Question

The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying "You can’t add apples and oranges." If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form [infinity]∑n=0anxn=a0+a1x+a2x2+a3x3+⋯, where the an are dimensionless constants for all n=0,1,2,⋯ and x is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency.

a) Because standard mathematical functions have infinite sums
b) Because the power series expansions involve dimensionless constants
c) Because dimensional consistency implies dimensionless arguments for mathematical functions
d) Because power series expansions cannot have dimensional constants

Answer 1

The requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency because standard mathematical functions can be expressed as power series expansions with dimensionless constants and powers of the argument.

Step-by-step explanation:

The requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency because standard mathematical functions, such as trigonometric functions, logarithms, and exponential functions, can be expressed as infinite sums called power series expansions.

In a power series, the terms are dimensionless constants multiplied by powers of the argument of the function. Since each term in the power series must have the same dimension, it implies that the arguments of standard mathematical functions must be dimensionless. Therefore, the requirement of dimensional consistency implies dimensionless arguments for mathematical functions.