Euler's Number (e): Definition, Value and Proofs

What Is Euler's Number?

Euler's number, written $e$, is the mathematical constant $e = 2.718281828459045\ldots$, the base of the natural logarithm and of the exponential function $e^x$. It is the number that continuous growth at a rate of 100% per unit of time produces from a starting value of 1. Its decimal expansion never ends and never repeats.

Brainly users ask it plainly: "What is 'e'? And where does it come from?" The short answer is that $e$ is one number reached by three different roads, and each road explains a different job it does:

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} + \cdots$$

and $e$ is the only base $a > 0$ for which the exponential function $a^x$ is its own derivative:

$$\frac{d}{dx}\left(e^{x}\right) = e^{x}.$$

Here $n$ is a positive integer that grows without bound, the dots mean the pattern of terms continues forever, and $\frac{d}{dx}$ is the derivative with respect to $x$. In the series, the $k$-th term is 1 divided by $k$ factorial, that is $\frac{1}{1 \cdot 2 \cdot 3 \cdots k}$, with the first term (for $k = 0$) equal to 1.

What Are The Three Ways To Define Euler's Number?

Euler's number has three standard definitions: a limit, an infinite series, and a derivative property. Textbooks pick one as the official definition and prove the other two from it, so all three are equally correct starting points.

Definition 1: Euler's Number As A Limit

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}$$

The base $1 + \frac{1}{n}$ shrinks toward 1 while the exponent $n$ grows. Those two pulls fight, and neither wins outright: the result settles at $2.71828\ldots$, not at 1 or at infinity. This is a limit at infinity of the indeterminate type "$1^{\infty}$", which is exactly why it cannot be read off by plugging in.

Definition 2: Euler's Number As An Infinite Series

$$e = \sum_{k=0}^{\infty} \frac{1}{1 \cdot 2 \cdot 3 \cdots k} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} + \cdots$$

The series converges fast, because each denominator is a product that grows quicker than any power. Eleven terms (up to $k = 10$) already give $2.7182818$, correct to 7 decimal places. This is the definition calculators and computers use, and it is the $x = 1$ case of the Maclaurin series

$$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \cdots$$

Definition 3: The Base Whose Exponential Is Its Own Derivative

Ask the BetterExplained question, "Why is e special? (2.718..., not 2, 3.7 or another number?)", and this definition answers it. For any base $a > 0$, the derivative of $a^x$ is a constant multiple of $a^x$ itself:

$$\frac{d}{dx}\left(a^{x}\right) = a^{x} \cdot \lim_{h \to 0} \frac{a^{h} - 1}{h}.$$

That constant is the slope of $y = a^x$ where it crosses the $y$-axis. For $a = 2$ it is about $0.6931$, and for $a = 3$ it is about $1.0986$. Somewhere between 2 and 3 sits the one base whose slope at the crossing is exactly 1, and that base is $e$. With slope 1, the multiplier disappears and $e^x$ becomes its own derivative.

This is the definition that takes longest to feel natural, and most textbooks state it as a fact and move on. The geometric picture helps: on the curve $y = e^x$, the slope of the tangent line at every point equals the height of the curve at that point.

Why Do All Three Definitions Give The Same Number?

The limit and the series meet through the binomial expansion. Expanding $\left(1 + \frac{1}{n}\right)^{n}$ gives a sum whose $k$-th term is

$$\frac{1}{1 \cdot 2 \cdots k}\left(1 - \frac{1}{n}\right)\left(1 - \frac{2}{n}\right)\cdots\left(1 - \frac{k-1}{n}\right).$$

As $n \to \infty$, every bracket tends to 1, so each term tends to the matching series term $\frac{1}{1 \cdot 2 \cdots k}$. A careful proof also controls the tail, but the matching term by term is why the two definitions agree.

The derivative definition links to the limit through logarithms. The slope constant $\lim_{h \to 0} \frac{a^{h} - 1}{h}$ equals $\ln a$, and $\ln a = 1$ exactly when $a = e$. The derivatives of exponential functions page works through the general rule $\frac{d}{dx}\left(a^{x}\right) = a^{x} \ln a$ that falls out of this.

What Is The Value Of $e$?

The value of Euler's number to 20 decimal places is

$$e = 2.71828182845904523536\ldots$$

For most school and college work, $e \approx 2.71828$ is plenty, and $e \approx 2.718$ is the common 3-decimal value.

