The Black Swan of Mathematics. Why elementary mathematics is illogical and incomplete.
A black swan is a highly improbable event with three principal characteristics: It is unpredictable; it carries a massive impact; and, after the fact, we concoct an explanation that makes it appear less random, and more predictable, than it was. For thousands of years, everyone knew all swans were white. Yet Karl Popper wrote ""No number of sightings of white swans can prove the theory that all swans are white. The sighting of just one black one may disprove it."
Guess what? The Black Swan lives in Australia!
For hundreds of years, people have been saying "multiplication is repeated addition" and ab = a added to itself b times." Why? The reason is Euclid defined multiplication that way in 300 BCE.
Yet the definition of multiplication cited for the past 445 years was NOT Euclid's. It was a London haberdasher's by the name of Henry Billingsley.
Who would have guessed the foundations of elementary mathematics education in the English language were buggy and broken?
Hundreds if not thousands of mathematics books are now proven wrong.
Yeah of course we know ab = a added to ZERO b times. It's just that Euclid wrote before zero and negative numbers were understood.
OK, take away the incorrect proof by reputation that ab = a added to itself b times. Euclid never said anything like that. Now let it be known our definition of multiplication is that of a London haberdasher, not Euclid and we can start to think for ourselves.
OK, let's substitute another Henry for haberdasher Henry Billingsley.
In an 1894 essay Sur La Nature du Raisonnement Mathematique, Henri Poincaré [] wrote:
DÉFINITION DE LA MULTIPLICATION
Nous définirons la multiplication par les égalités
a × 1 = a
a × b = [a × (b – 1)] + a
So ab = a added to itself b – 1 times.
Another Frenchman, Lacroix  wrote that multiplication on the positive naturals takes the form ab = a added to itself b – 1 times. For example, 16 multiplied by 4 or 16 x 4 is explained.. "the number 16 is repeated four times, and added to itself three times. 
OK, let's update the rest of arithmetic for the identity elements, zero and one, and it all becomes easy.
That's the Black Swan of Mathematics!
Proofs by Contradiction Reveal 445 Year Old Math Shock!
The truth about multiplication: From 300 BCE to 1637 to today...
 Nature Du Raisonnement Mathématique Henri Poincaré, in Revue de Métaphysique et de Morale, T. 2, No. 4 (Juillet 1894), P. 377, Published by: Presses Universitaires de Francehttp://www.jstor.org/stable/40891545
 Sylvestre François Lacroix also stated the obvious. On dit qu'un nombre est double quand il est ajoute a lui-meme une fois, ou ce qui revient au meme quand il est repete deux fois. On dit qu'il est triple quand il est ajoute a lui-meme deux fois, ou, ce qui est le meme chose, quand il est repete trois fois. This means; "We say that a number is twofold when added to itself once, or what amounts to the same when it is repeated twice. It is said to triple when it is added to itself twice, or what is the same thing, when it is repeated three times” From ‘Traité élémentaire d'arithmetique’ meaning, ‘A treatise of elementary arithmetic’, published 1797
 An Elementary Treatise on Arithmetic, Taken from the (French) Arithmetic of S. F. Lacroix, by John Farrar, P. 14, Printed by Hilliard & Metcalf, at the University Press, Cambridge, New England, USA, 1818.
BTW, I also have a new depiction of multiplication on the reals via a compass and straight edge construction, featured by the London Mathematical Society at: