Those who were banned can rejoin at this link: Join our discord server! https://discord.gg/CJ9Ks3WerR ... https://www.youtube.com/watch?v=3LlSnqcdNKM

Long before the concept of number was introduced, arithmetic had already been defined with ratios of magnitudes. You have learned rote fashion that: p/q ± r/s = (ps ± qr)/qs p/q -:- r/s = ps / qr p/q x r/s = pr / qs These operations are derived directly from the geometrical arithmetic with magnitudes. Much of the history is blurred as the arithmetic operators transition from operands that are magnitudes to ones that are numbers through the abstract unit whose size is irrelevant. A quotient is a ratio that is measured by any two magnitudes which share a common measure of divisor. It's not enough to call 2/4 a quotient unless both 2 and 4 have a common divisor. In the geometrical arithmetic operation, to say that the ratio p:q is a quotient means that both p and q have a common divisor d such that d and/or equal parts of d measure both p and q. The product (or multiplication) of two magnitude ratios is the quotient of either magnitude with the reciprocal of the other. It should come as no surprise that multiplication is defined in terms of division. Link to applet in this video: https://drive.google.com/file/d/14QpoXr3dtozkj0PbbiyMxUnYgq4bCL4Y Fraction arithmetic and similar triangles: https://drive.google.com/file/d/1vVhVTK3oJOc57asItXP07PRiHbgNg0qA Book 5, Proposition 12: https://drive.google.com/file/d/1oBBFAd0pymOTK9EcdWAGWd9sQd_ICjkZ Prologue to Euclid's Elements - A complete systematic derivation of Euclid's 5 requirements commonly misinterpreted as axioms: https://drive.google.com/file/d/1SYT-MbYtXUAYgwPeTgZ8QC8gxNpj-fap These are all available in one folder: https://drive.google.com/drive/folders/0B-mOEooW03iLd1Z2cVRtOElpYms Request to download the above folder free of charge and you'll gain access to all these applets and many others! Also, download my free eBook, the most important mathematics book ever written: https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO I am the great John Gabriel, discoverer of the New Calculus, the first rigorous formulation of calculus in human history. I am smarter than any other math academic alive today. If you want to learn more, nominate me for a TED talk. There is much more I have not shared because of the time constraints and also it is a pity that I cannot answer questions in real time from those who sincerely want to learn. I do know better than anyone else. ... https://www.youtube.com/watch?v=LOQRDRLvQz0

Unlike the idiot David Meel and his garbage article (https://www.academia.edu/65676754/Advanced_Mathematics_for_Secondary_Teachers_A_Capstone_Experience), I explain to you in sound terms how you got the operations of arithmetic from geometry: https://www.academia.edu/69488136/Theory_of_fractions_from_Book_5_of_Elements_for_Dummies No, you don't get to define p/q + r/s to be any old nonsense you like. p/q + r/s is exactly equal to (ps + qr)/(qs) and is geometrically derived. Thanks to the Ancient Greeks whose minds weren't polluted by set theoretic garbage! Thanks to their clarity of thought, we can perform the basic operations of arithmetic and in the case of mainstream academia, without any understanding thereof whatsoever! If you asked a fool like Meel to explain to you how the arithmetic operations are the same for all numbers, he will not be able to! His answer will be something to the effect of "It is defined that way." Why is 1/2 x 3/4 = 3/8 which is less than both the operands and why is 3 x 5 = 15 where 15 is greater than both the operands? Meel will not be able to explain to you how these operations can be done exactly the same way, regardless of the type of fraction - whether natural number or otherwise. His syphilis brain has been infected with the rot of Georg Cantor's set theory. To say as the ignoramus Meel says: Let ∼ denote the equivalence relation on F given by a/b ∼ c/d if and only if ad = bc. For the remainder of this section, when we use the word fraction, we shall mean an element of F. We define the operations of addition and multiplication on F by a/b + c/d = (ad + bc) / bd, and a/b · c/d = ac/bd. shows a complete lack of understanding. Firstly, the condition ad = bc is the reason a/b is EQUAL or PROPORTIONAL to c/d and the reason he can claim an "equivalence relation". Multiplication has already been assumed here. The dimwit then goes on to state how fractions are multiplied when he has already assumed the fact that ad=bc implies and is implied by a/b = c/d. There is so much circularity and lack of understanding that trying to correct fools like Meel is a lost cause. They are incorrigible idiots. The injection of non-mathematics such as "set theory" which includes fields, sets, elements, etc, facilitates the endless art of hand waving that is so prevalent among mainstream mathematics baboons. Thank me for enlightening you here: https://gofund.me/af8a5312 ... https://www.youtube.com/watch?v=vQYOHE6ZPsA

Are you ready to learn the fourth and final operation in working with ratios? Multiplying ratios may seem tricky, but it's actually just the reciprocal of division. First, we take the reciprocal of the second ratio. Then, we represent the ratios as sides of the same angle and convert them so that they have the same consequent. Finally, the product, or multiplication, is given by the quotient of the antecedents. For example, if we want to multiply the green-black ratio by the red-blue ratio, we would have a product of 1/2 multiplied by 1/3, which equals 1/6. But don't just take our word for it, try it out with any ratio of your own choosing! Want to get instant updates for the newest math around? Join our discord server! https://discord.gg/CJ9Ks3WerR Chase's channel: @AkashicLibrarians ... https://www.youtube.com/watch?v=Hz8nX6jOzYw