On the Surreal Character of Complex Numbers

  Not every quadratic equation has real solutions. When we assume a quadratic equation has solutions and follow the rules of algebra in some cases we end up with something surreal with a weird displacement from the mean of the two solutions. This conclusion is logically consistent with the initial assumptions. We have to extend the real numbers to allow for the imaginary numbers in order to guarantee a solution in all cases. The weirdness can be factored out by using the imaginary number i to represent it. Multiplying the two monomial factors for the roots takes us back to the original quadratic equation when we assume i²=-1. We can plot the solutions of the quadratic equation on a plane with a real axis and an imaginary axis. Like Gibbs we can say that real and imaginary numbers are apples and oranges. The situation with Hamilton's quaternions is even weirder.

Solving a Binomial Equation with the Mean and Difference of the Roots

  The mean and difference of the roots of a binomial equation allow us to solve the equation.

So the radical part gives us the difference between a root and the mean. When m²<c, δ²<0 which means δ is not a real number since its square is negative. It is still useful to maintain that δ us a number of sorts and express it as a multiple of the solution of i²=-1. Setting i²=-1 makes i a magic number and mathematicians magicians.

Supplemental (Jan 5): What is so curious about the imaginary numbers is their surreal character.

Some Historical Documents on Magic Squares

1705 La Hire - Construction de la Carres Magiques

1776 Euler - On Magic Squares (English)

1882 Lucas - Le Jeu du Taquin

1892 Rouse Ball - Magic Squares

1894 Maillet - Application of Group Theory to Magic Squares

1897 McClintock - Perfect Magic Squares

1899 Schubert - The Magic Square

1907 Veblen - On Magic Squares

1917 Andrews - Magic Squares

1917 Licks - diabolic or Nasik magic squares

1919 Cajori - Bessy 880 magic squares

What Might Be Odd About the Set of Operators

 The set of operators that was found seems a little unusual but finding magic squares is a kind of mathematical game played with numbers that in itself makes the problem unnatural. From the perspective of mathematical theory one has to ask how is this result peculiar. It differs from group theory slightly because there is a representation of the operations in terms of matrix multiplications and there are two distinct multiplications. Only two matrices and transposition were needed to represent all the operations that left the sums unchanged that we found by observation. The resulting operations were more complicated than the dihedral group because of the double shifts which left the sums unchanged too.

  The set of operations that we could use to find new magic squares came about in an effort to find unique magic squares. They tell us that the set of all magic squares can be partitioned into a class of sets containing simply related magic squares. The combinations of the two matrix operators, the two ways of doing the multiplications and transposition of the original magic square allowed us to generate a set of operations which could be performed and still leave the sums unchanged. We pointed out that the set of operators may not be complete because there was a simple process which produced a magic square that could be transformed into Dürer's by the permutation of the two middle rows.

  Magic squares may tell us something about the theory of equations and the multiplicity of solutions. Is our theory of equations complete? For example, we can form a quadratic equation by product of two linear equations x+q=0 and x+r=0 thus ax²+bx+c=
a(x+q)(x+r)=0. But the assumption that the quadratic is a perfect square doesn't always lead to a rational solution so we use imaginary numbers to represent how the solutions differ from a perfect square. A reflection in the complex plane also allows us to go from one complex solution to the other. The real solutions can also be transformed into one another by reflection across a plane through the perfect square.

  We can also ask how the vertical, horizontal and shift operations differ from a transposition. The former only involve permutations of the rows or the columns and their order can be exchanged. With ordinary addition a sum does not depend on the order in which the terms are added together. The same is true for vector addition. Transposition involves a change in context of the elements involved since rows are exchanged with columns and that affects the way we can combine transposition with the other operations.

  In 1886 Gibbs gave an address, On Multiple Algebra, to the American Association for the Advancement of Science which touches on similar issues dealing with matrix and vector algebra. The part about apples and oranges is interesting.

Edit (Jan 4): missing a in factors of quadratic.