# Is there a good notation for “ratio” comparable to the use of $\Delta$ for “difference”?

It is standard to use the symbol $\Delta$ to indicate a change in a quantity between two points on a curve, two rows on a table, and so forth. For linear functions, we write slope = $\Delta y / \Delta x$; this notation is used all over the place in Physics and Chemistry.

But when looking at exponential functions, the quantity that relates naturally to $\Delta x = x_2-x_1$ is not $\Delta y = y_2 - y_1$, but rather $\bf{\frac{y_2}{y_1}}$. This suggests that it would be helpful to have some kind of notation to signify "multiplicative change," in analogy with using $\Delta$ to signify "additive change".

Is there such a symbol? Is it used in any textbooks?

• I have never seen $\Delta y = \frac{y_2}{y_1}$ used. Mind providing a reference? The usual $\Delta y = y_2 - y_1$ is the only one I see in math, physics, chemistry, engineering... – Chris C May 4 '15 at 18:38
• I think you misunderstood what I wrote (or I miswrote it). My whole point is that $\Delta y$ means $y_2 - y_1$, and I'm wondering if there is an analogous (different) notation for $y_2/y_1$. – mweiss May 4 '15 at 18:40
• There is $\rho$ which is used more for variable density. There are texts which use a multiplicative scale instead of an additive scale in considering alternative forms of the calculus. I know of no widely accepted notation for the concept you consider. Gerhard "Living In A Null-Measure Set" Paseman, 2015.05.04 – Gerhard Paseman May 4 '15 at 19:10
• Maybe they use $\Delta \log$ instead. – Gerald Edgar May 4 '15 at 20:18
• $\Delta \log$ would do the mathematical work I have in mind, but in terms of instructional sequencing it puts the cart before the horse. I am thinking about the way in which one introduces exponential functions: You look at a table of values in which $x$ increments by some uniform amount, and say "Look, every time $x$ goes up by such-and-such, corresponding $y$ value multiplies by such and such". I was just thinking how odd it is that we can write "$\Delta x = 1$" to indicate that $x$ increases by $1$, but don't have an analogous way to indicate that $y$ increases by a constant multiplier. – mweiss May 4 '15 at 21:42

## 2 Answers

I likewise have needed such a thing—to converse coherently about transformation properties (actually symmetries) of power, exponential, and logarithmic functions. I did not find anything in textbooks or anything commonly used otherwise. I considered using $\rho$ for "ratio", but settled on $\sigma$ (and $\sigma_y$ as needed), because in conversing, "scaling" was more used than "ratio." Following are examples of summary statements.

An exponential function $f(x) = A b^x$ scales by the same factor over equal increments of the input. The base $b$ gives the factor over a unit increment. The base to the power of an input increment $\Delta x$ gives the output factor: $$\sigma_y = b^{\Delta x}.$$ The graph is invariant under the family of transformations $$(x,y) \rightarrow (x + \Delta, b^\Delta y)$$ where $\Delta$ can be any real number.

We use $\Delta x$, but "$\sigma y$" is clumsy, so I use a subscript. I try to avoid the subscript. I would write the transformation property ("equivariance") of the function as $$f(x + \Delta) = b^\Delta f(x),$$ provided students were ready for such symbolic stuff. (Usually some are, some aren't—a chance to "differentiate" instruction.) In making tables of values, often a column would be labeled with a $\sigma_y$ (or $\sigma_x$ for logarithmic and power functions).

With exponential functions I mostly avoid this $\sigma_y$ notation. If we have the exercise of finding the parameters for an exponential curve going through $(1,6)$ and $(3,30)$, say, I would encourage putting the data in a table and writing $\Delta x = 2$ on the $x$ side and $b^2 = 5$ somewhere, just saying 5 is the output scaling associated to an input increment of 2, rather than writing $\sigma_y = 5$. Solve for the base, and go from there.

I think its worth noting that the obelus symbol ($\div$) coincidentally looks like a vertical fraction and does denote division. http://en.wikipedia.org/wiki/Obelus

It could be used as a table header pretty easily:

$x \ \ \ y \ \ \div y$
$2 \ \ \ 7 \ \$
$4 \ \ 35 \ \ \ \ 5$

But could get confusing as an inline symbol: $\div y=5$ or $\frac{\div y}{\div x}=5/2$ don't quite look right, and try saying "Ten times the y ratio": $10\div y$ We'd get confused .

Maybe $\div$ with a triangle or circle around it? I don't know of a premade symbol like that though.