Mathematicians were disturbed, centuries ago, to find that calculating the properties of certain curves demanded the seemingly impossible: numbers that, when multiplied by themselves, turn negative.

Numbers thought to have no analogue in the real world have meaning at quantum scales. When you purchase through links on our site, we may earn an affiliate commission. Here’s how it works.

Our February Insights puzzle sent readers on a treasure hunt based on complex numbers. First we provided a bit of a primer, demonstrating that complex numbers (expressions of the form a + bi, where ...