Plotting Curves in Excel

  One has to create a table containing values of the coordinates x and y to plot a curve in Excel. This plot used smoothing to improve its appearance.

One can right click the chart and use "Select Data..." in the drop down menu to specify the horizontal and vertical data for the curve's series. One can also click on the axes and series to "format" the appearance of the chart. The only real solution for the cubic equation is x=-1.22249 as previously shown.

Supplemental (Feb 6): The axes are incorrectly labeled on the chart. horizontal is x and vertical is y.

Using Excel 2016 as a Mathcad

  Although Excel 2016 isn't optimized to function as a mathcad it has a limited capability for doing so. One can use it to take a look at the auxiliary parameters, u and v, used to solve the cubic equation. Here is a check to see that the values satisfy the equation 3uv+B₂=0.

An Excel chart can be used to give a visual confirmation of the relative position of these parameters.

The auxiliary equation allowed us to find the cube of u and the formula above relating u and v shows they are simply related.  The Excel chart allows one to plot a number of datasets at the same time. The circles in the plot above were generated by computing a table of x,y values and using the columns of data to specify a "series" for the chart. There's a video that explains how to create charts using spreadsheet data but doing plots like this involves a lot of trial and error.

Solving a Cubic Equation With Excel 2016

  Although complex numbers appear to be surreal and seem to involve something like Zeno's paradoxes they are quite useful. One can use Word 2016 to illustrate how one can use Cardan's method to solve a more general cubic equation. We first note that the coefficients of a cubic equation can be expressed in terms of its roots. We don't need a general coefficient for the cubic term since it can be eliminated by division.

One can also show that the coefficients are related to the sum of powers of the roots but this doesn't appear to be particularly useful.

One can use the mean of the roots to eliminate the coefficient, A, of the squared term in the equation by making the following substitution.

Then Cardan's method can be used to find an auxiliary equation for the solution for y.

One can solve this equation for u³ using the quadratic formula and find the cube roots to obtain solutions for u. If either B' or C' are zero the equation for u can be simplified further but we will assume that they are not. This method can be used in Excel 2016 employing the engineering functions to find the roots of a cubic equation.

After a little difficulty I was able to get an Excel scatter chart to plot these roots on a complex plane.

In the plot above the point z is equidistant from the three roots. Such a point only exists if the roots do not have a common line going through them. It can be determined as follows using the method of least squares. The asterisks indicate complex conjugates.

The expression for u above can be used to find z.

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.