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
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