Let's dive into one of the most important constants in mathematics—the number e. Known as Euler’s constant, this value is approximately 2.718 and it pops up everywhere, especially when dealing with exponential growth and decay. In fact, e is the base of the exponential function, written as $f(x) = e^x$, which is super useful in modeling real-world situations like population growth or compound interest.

In this article, we’ll break down what e really means, how to calculate it, and why it’s so important. Whether you're preparing for the SAT, ACT, or even the MCAT, understanding e is crucial. It’s used in all sorts of equations that pop up in calculus, probability, and statistics. We’ll also go through some examples to help you really grasp how it works. Ready to uncover the magic of e? Let’s get started!

Euler’s Number (e)

A mathematical constant is a key number whose value is fixed by an unambiguous definition. It is commonly denoted by a symbol (e.g., an alphabet letter) or by the names of mathematicians to make it easier to use across many problems. Euler’s number was first used by John Napier while working on logarithmic functions. The letter e was introduced by Leonhard Euler as the base for natural logarithms. Hence, it came to be known as Euler’s constant. The constant was also used by mathematician Jacob Bernoulli while studying compound interest.

The Euler’s constant is a non-terminating decimal which means the digits after the decimal point extend to infinite numbers. e is also an irrational number. It means that it cannot be represented as a ratio of integers. It is also a transcendental number, that is, it is not a root of any non-zero polynomial with rational coefficients.

Value of Exponential Constant

The exponential constant, or e, is a mathematical constant with a value given by

\(e=\sum_{n=0}^\infty{1\over{n!}}\)

OR

\(e=\lim_{n\rightarrow\infty}{(1+{1\over{n}})^n}\)

On solving these equations we can get the value of e upto a certain number of decimal places, 2.71828182845904523536028747135266249775724709369995… We round it off to 2.718 to use for calculation purposes. Natural logarithms have it as their base.

Steps to Calculate the Value of e

The exponential constant ‘e’ is the limit of \((1 + {1\over{n}})^n\) as n approaches infinity. It can be mathematically shown as,

\(e = \sum_{n=0}^{\infty } \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} …..\)

\(e = \sum_{n=0}^{\infty } \frac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1.2} + \frac{1}{1.2.3} + \frac{1}{1.2.3.4} + \frac{1}{1.2.3.4.5}…..\)

Where n! means factorial of n; n = 1, 2, 3, 4, …

Let’s try to find the value of e up to few places after the decimal.

\(e = \sum_{n=0}^{\infty } \frac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1.2} + \frac{1}{1.2.3} + \frac{1}{1.2.3.4} + \frac{1}{1.2.3.4.5}…..\)

\( \frac{1}{1.2} = \frac{1}{2} = 0.5\)

\(\frac{1}{1.2.3} = \frac{1}{6} = 0.166\)

\(\frac{1}{1.2.3.4} = \frac{1}{24}= 0.04166\)

\(\frac{1}{1.2.3.4.5} = \frac{1}{120} = 0.008333\)

Substituting the values back in the main equation we get,

\(e = 1+ 1 + 0.5 + 0.166 + 0.041666+ 0.008333\) = 2.716667

Importance of e

The most typical use of e is to account for population growth in practically any evolving population. This is due to the fact that it may be used as a parameter to generate an exponential function, which has numerous real-world applications in areas such as inflation, compounding interest, probability distributions and fourier transforms.

In calculus, physics, and engineering, the value of e has numerous uses. It is used to find derivatives and integrals in calculus. It is used in physics to calculate change and decay rates. In the designing of electrical circuits and spring mass systems, e appears in the solution of the differential equations. It also appears in Newton’s law of cooling. It’s used in engineering to figure out vibrational frequencies and energy levels.

Solved Examples related to the Value of e

Let’s see some solved examples on Value of e.

Solved Example 1: Find the value of r if: \(re^{-3} = e^{-2}(lne^{-2})^2\)

Solution:

\(re^{-3} = e^{-2}(lne^{-2})^2\)

From the properties of logarithm we know that, \(log_{10}a^n = nlog_{10}a\)

\(re^{-3} = e^{-2}(-2lne)^2\)

lne = 1, as the base is same.

\(re^{-3} = e^{-2}(-2)^2\)

\(re^{-3} = e^{-2}4\)

\(r = e^{-2 + 3}4\)

\(r = e\times4\)

\(r = 10.87\).

Solved Example 2: Find the derivative of \(y=e^x\)

Solution:

\(y=e^x\)

Let’s take log to base e on both sides

\(lny = lne^x\)

From the properties of logarithm we know that, \(log_{10}a^n = nlog_{10}a\)

\(lny = xlne\)

lne = 1, as the base is same.

lny = x

Differentiating on both the sides,

\({1\over{y}}{dy\over{dx}} = {d\over{dx}}{x}\) …….. \([{d\over{dx}}logx = {1\over{x}}]\)

Thus, \({dy\over{dx}} = y(1) = y …….. [{dx\over{dx}}=1]\)

\({dy\over{dx}} = e^x\)

Solved Example 3: Evaluate \(f(x) = e^x\) to the nearest thousandth for each value of x below.

Solution:

\(f(x) = e^x\)

x = 2

\(e^2 = 7.389\)

x = ½

\(e^{1\over2} = 1.649\)

x = -1

\(e^-1 = 0.368\)

x = 6

\(e^6 = 403.429\)

\( x = {1\over{3}} \)

\(e^{1\over{3}} = 1.396\)

x = -2

\(e^{-2} = 0.135\)

Conclusion

To wrap things up, e is a super important constant in math that pops up in all sorts of real-world situations, like population growth and interest rates. Understanding how e works is key for tackling problems in calculus, physics, and beyond. Whether you're prepping for the SAT, ACT, or MCAT, getting comfortable with e will definitely give you a boost in solving exponential-related problems. Keep practicing, and you'll master it!