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?

  • $\begingroup$ 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... $\endgroup$
    – Chris C
    Commented May 4, 2015 at 18:38
  • 6
    $\begingroup$ 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 $y_2/y_1$. $\endgroup$
    – mweiss
    Commented May 4, 2015 at 18:40
  • $\begingroup$ 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 $\endgroup$ Commented May 4, 2015 at 19:10
  • 3
    $\begingroup$ Maybe they use $\Delta \log$ instead. $\endgroup$ Commented May 4, 2015 at 20:18
  • 1
    $\begingroup$ $\Delta \log$ 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. $\endgroup$
    – mweiss
    Commented May 4, 2015 at 21:42

2 Answers 2


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.