In mathematics, there are few numbers that are used quite often such as $\pi$, which is used in arithmetic, trigonometry, etc. Similarly, Euler’s constant ($e$) is also one such important number. It is an irrational number, but is applied in several different fields unlike other irrational numbers. In this article, you’ll know what does e mean in math and why it is so important.

## What Does e Mean in Math and From Where It From?

The number represented by $e$ (and not $E$) was discovered by mathematician Leonard Euler as a solution to a financial problem posed by another mathematician, Jacob Bernoulli.

The problem was similar to the one described below.

Suppose you put ₹$1,000$ in a bank that pays $100\%$ annual compound interest and leave it there for a year. You’ll have ₹$2,000$. Now suppose the interest rate is half that, but the bank pays it twice a year. At the end of a year, you’d have ₹$2,250$. Now suppose the bank paid only $8.33\%$, which is $\frac{1}{12}$ of $100\%$, but paid it $12$ times a year. At the end of the year, you’d have ₹$2,613$. The general equation for this progression is: $ \left(1 + \frac{r}{n} \right)^{n}$ (Compound Interest formula), where $r$ is rate of interest and $n$ is the payment period.

It turns out that, as n approaches infinity, the result gets closer and closer to $2.7182818284 \left(e \right)$. This is how Euler discovered it. The maximum return you could get on an investment of ₹ $1,000$ in one year would be in case of ‘**compounded continuously**’ which is ₹ $2,718.28$.

Continuous compounding is the mathematical limit that compound interest can reach if it’s calculated and reinvested into an account’s balance over a theoretically infinite number of periods in a year basically every nanosecond (in fact, much much faster than that). While this is not possible in practice, the concept of continuously compounded interest is important in finance.

## Significance of Euler’s Constant (e)

If you plot a graph of $y = e^{x}$, you’ll get a curve that increases exponentially, just as you would if you plotted the curve with base $10$ or any other number. For that reason, exponents with $e$ as a base are known as natural exponents.

The curve $y = e^{x}$ has two special properties.

- The value of $y$ equals the value of the slope of the graph at that point.
- The value of $y$ equals the area under the curve up to that point.

This makes e an especially important number in calculus and in all the areas of science that use calculus.

## Applications of e

Euler’s constant, $e$ has many real life applications, such as:

- It is used in growth and depreciation problems, such as population models.
- The base rate growth shared by all continuously growing processes
- In calculus, to find slopes and areas under curves.
- In physics it is used in equations of waves, such as light waves, sound waves and quantum waves.

## Conclusion

The number $e$, like $\pi$, is an irrational number, because it has a non-recurring decimal that stretches to infinity. It’s one of the most useful numbers in mathematics finding a place in many applications.

## Practice Problems

**Which of the following is true?**

- $ \log x = \log_{10}x$
- $ \log x = \log_e x$
- $ \ln x = \log_{10} x$
- $ \ln x = \log_e x$

**Which of the following is true?**

- $e^{x}$ is inverse of $ \log x$
- $e^{x}$ is inverse of $ \ln x$

**Which of the following is a graph of $y = e^{x}$?**

**Which of the following is a graph of $y = \ln x$?**

**After how many years an amount invested at $8 \%$ p.a. compounded continuously will (Use the formula $A = Pe^{rt}$)**

- Double itself
- Tiple itself

**At what rate of interest will an amount double itself in 3 years compounded continuously? (Use the formula $A = Pe^{rt}$)**

## Recommended Reading

- Difference Between Axiom, Postulate and Theorem
- How Was The Area Formula for a Circle Derived?
- Understanding Successive Discount

Image Credit: Increase sales vector created by pch.vector – www.freepik.com