Math and Units

Now is a good time to take a little side trip to discuss math and units, as I'll use be using some math in this blog from time to time (and already have a bit in an earlier post).

Sometimes I will say X is proportional to Y. For example, for a boingy spring, force (F) is proportional to how far the spring is pulled (x - we'll call it d for distance).

That means that as the quantity 'd' (distance the spring is pulled) gets bigger, the force (F) gets bigger by the same factor. So if d doubles, the force doubles. If d is 5 times greater, the force is 5 times greater. In math, this is stated as

• F = C*d

Here '*' is the symbol used to represent multiplication (common in computer languages because the times symbol 'x' is too easily confused with the letter 'x'). "C" is a constant of proportionality. It's some fixed number, such as 12 for example. Really, all that the C is useful for is converting from one unit to another. There is always a way of defining a new unit such that the C goes away.

For example, energy (E) is force (F) applied over a distance (d)

• E = F*d

This means that energy is proportionate both to force and to distance. If force doubles, energy doubles. If distance doubles, energy doubles. If they both double, energy doubles and doubles again, so it is 4 times greater. If we want to express Energy in food calories, force in pounds, and distance in inches, this becomes

• E(in food calories) = 0.000027 * F(in pounds force) * d(in inches)

Useful if you are lifting weights to lose weight!

However, if we use SI (System International) units, where the base units are Kilograms, Meters, and Seconds, then energy is expressed in Joules, force is expressed in Newtons, and distance is expressed in Meters, and in this case the constant goes away, getting us back to

• E(in kg) = F(in N) * d(in m)

No constant is necessary.

The moral of the story is that the constant can always be done away with if we define the units appropriately. So say I really like pounds and inches, then I can define a new unit of energy called the Dan, and tell you that 1 Dan is defined as 1 lb*in, such that

• E(in Dan) = F(in lb) * d(in in)

Now are you are left with is the problem of converting from Dans into more familiar energy units, but that was what the constant was for:

• 1 Food Calorie = 0.000027 Dans

Units are never fundamentally important to understanding something, though, they are good for calculating things, so we won't worry about them too much in this blog.

We also sometimes say that X is inversely proportional to Y,  so that if Y doubles, X halves. In math,

• X = 1/Y

where the symbol '/' represents division.

There are other, higher order relationships as well where something varies as the square or cube of something else. For example, the force is gravity is proportional to the mass of each of the two things, and inversely proportional to the distance squared. With appropriate choice of units,

• F = m1 * m2 / d^2

The '^' symbol means "to the power of", so d^2 means d squared, or d*d. So if the distance doubles, the force falls off by a factor of 4.

Raising something to the power of a negative number is like dividing by that thing, so this can be re-stated as

• F = m1 * m2 * d^(-2)

There we go. That's the math we will need for the time being.