Gadget by The Blog Doctor.

Monday, September 24, 2007

Book Review: Imagining Numbers by Barry Mazur

The full title of the book is:
Imagining Numbers
(Particularly the square root of Minus Fifteen)

In his preface to the book Mazur states that the book is "written for people who have no training in mathematics ... but who may wish to experience an act of mathematical imagining and to consider how such an experience compares with the imaginative work involved in reading and understanding a phrase in a poem." A major aim of the book then is to draw analogies between the imaginative process of mathematics and poetry.

Mazur returns regularly to parts of the tone poem "Whatever It Is, Wherever You Are" by John Ashbery. In pondering the inventors of writing Ashbery writes:

To what purpose did they cross-hatch so effectively, so that the luminous durface that was underneath is transformed into another, also luminous but so shifting and so alive with suggestiveness that it is like quicksand, to take a step there would be to fall through the fragile net of uncertainties into the bog of certainty ...

I would like some of the "bog of certainty" in Mazur's description of the relationship between poetical and literary imagining and the mathematical variety. I have an interest in poetry and literature on the one hand and mathematics and science on the other, but if Mazur provides an explanation of the relationship between imagining in the humanities and the sciences then it is beyond my ken. This book can be seen as a contribution to breaking down the division between the two cultures described by C. P. Snow - the Humanities and the Sciences. Snow might have overstated his case but something of a division still exists. Even if the book does not finally succeed in drawing a convincing analogy between creativity in the sciences and creativity in the humanities, the erudition of Mazur in both literature and mathematics show that it is possible for a modern expert in the sciences to have and interest and skills in the humanities.

Consequently the rest of this post will concentrate on the mathematical ideas in the book. Mazur's description of the development of the idea of complex numbers by a range of mathematicians over three centuries is interesting and compelling.

A brief note on typology. Square foots will be mentioned frequently in this post. I will not use the conventional square root symbol - √ - as it will require too many graphics, instead I will represent square roots using the computer terminology, so that the square root of 4 will appear as: sqrt(4).

As Mazur describes (p39) the Italian mathematician Girolamo Cardono (1501-76) in his Ars Magna coming across an equation that contains (5 + sqrt(-15)) × (5 - sqrt(-15)) . There are three possible reactions to this piece of maths, but to make the discussion clearer I will use sqrt(-4).
A reaction from those who have little understanding of or interest in maths would be "what is all the fuss about? the answer is obviously -2" . Those who have a reasonably sophisticated understanding of maths would think 2i . Those who took some notice in their middle secondary school maths classes would be saying "that's impossible, there is no such thing as the square root of a negative number!" As i described in this post multiplying two negative numbers together will result in a positive one. If you multiply -2 by -2 the answer is +4, so there seems to be no solution to sqrt(-4). The problem for that point of view is that if you ignore the objection to square roots of negative numbers and just multiply out Cadano's expression you end up with the positive whole number 40, as demonstrated below:
(5 + sqrt(-15)) × (5 - sqrt(-15))
= 25 -5 × sqrt(-15) +5 × sqrt(-15) -(sqrt(-15)× (sqrt(-15)
-5 × sqrt(-15) +5 × sqrt(-15) cancel each other out os
= 25 -(-15)
= 25 +15
= 40
It is surprising that such an unusual expression evaluates to the simplest type of number - a positive integer. Although sqrt(-15) seems initially nonsensical if it is accepted without thinking too much about what it means it can produce acceptable results.

If we multiply the number line by -1 we rotate it by 180 degrees.

Writing sqrt(-1) is a little clumsy so mathematicians developed a notation for it" i . Sqrt(-4) can be written as sqrt(4) × sqrt(-1) which becomes 2i when sqrt(-1) is written as i . Numbers that contain i are called imaginary numbers but we can generalise the function by adding a real number to form a +bi. These are called complex numbers.

Previously we discussed multiplying by -1. What happens if we multiply a complex number by i? Lets see: i × (a + bi) = ai + b × i × i = ai - b (as i × i = -1). If you plot these numbers on a cartesian plane, with the real numbers on the X-axis and the imaginary ones on the Y-axis then it should be clear that multiplying by i is equivalent to rotating by 90 degrees. Mathematicians plot complex numbers on the complex plane with the horizontal axis for the real portion of the number and the vertical axis for the imaginary numbers. This makes sense if multiplying by i is equivalent to rotating by 90 degrees.

What happens if we multiply the complex number by i a second time. This is obviously another 90 degree rotation resulting in a cumulative 180 degree rotation - the same as multiplying by -1. That is multiplying by i2 produces the same result as multiplying by -1. This might not be particularly surprising as i2 = -1. We seem to have arrived at the same point from two different directions.

Many mathematicians use complex numbers without concerning themselves with the question of what imaginary numbers look like but for those or us who have less mathematical skill Mazur's description of i as a rotation of 90 degrees on the complex plane is satisfying.

This post just scratches the surface of the area of mathematics called Complex Analysis. Other reasons to believe in the reality of complex numbers is that complex maths is essential to a number of important areas of physics including Quantum Mechanics. Some time or other (when I learn something about it) I hope to further explain that claim.

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