“I’ve got another solution to Larry’s challenge for you, Vinnie.”

“Will I like this one any better, Sy?”

“Probably not but here it is. The a‑line has unit length, right?”

“That’s what the picture says.”

“And the b‑line also has unit length, along the i‑axis.”

“Might as well call it that.”

“And we’ve agreed it’s a right triangle because all three vertices touch the semicircle. A right triangle with two one‑unit sides makes it isosceles, right, so how big is the ac angle?”

“Isosceles right triangle … 45°. So’s the bc angle even though it don’t look like that in the picture.”

“Perfect. Now let me split the triangle in two by drawing in this vertical line.”

“That’s its altitude. I told you I remember Geometry.”

“So you did. Describe the ac triangle.”

“Mm, it’s a right triangle because altitudes work that way. We said the ac angle is 45° which means the top angle is also 45°.”

“What about length c_a?”

“Gimme a sec, that’s buried deep. … If the hypotenuse is 1 unit long then each side is √½ units.”

“Vinnie, I’m impressed. So what’s the area of the right half of the semicircle?”

“That’s a quarter‑circle with radius c_a which is √½ so the area would be ¼πc_a² which comes to … π/8.’

“Now play the same game with the b‑side.”

“Aw, geez, I see where you’re going. Everything’s the same except c_b has an i in it which is gonna get squared. The area’s ¼πc_b² which is … ¼π(i√½)² which is –π/8. Add ’em both together and the total area comes to zero again. … Wait!”

“Yes?”

“You said the i‑axis is perpendicular to everything else.” <sketching on the back of the card> “That’s not really a semicircle, it’s two arcs in different planes that happen to have the same center and match up at one point. The c‑line’s bent. Bo‑o‑o‑gus!”

“I said you wouldn’t like it. But I didn’t quite say ‘perpendicular to everything else‘ because it’s not. The problem is that the puzzle depends on a misleadingly ambiguous diagram. The only perpendicular to i that I worked with was the a‑line. You’ve also got it perpendicular to part of the c‑line.”

“Sy, if these i-number things are ambiguous like that, why do you physics and math guys spend so much time with them? And do they really call them i‑numbers?”

“Ya got me. No, they’re actually called ‘complex numbers,’ because they are complex. The problem is with what we call their components.”

“Components plural?”

“Mm‑hm. Mathematicians put numbers in different categories. There’s the natural numbers — one, two, three on up. Extend that through zero on down and you have the integers. Roll in the integer/integer fractions and you’ve got the rational numbers. Throw in all the irrationals like pi and √2 and you’re got the full number line. They call that category the reals.”

“That’s everything.”

“That’s everything along the real number axis. Complex numbers have two components, one along the real number line, the other along the perpendicular pure i‑number line. A complex number is one number but it has both real and i components.”

“If it’s not real, it’s imaginary, HAW!”

“More correct than you think. That’s exactly what the ‘i‘ stands for. If Descarte had only called the imaginaries something more respectable they wouldn’t have that mysterious woo‑quality and people wouldn’t have as much trouble figuring them out. Complex numbers aren’t mysterious or ambiguous, they’re just different from the numbers you’re used to. The ‘why’ is because they’re useful.”

“What good’s a number with two components?”

“It’s two for the price of one. The real number line is good for displaying a single variable, but suppose one quantity links two variables, x and y. You can plot them using Descartes’ real‑valued planar coordinates. Or you could use complex numbers, z=x+iy and plot them on the complex plane. Interesting things happen in the complex plane. Look at the difference between e^x and e^ix.”

“One grows, the other cycles.”

“Quantum mechanics depends on that cycling.”

~ Rich Olcott