A Quora reader asks, "Is there an intuitive way to understand why Euler's number is 2.718, especially for non-mathematicians?" The intuitive route is the compounding table below. Each row splits a year of 100% interest into $n$ equal periods, so the growth factor is $\left(1 + \frac{1}{n}\right)^{n}$:

The jumps shrink fast. Going from yearly to monthly adds more than 0.6, while going from hourly to every minute adds less than 0.0002. The growth is capped, and the cap is $e$.

Why Must $e$ Lie Between 2 And 3?

The series gives a two-line proof. The first two terms already add to 2, so $e > 2$. For the upper bound, each denominator $1 \cdot 2 \cdot 3 \cdots k$ is at least $2^{k-1}$, so

$$e < 1 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1 + 2 = 3,$$

where the bracketed tail is a geometric series with sum 2. Jacob Bernoulli reached the same "between 2 and 3" bound in 1683.

Where Does $e$ Come From? The Compound-Interest Origin

$e$ was found by asking a money question, not a geometry question. A Brainly.in user asks "what the origin of e in maths", and the honest answer starts in 1683, when Jacob Bernoulli studied compound interest. His question was what happens to a loan as interest is added more and more often. According to the MacTutor history of the number e, Bernoulli tried to find the limit of $\left(1 + \frac{1}{n}\right)^{n}$ as $n$ grows and showed that the limit lies between 2 and 3.

The problem $e$ solves is a real one. Without it, every lender, bank and scientist modelling smooth growth would face a different formula for every compounding schedule:

• The discrete version. Principal $P$ at annual rate $r$ compounded $n$ times a year for $t$ years grows to $P\left(1 + \frac{r}{n}\right)^{nt}$. The standard interest formula covers this case.

• The continuous version. Let $n \to \infty$ and the same expression becomes $P e^{rt}$, one clean formula with no schedule in it. Euler's number is what is left when the schedule disappears.

• The general lesson. Any quantity that grows in proportion to its current size, without pauses, grows like $e^{kt}$. Bacteria, radioactive atoms and cooling coffee all follow this law, which is why the exponential growth and decay models are built on $e$ rather than on 2 or 10.

The order of discovery is worth noticing. Natural logarithm tables appeared in 1618 in an appendix to John Napier's work, and Nicolaus Mercator used the term "natural logarithm" in 1668, yet nobody named the base. The number sat underneath logarithms for decades before the interest problem forced it into view.

Is $e$ Irrational, And Is It Transcendental?

Yes to both. Euler's number is irrational, meaning it cannot be written as a fraction $\frac{p}{q}$ of whole numbers, and it is transcendental, meaning it is not the root of any polynomial equation with integer coefficients. Like pi, its decimal form never ends and never repeats.

A Short Proof That $e$ Is Irrational

The argument now usually credited to Joseph Fourier uses only the series. Suppose, for contradiction, that $e = \frac{p}{q}$ for positive whole numbers $p$ and $q$. Multiply both sides by the product $N = 1 \cdot 2 \cdot 3 \cdots q$ (that is, $q$ factorial):

$$N e = N\left(1 + 1 + \frac{1}{2} + \cdots + \frac{1}{1 \cdot 2 \cdots q}\right) + N\left(\frac{1}{1 \cdot 2 \cdots (q+1)} + \cdots\right).$$

• The left side is a whole number, because $N e = N \cdot \frac{p}{q}$ and $q$ divides $N$.

• The first bracket times $N$ is a whole number, because every denominator up to $1 \cdot 2 \cdots q$ divides $N$.

• The tail times $N$ is not a whole number. It equals $\frac{1}{q+1} + \frac{1}{(q+1)(q+2)} + \cdots$, which is positive and smaller than the geometric sum $\frac{1}{q+1} + \frac{1}{(q+1)^{2}} + \cdots = \frac{1}{q} \leq 1$.

A whole number cannot equal a whole number plus something strictly between 0 and 1. So no such $p$ and $q$ exist, and $e$ is an irrational number.

Who Proved $e$ Is Transcendental?

Charles Hermite proved in 1873 that $e$ is not an algebraic number, which is the formal meaning of transcendental. Irrationality only rules out degree-one equations like $q x - p = 0$, while transcendence rules out every polynomial equation with integer coefficients at once. Most people credit Euler with the first proof that $e$ is irrational, more than a century earlier.

