## The mathematics of weight loss

There’s something about the following that doesn’t seem quite right:

The mathematics of weight loss is, in fact, quite simple, involving only subtraction. “Take in fewer calories than you burn, put yourself in negative energy balance, lose weight,” says Braun, who has been studying exercise and weight loss for years. (from this recent article)

I’ve read variations of that statement for years, and always assumed it to be a basic truth. However, after reading that article it occurred to me that there’s nothing relativistic happening at the macro scale in the human body: energy is not converted into mass, and mass is not converted into energy. So why the focus on calories? Dieters are trying to lose body mass, not energy.

Which weighs more: a ton of feathers or a ton of bricks? I’ll bet you fell for that one the first time you heard it (probably back in elementary school if you’re like me). The dieting analogy would be as follows: which will cause you to gain more weight: eating a pound of spinach or a pound of steak? You’ll be one pound heavier after consuming either one of course.

Which is not to say that eating steak is the same as eating spinach. The spinach contains more water, which your body will eventually excrete (unless you happen to be dehydrated). But why not think in terms of a mass balance instead of an energy balance? A simple mass balance for the human body would look something like this:

$\Delta m = \Delta m_i - \Delta m_o$

where m is your body mass, mi is what you drink and eat, and mo is what you excrete. What this says in words is that the change in body mass over a given period of time equals what you consume, minus what you excrete.

That is the basic equation of weight loss, and it’s not a simple subtraction problem because there are two unknowns. If I eat a pound of steak, nobody can tell me precisely how much of that pound I will eventually excrete and how much my body will retain. It all depends, certainly, on how much exercise I do. But it also depends on my rate of metabolism — that is, how many calories my body consumes during its resting state. It also depends, I’m sure, on what my body happens to need at the moment. If I’m already full, my body doesn’t need the steak, so most of the mass will be excreted. But if I’m hungry and haven’t eaten in a while, my body will retain as much of that mass as it can and convert it into nutrients for my cells.

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