Losing the Units: Explaining Why Exponentials are Dimensionless

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In summary, the conversation discusses the dimensionless nature of exponential and trigonometric functions, and how they are used in physics. It is not strictly correct to assign dimensions to the arguments of these functions, but in some cases, it is accepted to do so for practical purposes.
  • #1
starzero
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Hi All and sorry if this is too easy a question but here goes...

Sines, Cosines and the rest of the trig functions are the ratio of two lengths and thus are dimensionless quantities.

That is if I plug in a value for t in sin(ωt) there are no units.

For example the solution of

x'' + ω^2 x = 0 with x(0) = x0 and x'(0) = v0 is given by

x(t) = x0 cos(ωt) + v0/ω sin(ωt)

The units come from the initial conditions not the sine or cosine.


So here is the question...

Same is true ( I believe ) when using the natural exponential function exp(t).

How does one simply explain this.

I tried to reason it out using eulers formula exp(iω) = cos(ω) + i sin(ω) figuring that again we get ratios of lengths,
however in the case where the real part is non-zero we get another exponential (which is not the ratio of lengths)

exp(a +ib) = exp(a)(cos(b) + i sin(b))

Is there a simple explanation as to why we "lose the units" when using the exponential function?
 
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  • #2
It is not strictly correct to supply any argument with dimensions to an exponential or trig function. Example: [itex]\omega t[/itex] has no dimensions (frequency is 1/time, and time/time is dimensionless).

Why is it incorrect? Consider a series expansion of the exponential function.

[tex]e^x = 1 + x + \frac{1}{2} x^2 + \ldots{}[/tex]

[itex]x[/itex] must be a dimensionless parameter. If it had, say, dimensions of length, how would we add length to 1 and to length squared?

Nevertheless, in physics this strict need is sometimes ignored. As an example, you should take as implicit that [itex]e^t[/itex] is in fact [itex]e^{t/\tau}[/itex] where [itex]\tau[/itex] is 1 in whatever units of time you're working with.
 
  • #3
Muphrid said:
It is not strictly correct to supply any argument with dimensions to an exponential or trig function. [..]
Nevertheless, in physics this strict need is sometimes ignored. As an example, you should take as implicit that [itex]e^t[/itex] is in fact [itex]e^{t/\tau}[/itex] where [itex]\tau[/itex] is 1 in whatever units of time you're working with.
Yes indeed, and it may be useful to give an example (of not ignoring this):
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capdis.html
 
  • #4
Thank you both for your fast insightful and illustrative replies.
 
  • #5




The reason why exponential functions are dimensionless is because they are related to ratios of lengths, just like the trigonometric functions. This can be seen in Euler's formula, where the exponential function is expressed as a combination of cosine and sine functions, both of which are dimensionless.

In addition, the exponential function can also be thought of as the limit of a geometric sequence, where the ratio of consecutive terms approaches a constant value. This constant value, which is the base of the exponential function, is also dimensionless.

Furthermore, the exponential function is often used to model growth or decay processes, where the units of time cancel out in the exponential term. This is because the rate of change is constant and the units of time are essentially "lost" in the calculation.

Overall, the key factor in understanding why exponentials are dimensionless is the concept of ratios and limits. By looking at the fundamental properties of the function, we can see that it is inherently dimensionless and this is why it is a powerful tool in many scientific fields.
 

What is the concept of "Losing the Units" in exponentials?

"Losing the Units" refers to the phenomenon in exponentials where the units attached to a quantity are cancelled out, resulting in a dimensionless value.

Why do exponentials result in dimensionless values?

Exponentials involve raising a number to a power, which essentially multiplies the number by itself a certain number of times. This repeated multiplication results in the units being cancelled out, leaving only a numerical value.

Can you provide an example of "Losing the Units" in exponentials?

One example is the equation for radioactive decay, where the exponential function is used to calculate the amount of remaining radioactive material. The units of time and the units of the radioactive material cancel out, leaving only a numerical value.

What is the significance of "Losing the Units" in scientific calculations?

"Losing the Units" allows for simplification and easier interpretation of mathematical equations. It also highlights the idea that certain physical quantities, such as radioactive decay rate, are independent of the units used to measure them.

Are there any exceptions to "Losing the Units" in exponentials?

Yes, there are certain situations where the units are not cancelled out in an exponential equation. For example, in the ideal gas law equation, the units of pressure and volume do not cancel out, but rather combine to form the units of energy.

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