Why Is 2.718 Called Euler's Number And Not Bernoulli's?

The Quora question "Why is 2.718.. Euler's number? Isn't that unfair to Bernoulli?" has a fair answer: Bernoulli found the limit, but Euler named the number, computed it and built the theory around it. The MacTutor record gives the sequence:

• 1683: Jacob Bernoulli meets the limit $\left(1 + \frac{1}{n}\right)^{n}$ while studying compound interest and bounds it between 2 and 3.

• 1690: Gottfried Leibniz, writing to Christiaan Huygens, uses the letter $b$ for the number, the first time it appears as a number in its own right.

• 1731: Leonhard Euler uses the letter $e$ in a letter to Christian Goldbach.

• 1748: Euler's Introductio in Analysin infinitorum gives the full treatment: the series, the limit, and a value of $e$ to 18 decimal places.

Bernoulli is far from forgotten. The Bernoulli numbers in number theory carry his name, so the credit is split rather than lost. (Euler, for his part, has so many results named after him that "Euler's number" can confuse; the next section sorts that out.)

How Is Euler's Number Different From Euler's Constant And Euler's Identity?

Euler's number $e \approx 2.71828$ is a different quantity from Euler's constant $\gamma \approx 0.57722$, and both differ from Euler's identity, which is an equation rather than a number. The three share a name and nothing else at first sight:

Euler's identity uses Euler's number, so the two are linked: it is the case $\theta = \pi$ of Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$. Euler's constant $\gamma$ measures how far the sum $1 + \frac{1}{2} + \cdots + \frac{1}{n}$ runs ahead of $\ln n$, a story that belongs to the harmonic series.

How Does $e$ Connect To The Natural Logarithm And $e^x$?

The natural logarithm $\ln x$ is the logarithm with base $e$, so $\ln x$ answers the question "what power of $e$ gives $x$?". That makes $e^x$ and $\ln x$ inverse functions:

$$e^{\ln x} = x \quad (x > 0), \qquad \ln\left(e^{x}\right) = x, \qquad \ln e = 1.$$

Geometrically, the graphs of $y = e^x$ and $y = \ln x$ are mirror images in the line $y = x$. The curve $y = e^x$ passes through $(0, 1)$, stays above the $x$-axis, and has the $x$-axis as a horizontal asymptote on the left; the curve $y = \ln x$ passes through $(1, 0)$ and has the $y$-axis as a vertical asymptote.

The calculus facts follow from Definition 3:

$$\frac{d}{dx}\left(e^{x}\right) = e^{x}, \qquad \int e^{x},dx = e^{x} + C, \qquad \frac{d}{dx}\left(\ln x\right) = \frac{1}{x} \quad (x > 0).$$

Here $C$ is the constant of integration. For composite exponents, the chain rule adds the derivative of the inside: $\frac{d}{dx}\left(e^{3x}\right) = 3e^{3x}$ and $\frac{d}{dx}\left(e^{x^{2}}\right) = 2x,e^{x^{2}}$. The integration of exponential functions page extends the integral side to forms like $\int e^{ax},dx$.

Examples Of Euler's Number

Example 1

Use the first six terms of the series to estimate $e$, and find the error.

$$e \approx 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120}$$

$$= 1 + 1 + 0.5 + 0.16667 + 0.04167 + 0.00833$$

$$= 2.71667$$

$$\text{error} = 2.71828 - 2.71667 = 0.00161$$

Final answer: $e \approx 2.7167$, about $0.0016$ below the true value.

Example 2

Evaluate $\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{n}$.

A tempting first attempt: as $n \to \infty$, the base $1 + \frac{2}{n}$ tends to 1, and 1 to any power is 1, so the limit "is" 1.

Test that claim with $n = 100$:

$$\left(1 + \frac{2}{100}\right)^{100} = 1.02^{100} \approx 7.2446$$

That is nowhere near 1, and it grows as $n$ grows. The flaw: the base and the exponent change together, so the base never actually equals 1 while the exponent is still finite. The form is "$1^{\infty}$", which is indeterminate.

