# Measurement units and complex numbers.

1. Mar 11, 2006

### Orefa

I am programming a module used to convert measurement units. This will be part of a system that supports complex numbers. I never use complex numbers in my field but of course engineers and physicists do so I thought I should ask a couple of questions first.

Q1. Is there anything unusual about converting something like (5 + 2i) meters/sec into miles per hour? I think it is the same process as with plain old real numbers (multiply by ~2.237 in this particular case) but are there conditions specific to complex values and not reals?

Q2. Is there any mesurement unit(s) that uses a complex conversion factor instead of just a real ratio? In other words, would the software ever need to indicate a conversion ratio as a complex value instead of just a real?

2. Mar 11, 2006

### topsquark

Without getting too Mathematical about it, basically a physical quantity is a "number" with a unit attached to it. "number" means whatever number system you are using, so the unit for a "complex velocity" will still be in m/s, or cm/s, or furlongs/fortnight or whatever your unit system uses. Matrices also work the same way, in case you are wondering. Most people, though, tend to attach the unit to the numbers that make up the matrix rather than apply it to the matrix itself. But there is nothing wrong in doing so.

For the above reasons I can't think of any sort of a "complex" unit. Even if there is a way to define such a thing, I would think that the imaginary nature of the unit would simply be defined as part of the unit itself. For example, you can technically think of a kilogram as being somehow at "right angles" to a meter, and define a kilogram as an "imaginary meter" or $$kg=\sqrt{-1}m$$. But the unit would still work the same way as a kilogram would, so I can see no advantage in doing so.

-Dan

3. Mar 12, 2006

### Orefa

Nothing to be concerned about then. Thanks Dan.

4. Mar 13, 2006

### BobG

If you're multiplying by a scalar (which a conversion factor is), just multiply each component by the conversion factor.

As a general rule, you can think of a complex number as a vector. The real component lies on the x-axis and imaginary component lies on the y-axis. That only gives you a two-dimensional vector.

One of the problems in defining three dimensional vectors is that it's hard to define multiplication of two numbers that have a real part and two imaginary parts. The solution is quaternions. They have a real part and three imaginary parts. A true three-dimensional vector has three imaginary parts with the real part set to zero. The real part of the quaternion can come into play if you move into designing three dimensional animations used in modern video games or simulations.

5. Mar 15, 2006

### Markjdb

Just as a piece of random information, I seem to recall reading about a hypothetical particle called the tachyon which has an imaginary rest mass, which has the effect of forcing it to move at superluminal velocities, although never actually accelerating or decelerating past c.