e^t, where t is time. Both the exponential function and its argument must be dimensionless. ?
Clearly the function has a "t" as a unit of time. How can this equation be valid. Can anyone help me understand this paradox?
- VamanLv 71 month ago
This equation is not valid. The term in the exponent shoud be dimensionless. You need e^(wt) or e^(t/T) type.
- DixonLv 71 month ago
You are correct, raising to a power means the power must be just a number. Typically you will see t/τ where τ is some kind of "time constant" in the same units as t. So if t is in years and τ = 100 years, this would effectively give you e^(number of centuries)
- husoskiLv 71 month ago
Yes, you're right. Technically, it's not valid. Generally, in physical sciences you'll find that a formula involving exponential growth in time has coefficients with units that cancel out or apply dimensionality as needed; something of the form:
f(t) = A e^(bt)
The units of b will be such that bt is dimensionless, while the dimensions of A will be those expected for the function result. The same is true for powers with a variable exponent or with a variable base and an irrational exponent, as well as for logarithms, trig functions and other special functions.
Something similar also happens in polynomials, where an equation like
f(t) = at^2 + bt + c
....had different dimensonality on each constant so that the terms all have the dimensionality required for the function result.
In math textbooks, though, dimensionality is often ignored--even in so-called "real world" problems.