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July 2009The Scope Of EquationsEquations and mathematical expressions sometimes have logical boundaries. For example, is it well known that any attempt to take the square root of a negative number produces an "imaginary number". This is a case where an input value is clearly outside of the scope of the equation. It has gone beyond the point where it produces a reliable result. There are also many less obvious limits however. There is often a difference in the way that computer programmers and physicists view equations. In software development, the "scope" of an equation is always a matter of concern. If a theoretical physicist makes a mistake with the scope of a function, then in a year, or a decade or a century, someone will publish a correction. If a computer programmer makes a mistake in the scope of a function, something goes "bang", someone loses money or planes fall from the sky. Some well-known examples of results in physics that are commonly read for out-of-scope values include the Lorentz Factor and density calculations. In the case of the Lorentz Factor values of v >= c are out of scope. Similarly values of 0 volume are outside the scope of density equations. Why? Because zero volume is not a volume - it's a point. An equation that works for a 3-dimensional value does not work for a 0-dimensional one. Division by zero is generally avoided by setting a limit at zero, although the reasoning behind this is often overlooked. Division by zero can and should be avoided because there is no number. There is no information content in the result - that is why computer programmers discard it. If zero is a number then we would and should not discard it. The fact that we can and do ignore the results of division by zero is that zero is not a number and is outside the scope of the equation. Type 2 infinities (i.e. resulting from division by zero) are typically the result of an equation that has gone out of scope. Results of zero may also come from an equation that has gone out of scope, although this is often not as obvious. The main point is that there is no information contained in the result - it is the absence of any value, not an actual number.
Regards, AJ Corcoran
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