The rescue is to reshape the expression into the $e$ limit. Let $m = \frac{n}{2}$, so $\frac{2}{n} = \frac{1}{m}$ and $n = 2m$:

$$\left(1 + \frac{2}{n}\right)^{n} = \left(1 + \frac{1}{m}\right)^{2m} = \left[\left(1 + \frac{1}{m}\right)^{m}\right]^{2}$$

$$\lim_{m \to \infty} \left[\left(1 + \frac{1}{m}\right)^{m}\right]^{2} = e^{2}$$

Final answer: $e^{2} \approx 7.3891$.

The general pattern is $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^{n} = e^{a}$.

Example 3

Invest 1000 at 5% a year for 10 years. Compare annual, monthly and continuous compounding.

$$\text{Annual: } 1000\left(1.05\right)^{10} \approx 1628.89$$

$$\text{Monthly: } 1000\left(1 + \frac{0.05}{12}\right)^{120} \approx 1647.01$$

$$\text{Continuous: } 1000,e^{0.05 \times 10} = 1000,e^{0.5} \approx 1648.72$$

Final answer: about 1628.89, 1647.01 and 1648.72. Continuous compounding earns the most, but only 1.71 more than monthly.

Students who first meet $P e^{rt}$ almost always type the rate as $r = 5$ instead of $r = 0.05$. The calculator then reports $1000,e^{50}$, a number with 25 digits, which is the quickest way to spot the error.

Example 4

Find the equation of the tangent line to $y = e^{x}$ at $x = 1$.

$$\frac{dy}{dx} = e^{x}, \quad \text{so the slope at } x = 1 \text{ is } e$$

$$\text{point: } (1, e)$$

$$y - e = e(x - 1)$$

$$y = e,x$$

Final answer: $y = e,x$. The tangent passes through the origin, a neat check that slope equals height on $y = e^x$.

Example 5

Solve $e^{2x} = 10$ for $x$, to 4 decimal places.

$$\ln\left(e^{2x}\right) = \ln 10$$

$$2x = \ln 10$$

$$x = \frac{\ln 10}{2} \approx \frac{2.302585}{2}$$

$$x \approx 1.1513$$

Final answer: $x \approx 1.1513$. Check: $e^{2.3026} \approx 10$.

Example 6

Evaluate $\lim_{n \to \infty} \left(1 - \frac{3}{n}\right)^{2n}$.

$$\left(1 - \frac{3}{n}\right)^{2n} = \left[\left(1 + \frac{-3}{n}\right)^{n}\right]^{2}$$

$$\lim_{n \to \infty} \left(1 + \frac{-3}{n}\right)^{n} = e^{-3}$$

$$\text{limit} = \left(e^{-3}\right)^{2} = e^{-6}$$

Final answer: $e^{-6} \approx 0.0025$. A minus sign inside the bracket turns growth into decay, and the limit lands below 1.

Where Is Euler's Number Used In Real Life?

Euler's number appears wherever a quantity changes at a rate proportional to its own size, and in a few counting problems where nobody expects it:

• Finance: continuously compounded interest $P e^{rt}$, and the rule that money at 7% continuous growth doubles in $\frac{\ln 2}{0.07} \approx 9.9$ years.

• Radioactive decay and carbon dating: the amount left after time $t$ is $N_0 e^{-kt}$, where $k$ is the decay constant.

• Biology and medicine: bacterial growth, and the decline of drug concentration in the blood, both follow $e^{kt}$ models.

• Physics and engineering: a charging capacitor, a cooling object, and a damped spring all have $e$ in their formulas.

• Statistics: the normal distribution curve is built from $e^{-x^{2}/2}$.

• Probability: if $n$ people draw names from a hat for a gift exchange, the chance that nobody draws their own name tends to $\frac{1}{e} \approx 0.3679$. With 10 people it is already $0.3679$ to 4 decimal places.

Why Do Students Get Euler's Number Wrong (And What Is The Fix)?

Mistake 1: Treating $1^{\infty}$ As 1

Where it slips in: Limits of the form $\left(1 + \frac{a}{n}\right)^{n}$, and any question that asks for the value of $\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{n}$.

Don't do this: Send the base to 1 first, then raise 1 to the power, and answer 1.

The correct way: Keep the base and the exponent together. Rewrite the expression as $\left[\left(1 + \frac{1}{m}\right)^{m}\right]^{a}$ and use $\lim_{n \to \infty}\left(1 + \frac{a}{n}\right)^{n} = e^{a}$.

