainstream mathematics academics who are for the most part absolute morons, never did quite understand the purpose of the level term. They knew how to calculate it, but did not understand its significance. As a result, you have millions of stupid teachers still calculating class grade averages which tell the teacher nothing about the students' progress or his ability to teach.
"I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives." -Tolstoy
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https://www.youtube.com/watch?v=tlb6XTfDehA
The only geometric object that has slope as attribute is the straight line. All straight lines have slopes.
The applet for this lesson can be accessed here:
http://nccourse.weebly.com
The article can be accessed here:
https://drive.google.com/open?id=0B-mOEooW03iLYnNjVzJ3emFFS2s&authuser=0
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https://www.youtube.com/watch?v=6hinFG-4V3I