Following up on his previous post, “Is Math a Gift From God?” — calculus students say, “No!” — Jason Rosenhouse has a new essay for your delectation, “Is God Like an Imaginary Number?” Again, the short answer is, “Nope.” The longer answer will take us into the history of mathematics, the role of mysticism in theology and the relationship between science and verbal description.

Rosenhouse sets himself the task of fisking an essay in the religious periodical *First Things,* by a “Junior Fellow” of that publication named Amanda Shaw. Shaw’s thesis is that the notion of God is akin to that of an imaginary number, and moreover that the same closed-minded orthodoxy which rejected the latter from mathematics for oh so many years is unjustly keeping the former out of science. I find this stance to be, in a word, ironical: if you’re looking for dogmatism and condemnations of the heterodox, your search will be much more rewarding if you look among the people who reject scientific discoveries because they are inconsistent with a Bronze Age folk tale than if you search through science itself!

Still, it’s a fun chance to talk about history and mathematics.

**PART A: COMPLEX NUMBERS**

As I described earlier, “imaginary” and “complex” numbers arise naturally when you think about the ordinary, humdrum “real numbers” — you know, fractions, decimals and all those guys — as *lengths on a number line.* In this picture, adding two numbers corresponds to sticking line segments end-to-end, multiplication means *stretching* or *squishing* (in general, *scaling*) line segments, and negation means flipping a segment over to lie on the opposite side of zero. Complex numbers appear when you ask the question, “What operation, when performed twice in succession upon a line segment, is equivalent to a negation?” Answer: *rotating* by a quarter-turn!

Historically, mathematicians started getting into complex numbers when they tried to find better and better ways to solve real-number equations. Girolamo Cardano (1501–1576), also known as Jerome Cardan, posed the following problem:

If some one says to you, divide 10 into two parts, one of which multiplied into the other shall produce […] 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion.

Writing this in more modern algebraic notation, this is like saying [tex] x + y = 10 [/tex] and [tex] xy = 40 [/tex], which we can combine into one equation by solving for [tex] y [/tex], thusly:

[tex] xy = x(10 – x) = 40.[/tex]

In turn, shuffling the symbols around gives

[tex] x^2 – 10x + 40 = 0,[/tex]

which plugging into ye old quadratic formula yields

[tex] x = \frac{10 \pm \sqrt{100 – 160}}{2}, [/tex]

or, boiling it down,

[tex] x = 5 \pm \sqrt{-15}. [/tex]

Totally loony! Taking the square root of a *negative number?* Forsooth, thy brains are bubbled! Oh, wait, didn’t we just realize that we could maybe handle the square root of a negative number by moving into a two-dimensional plane of numbers? Yes, we did: that’s the prize our talk of flips and rotations won us!

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