The rusher makes this mistake most. The first instinct on $\left(1 + \frac{2}{n}\right)^{n}$ is to say the base goes to 1, so the answer is 1, and the habit that fixes it is plugging in one large value such as $n = 100$ before writing anything down.

Mistake 2: Confusing $e$ With Euler's Constant Or The Calculator's "E"

Where it slips in: Questions that say "Euler's constant", and calculator screens that show results like 2.5E3.

Don't do this: Use $0.5772$ when the question means $e$, or read 2.5E3 as $2.5 \times e^{3}$.

The correct way: Euler's number is $e \approx 2.71828$, Euler's constant is $\gamma \approx 0.57722$, and a calculator's "E" means "times 10 to the power", so 2.5E3 is 2500.

Mistake 3: Dropping The Chain-Rule Factor

Where it slips in: Derivatives such as $\frac{d}{dx}\left(e^{3x}\right)$ or $\frac{d}{dx}\left(e^{x^{2}}\right)$.

Don't do this: Write $\frac{d}{dx}\left(e^{3x}\right) = e^{3x}$ because "$e^x$ is its own derivative".

The correct way: Only $e^x$ itself is its own derivative. For $e^{g(x)}$, multiply by $g'(x)$: $\frac{d}{dx}\left(e^{3x}\right) = 3e^{3x}$.

The memorizer is the student most exposed here. Someone who learned "e to the x stays the same" as a slogan applies it to every exponent; the student who learned why (the slope constant equals 1 only for base $e$ and input $x$) sees at once that $e^{3x}$ climbs three times as steeply.

The real-world version of Mistake 1 cost a stock exchange half its index. In January 1982 the Vancouver Stock Exchange started a new index at 1000 and, after each of roughly 3000 daily updates, truncated it to three decimal places. Each cut was tiny, exactly the "it is basically 1, so it does not matter" thinking behind treating $1^{\infty}$ as 1, but repeated thousands of times a day it compounded. By November 1983 the index read 524.811, and the correction raised it to 1098.892 (Vancouver Stock Exchange, Wikipedia).

Practice Problems On Euler's Number

1. Estimate $e$ using the first five terms of the series, $1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24}$, to 4 decimal places.

2. Evaluate $\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{3n}$.

3. Find the value of 500 invested at 4% a year, compounded continuously, after 5 years.

4. Differentiate $x,e^{x}$, and evaluate $\int_{0}^{\ln 2} e^{x},dx$.

5. Solve $e^{x} = 5$ to 4 decimal places.

6. Evaluate $\lim_{n \to \infty}\left(1 + \frac{1}{3n}\right)^{n}$ to 4 decimal places.

Answer to Question 1: $2.7083$.

Answer to Question 2: $e^{3} \approx 20.0855$.

Answer to Question 3: $500,e^{0.2} \approx 610.70$.

Answer to Question 4: By the product rule, $(x + 1)e^{x}$; the integral is $e^{\ln 2} - e^{0} = 2 - 1 = 1$.

Answer to Question 5: $x = \ln 5 \approx 1.6094$.

Answer to Question 6: $e^{1/3} \approx 1.3956$.

Key Takeaways

• Euler's number $e \approx 2.71828$ is the base of the natural logarithm and of the exponential function $e^x$.

• It can be defined as the limit of $\left(1 + \frac{1}{n}\right)^{n}$, as the series $1 + 1 + \frac{1}{2} + \frac{1}{6} + \cdots$, or as the one base whose exponential is its own derivative.

• Jacob Bernoulli met it through compound interest in 1683, and Euler named it and built its theory.

• The number $e$ is irrational and transcendental, so its decimals never end or repeat.

• The most common mistake is treating the $1^{\infty}$ form as 1 instead of rewriting it as a power of $e$.

Where To Go From Here

Work through Practice Problems 1 to 6 in order; if Problem 2 or Problem 6 trips you up, return to Example 2 and copy its substitution step, and if Problem 4 does, reread "How Does $e$ Connect To The Natural Logarithm And $e^x$?". Want a live Bhanzu trainer to walk through more Euler's number problems? Book a free demo class.

Read More

• Taylor series and how functions become polynomials

• Infinite series and their partial sums

• Logarithms and the rules for working with them

• Derivatives of logarithmic functions

• Hyperbolic functions built from e^x

• The harmonic series and why it diverges