Why must exponents be dimensionless?

[cocopops12]

suppose we have a^b
why must 'b' be dimensionless?

Mathematicians have defined crazy things over the centuries
so why haven't they defined this one?

 

[HallsofIvy]

"Dimensions", in the sense that you are using the word (meters, kilograms, degrees Celcius) are not mathematical objects, they are physical. If you are asking why no physics formula, with exponents, has no units on the exponent, you will have to ask a physicist.

 

[cocopops12]

I see, thank you sir.

 

[trollcast]

I see, thank you sir.

If x is a variable then you do something like:

$$e^x=\sum_n \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...$$

Now does that sum make sense if x has a dimension?

However the exponent can contain variables with dimensions but they must cancel to give a dimensionless number:

eg. $$M(t)=M_oe^{-\lambda t}$$

 

[Mark44]

suppose we have a^b
why must 'b' be dimensionless?

If we restrict our attention to exponents that are positive integers, then an exponent means repeated multiplication. For example, x² = x * x, and x³ = x * x * x.

The volume of a cube whose edge length is s is V = s³ = s * s * s. The units are tied to the variable s. All the exponent does is keep track of how many factors of s are present.

 

[Whovian]

$$e^x=\sigma_n \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...$$

I think you meant $$e^x=\sum_n \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...$$

\Sigma works, though \sum tends to work a little better.

 

[trollcast]

I think you meant $$e^x=\sum_n \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...$$

\Sigma works, though \sum tends to work a little better.

Oops, good trick with the \sum, I always wondered how to get the sigma bigger.

Fixed it now

 

[Ferramentarius]

There are matrix exponentials for a given matrix X of nxn dimensions defined similarly to the ordinary exponential of a number.

e^X = $\sum$$^{∞}_{k=0}$ $\frac{1}{k!}$ X^k

 

[Mark44]

Oops, good trick with the \sum, I always wondered how to get the sigma bigger.

Fixed it now

Greek letters have upper and lower case forms: sigma is lowercase ($\sigma$) and
Sigma is uppercase ($\Sigma$).