You’re probably here because you saw a weird symbol in a textbook or a calculator spat out a number you didn’t expect. It happens. Math has a way of looking like an alien language until someone actually breaks it down. Honestly, e squared—written as $e^2$—is one of those things that sounds intimidating but is basically just a very specific, very important number.
It’s roughly 7.389.
That’s the short answer. But if you’re trying to understand why your physics professor is obsessed with it, or why it keeps popping up in your finance app, there's a lot more under the hood.
What Exactly Is "e" Anyway?
Before we talk about squaring it, we have to talk about $e$. Most people know $\pi$ because of circles. But $e$ is the "natural" constant. It’s roughly 2.71828. It was "discovered" (though some argue invented) by Jacob Bernoulli while he was looking at compound interest.
Imagine you have a dollar. A bank gives you 100% interest. If they credit it once a year, you have two bucks. If they credit it every six months, you get a little more because of the compounding. If they credit it every second—or every microsecond—you don't get infinite money. You hit a limit. That limit is $e$.
It’s the speed limit of growth.
Squaring the Constant
So, when we talk about e squared, we are taking that fundamental growth constant and multiplying it by itself.
$$e^2 \approx 2.71828 \times 2.71828 \approx 7.38905609893$$
In practical terms, $e^2$ represents what happens when something grows at a 100% continuous rate for two units of time. Think of it as "double-strength" natural growth.
Why Does e Squared Matter in the Real World?
You don't just see this in math homework. It’s everywhere.
Take statistics. If you’ve ever looked at a Bell Curve (the normal distribution), you’re looking at $e$. The formula for that curve uses $e$ raised to a power. When you’re calculating things like "standard deviations," you often run into $e^2$ tucked away in the probability density functions. It’s the math that tells us how likely it is that a "once in a lifetime" flood will actually happen this year.
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In physics, it’s even cooler. Radioactive decay follows an exponential path. If you want to know how much of a substance is left after a specific amount of time, you’re using $e$. Specifically, $e^2$ often pops up when looking at the "mean life" of a particle or the damping of an oscillation in a car’s suspension system.
The Finance Connection
Money is where most people actually feel the effects of $e$. If you have an investment that is continuously compounded, the formula is $Pe^{rt}$.
- $P$ is your principal (initial money).
- $r$ is the rate.
- $t$ is time.
If your interest rate is 100% (we can dream, right?) and you leave it for 2 years, your money grows by a factor of $e^2$. You’d turn $1,000 into roughly $7,389. That is the power of compounding that Bernoulli was obsessing over.
Common Misconceptions About e Squared
A lot of students confuse $e$ with a variable like $x$ or $y$. It isn't. It's a constant, like the speed of light. You can't change it.
Another big one? People think $e^2$ is just "roughly 7." In high-level engineering, those decimals matter. If you’re building a bridge or a circuit board and you round $e^2$ down to 7, the thing might literally fall apart or short circuit. Precision is the whole point of using transcendental numbers in the first place.
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Also, don't confuse it with $2^e$. That’s a totally different number. Order matters.
How to Calculate e Squared on Any Device
Most people don't memorize 7.389. You shouldn't either.
On a standard scientific calculator, you usually see an $e^x$ button. You hit that, type "2," and hit equals. Done. If you're on a Mac or iPhone, flip your calculator sideways to get the scientific view. On Google, you can literally just type "e^2" into the search bar, and it’ll give you the answer to more decimal places than you’ll ever need.
In Excel or Google Sheets, the formula is =EXP(2).
The Nuance of Natural Logarithms
You can't talk about e squared without mentioning the natural log ($ln$). They are two sides of the same coin. If $e^2 = 7.389$, then the $ln(7.389) = 2$.
Scientists use this to "undo" growth. If they know how much bacteria is in a petri dish now, and they know the growth rate, they use the natural log to figure out how long it’s been growing. It’s like hitting the rewind button on a video.
Real-Life Example: Cooling Coffee
Newton’s Law of Cooling uses $e$. If you leave a hot cup of coffee in a room, it doesn't cool down at a steady rate. It cools fast at first, then slows down as it gets closer to room temperature. This is an exponential decay. If you were to track the temperature difference over two time constants, you’d be dealing directly with the ratio defined by $e^2$.
Actionable Steps for Mastering Exponential Math
If you actually want to use this knowledge rather than just reading about it, try these three things:
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- Check your 401k or Savings: Look for the words "compounded continuously." If you see them, know that $e$ is the engine driving your interest.
- Practice the "Rule of 72": While not exactly $e$, it’s a mental shortcut for exponential growth. Divide 72 by your interest rate to see how long it takes to double your money. It’s the "civilian" version of $e$-based math.
- Visualize the Curve: Go to a site like Desmos and type in $y = e^x$. Look at how steep it gets when $x$ hits 2. That vertical jump from $x=1$ to $x=2$ is the jump from 2.7 to 7.3. It shows you why "exponential growth" is such a scary/exciting term in news headlines.
Understanding $e^2$ isn't about being a math genius. It's about recognizing the pattern of how the world actually works—from the way heat leaves your coffee to the way your debt grows if you don't pay off that credit card. It’s the constant of change.