Cubic Polynomials and Complex Numbers

My goal in this post is to show you one of the solutions to the equation

x^3 = 15 x + 4\ .

In fact, that equation has three solutions – it’s a cubic! – and all three solutions are real. But getting them will require complex numbers. Along the way, we’ll talk briefly about negative numbers and the quadratic equation, too.

Let’s dive right in.

We can treat that equation as a function, writing

y = x^3 - 15 x - 4

and graph it. Mathematica does it so easily…

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Happenings – 2011 Aug 28

Yes, today is Sunday. I’m a day late.

Yesterday morning I was in a hotel room in Houston, Texas… having traveled for work. I got home at 4 PM, and although I considered writing a blog post yesterday evening, I was too tired.

As for the plane flights, I was either looking at the ground we flew over, or reading fiction… no mathematics at all got done on the flight, and almost none during the week.

Well, someone there did ask an interesting question, I think because they were in the presence of a mathematician who appeared to be able to speak English.

Do negative numbers and complex numbers really exist?
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Regression 1 – Example 5: A Toyota Car

Edit: 2011 Nov 26. I computed a correlation matrix of the parameters when I meant to compute the correlation matrix of the data. Find “edit”.

Let’s do another regression, okay? I’m sick of the Hald data. It’s been so long since we did something other than multicollinearity… you might refresh your recollection of the earlier examples and the material leading up to them.


This data comes from Ramanathan, dataset 3-7 in both the 4th and 5th editions, but 3-6 in the 3rd. See this post for information about obtaining his data.

From the description in the 4th ed. data, I infer that this is data for one vehicle over a period of 14+ years.

DATA3-7: Data for a Toyota station wagon (57 observations)
cost = cumulative repair cost in actual dollars (11 – 3425)
age = age of car in weeks of ownership (Range 5 – 538)
miles = miles driven in thousands (Range 0.8 – 74.4)
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Happenings – 2011 Aug 20

If you’ve been following this blog for a while, you know what this picture implies.

When I started drafting this – before noon – I had not yet been advised of a magnitude 7.0 quake that occurred in the Vanuatu region. By noon, however, I had found that my forecast for the month has been fulfilled: there has been an earthquake of magnitude at least 7 somewhere on earth in the month of August.

(Considering the potential loss of life and tsunami damage, I would be happy to have been wrong… but my forecast didn’t cause it.)

Here’s their usual picture:

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Happenings – 2011 Aug 13

Earlier this year, as I said when it happened, someone asked me if there was a book about string theory at a popular level. I had thought I was going to have to recommend one fairly challenging book (Brian Greene’s “The Elegant Universe”) and one anti-string book (Lee Smolin’s “The Trouble with Physics”).

There is an elementary book. As far as I can tell, there is only one… but that could change.

And where would we look for such a book?
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Regression 1: Multicollinearity in the Centered Hald Data

Edit, 2011 Nov 25: added a link to the Norms and Condition Numbers post. Find “Edit”.

Let us put the Hald data to rest, by making a transformation intermediate between the raw data and the standardized data. We will merely center the data, by subtracting the means of each variable.

Some things will change, but some things will not. I learned a lot from both.

In particular, we will see – as we did with the standardized data – that the relationships among the independent variables remain. The details change, but we still have a very strong relationship among X1, X2, X3, and X4… a strong relationship between X2 and X4… and a weak relationship between X1 and X3.

These relationships will be identified, as before, in a couple of ways. The VIFs, variance inflation factors – which I recast as RSquareds using the equation

R^2 = 1 - \frac{1}{\text{VIF}}

which came from the definition in this post

\text{VIF} = \frac{1}{1-R^2}

are what would show me that these three relationships among the raw variables still hold at the same strength, even when we center or standardize the data.
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Happenings – 2011 Aug 6

What’s been happening? Not all that much, or so it seems.

What am I doing? Probably more than I give myself credit for.

The first thing that isn’t happening is: magnitude 7 earthquakes. Not one so far in August, not anywhere on earth.

I finished Ronan’s book, “Symmetry and the Monster”. It was fun, if a little unsatisfying… I could have used a little more mathematics.

I did find in it a verbal description of what it means for a group to be k-transitive on a set. Marvelous! Of course, it says exactly what the mathematical definition did – but I, poor fool, had not been able to phrase the mathematics in words that made sense. And, hard as it may be to believe, I need mathematics definitions to make sense in words.
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Norms and Condition Numbers

We have had a few occasions to talk about norms of vectors. I want to take a look at them, and to also look at norms of matrices, and then at condition numbers of matrices.

Vector Norms

Let me jump right in. Here’s a 2-dimensional vector:

we get the Euclidean length of the vector – it’s the length of the hypotenuse of a 3-4-5 right triangle.

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