Numbers that forced us off the number line

 kw: book reviews, nonfiction, mathematics, complex numbers, introduction

It is rare for me to post the cover of a book I am reviewing. In this case, when I saw in the subtitle "square root of minus fifteen", I just had to show it. The picture made me think, "What is the bee^th root of a yellow tulip?" That's a lot harder to imagine than any even root of a minus number!

I found Imagining Numbers: (particularly the square root of minus fifteen) by Barry Mazur to be a very thorough leading-by-the-hand-gently introduction to "imaginary" numbers (those based on the square root of minus one), and the "complex numbers" that derive from them, . A Complex Number is a two-part quantity; one part is "real" and the other part is a real number times the square root of minus one, called i or j. Thus 1+j and 3-2.4j are complex numbers.

I was introduced to the concept of a number named i (for imaginary) in a high school math class, but in college I primarily studied engineering, where I learned that engineers prefer it to be named j, to get away from the notion of "imaginary." However, for the graphical expression of complex numbers, the horizontal axis is equated to the number line, and is called the R or Real axis, and the vertical axis is called the I or Imaginary axis; there's no getting away from it. One may call the R part the "scalar" part of a complex number, but the other part just doesn't have a good alternative to "imaginary". This paragraph touches on concepts that occupy 2/3 of this book. The latter third focuses on graphical representation.

I was introduced to the graphical expression of complex numbers this way: First it was emphasized that 1 has two square roots, 1 and -1. Then, by analogy, we learned that -1 also has two square roots, j and -j (which is -1×j). Before going further, we got to experiment with multiplying various quantities with j. Then we were asked to imagine how j or -j could be "halfway" between 1 and -1, without being equal to zero. Finally, someone asked, "Then where does that go on the number line?" At that point we were shown that a second number line crosses the usual number line at right angles, forming a Cartesian coordinate system in which the x-direction was the R part and the y-direction was the I part of a complex number. Furthermore, going from 1 to j to -1 to -j and back to 1 again was seen to be a rotation. Doing some multiplication and addition of various elementary complex numbers with one another showed us how they had graphical analogues. Complex numbers are an alternative notation for a polar coordinate system, one based on distance-plus-angle rather than horizontal-plus-vertical. This fixed the concept in our minds. Understanding this was essential to getting the hang of engineering calculus.

Dr. Mazur's genius is in understanding that anyone who can do basic algebra can learn to understand complex numbers. This book takes elementary, easy steps, first through the history of how i was very gradually understood to be something very useful, and not at all "imaginary," and then through the way graphical representation that helps us get the concept and fix it in our minds. Complex numbers and complex analysis are essential for engineering, particularly when cyclical processes are being designed or analyzed.

I must admit to a bit of ennui at times. The author tells us several times that the book is really written for those who don't already understand complex numbers. Anyone who has not delved for decades into engineering math, as I have, will probably find the book a bit challenging, but not boring, and it will draw one along to take in concept after concept.

I must admit, I never did find out why the square root of minus fifteen is emphasized in the subtitle…