Most people are familiar with Leonhard Euler and his various contributions to mathematics. He is credited for introducing many modern notations such as f(x), the notations for trigonometric functions, as well as the notation for imaginary numbers being \(i\). Most notably, he is credited with the number \(e\), hence the name Euler’s number, though the discovery was not solely his own. However, Euler did manipulate the number in ways allowing him to discover more unique properties of it. For example, he was the first to use it as the base of logarithm leading to the discovery of the natural logarithm. \(e\) is arguably the most useful number in mathematics, especially calculus, as well as other STEM fields. It has a special property where, given the function \(f(x)=e^x\) , the slope and the area from negative infinity to a certain x value \((\int_{-\infty }^{x } e^{t}dt = e^{x})\) equals f(x) at that point. In other words, \(\frac{\mathrm{d} }{\mathrm{d} x} e^{x}\) and \(\int e^{x}dx\) are both equal to \(e^{x}\), allowing it to be the only function to satisfy the differential equation \(\frac{\mathrm{d}y }{\mathrm{d} x}=y\). For these reasons, \(e^{x}\) is known as the natural exponential function and comes up in many fields of STEM including physics, chemistry, finance, electronics, computer science, and more.

However, Euler also discovered something arguably more fascinating known as Euler’s formula. He discovered this while playing with the Maclaurin series expansions of \(e^{x}\), \(\sin (x)\), and \(\cos (x)\). More broadly, the Taylor series approximates a function using higher order derivatives at \(x = a\) with the given general formula: \(\sum_{n=0}^{\infty } \frac{{f}^{(n)}(a)}{n!}(x-a)^{n}\) . The Maclaurin series is just evaluated for \(a = 0\) and is equivalent to the Taylor series if the radius of convergence is infinite. For functions with an infinite radius of convergence, such as those Euler was using, the Maclaurin series expansion, or their power series, becomes more exact as the number of terms approaches infinity. The power series Euler utilized included:

\(e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\)

\(\cos(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n}}{(2n)!}\)

\(\sin(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n+1}}{(2n+1)!}\)

Euler curiously substituted \(x\) with \(ix\) and expanded the power series. What he got was an infinite complex series. He noticed that all the even powers within the series, \((x^{2},x^{4},x^{6}…)\) got rid of the \(i\), since \(i^2=-1\). Euler isolated the even powers of \(x\) and noticed it is equivalent to the power series of \(\cos (x)\). Furthermore, since all the odd powers \((x,x^{3},x^{5}…)\) still included \(i\), he factored it out and the remaining infinite series was equal to the power series of \(\sin (x)\). From this, he discovered the famous Euler’s formula: \(e^{i\theta}=\cos(\theta)+i\sin(\theta)\).

On the real plane, \(e^{ix}\) looks as a normal \(\cos (x)\) function. However, on the complex plane, it takes the curve of \(\sin (x)\). On a three-dimensional plane containing an x, y, and an imaginary axis, the graph takes the shape of a spiral. This formula manages to relate algebra, trigonometry, and imaginary numbers.

Additionally, Euler’s identity is derived from this formula. If we were to plug in \(\Pi \) into the function, we get \(e^{\pi i}=\cos(\pi)+i\sin(\pi)\). This evaluates to -1, giving the famous identity \(e^{\pi i}=-1\) or \(e^{\pi i}+1=0\).