Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

From wikipedia: """In 1885, Hermann Ebbinghaus discovered the exponential nature of forgetting. The following formula can roughly describe it: $R = e^{-t/s}$ where $R$ is memory retention, $S$ is the relative strength of memory, and $t$ is time."""

I'd just like to understand what this math is trying to say. Such that if I want $100\%$ retention level of something, thus as my relative strength is less the more time it will take to remember something. But also that as my retention increases, the less amount of time it takes to have strength of memory reviewed.

So I suppose I can articulate it, but how can I use the equation to roughly calculate how I'm doing remembering something?

Wolfram Alpha puts $t = -s \log(R)$ and gives a funky 3D graph. But if I use $R=1$ then $t=0$. If $R=100$, $t=-900$. So a bit confusing. I think the wikipedia graphic is more germane to understanding that as the y axis is R and that the graph exponentially decays, but at a different rate for each new iteration. Kicking the can down the road so to speak...

So let's go for a low, medium and high retention graph based on hours in a day. How can this be used in the formula to determine the time needed to study something again?

Let's say I have a class at 8am and I want to review it. I'll have to review it early on, then increasingly less. I'm just trying to ballpark when the best time to study is based on this decay graph.

Hope that makes sense.

share|cite|improve this question
The formula doesn't measure how long it will take you to remember something, it measures how long you will remember something (that is, how long it takes you to forget it). The larger the value of $s$, the slower the decay is, which means, the stronger your relative strength, the longer you will remember it. – Arturo Magidin Jun 20 '11 at 3:05
It seems to me that $R$ is intended to be a percentage - it's the fraction retained - so putting in $R=100$ (or any value of $R$ not between 0 and 1) is guaranteed to give nonsensical results. – Gerry Myerson Jun 20 '11 at 3:12
@Arturo Magidin : thank so much, that helps a lot. Copy paste your comment into an answer and i'll check it. Cheers – sf2k Jun 20 '11 at 17:07
@Gerry Myerson: thank you, that makes a lot more sense – sf2k Jun 20 '11 at 17:08
Gerry you too. Just a note for viewers that both comments above are the answer for me, and helped my understanding. – sf2k Jun 20 '11 at 17:24
up vote 1 down vote accepted

As requested (and a bit of an expansion, including Gerry Myerson's further explanations).

It seems that you are misinterpreting the formula in question: it does not describe how long it takes for you to remember something, but rather how long it takes for you to forget it once you already know it.

Here, $R$ is memory retention, and is expressed as a percentage, $0\leq R\leq 1$, with $R=p$ meaning that you still remember p% of what you originally learned. The function $e^{-t/s}$ is decaying, with value $1$ at $t=0$ (when you start, you remember everything). The larger the value of $s$ (the stronger your "relative memory strength"), the slower the decay: for example, for $s=1$ it takes until $t=\ln(2)$ to forget half of what you had memorized, but for $s=2$ it takes double that time, to $t=2\ln(2)=\ln(4)$). In other words, the stronger your relative memory strength, the longer you will remember more of what you had memorized at time $t$.

So this is not something that will tell you how long you need to study something in order to memorize it; it's meant to model how long it takes you to forget it once you have already memorized it.

share|cite|improve this answer

If you are really interested in optimizing your study time, I suggest a program called Anki. The short description is that Anki is a flashcard system that runs an algorithm to determine which cards need reviewing and which don't. I used it to study for my comprehensive exams and found it quite helpful.

share|cite|improve this answer
Yes thanks. That and Mnemosyne are popular. However it's not about using software to do this, for me I was asking about understanding the math formula. Cheers – sf2k Jun 20 '11 at 17:15

I think the theory is not robust enough to answer questions like how many minutes of study the night before is worth one minute of study just before class.

Is this a problem in the Calculus of Variations?

share|cite|improve this answer
no it's more a general interest question, although this type of exponential question format is seen all the time. I was trying to understand the why more. We just pump out answers, it's nice to know what's going on ;) Sounds like Calculus of Variations would be a similar topic, so I'll check that out. Thanks. – sf2k Jun 20 '11 at 18:33
Another exponential phenomenon is charge and discharge of a capacitor. – phv3773 Jun 20 '11 at 18:48

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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