Imagine winding the hour hand of a clock back from 3 o’clock to noon. Mathematicians have long known how to describe this rotation as a simple multiplication: A number representing the initial ...

Now, on to the subject of complex numbers. We have stayed away from this subject until now simply because we did not want to confuse things. Initially, it is easy to think of signals as “real valued.” ...

All kinds of parameters can be described using complex numbers of the form real + j * imaginary. The numerical values of real and imaginary can vary over very wide ranges. Trying to graph them ...

The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically ...

A century ago, the strange behavior of atoms and elementary particles led physicists to formulate a new theory of nature. That theory, quantum mechanics, found immediate success, proving its worth ...

Physicists construct theories to describe nature. Let us explain it through an analogy with something that we can do in our everyday life, like going on a hike in the mountains. To avoid getting lost, ...

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 ...

If standard quantum theory holds up, imaginary numbers are critical. When you purchase through links on our site, we may earn an affiliate commission. Here’s how it works. Imaginary numbers are ...

Jeremiah Bartz owns shares in Nvidia. A shard of smooth bone etched with irregular marks dating back 20,000 years puzzled archaeologists until they noticed something unique – the etchings, lines like ...

An international team of researchers shows through a concrete theoretical experiment that the prediction by standard complex quantum theory cannot be expressed by its real counterpart and ratifies its ...