Introduction

When we introduced the circular exponential, the trigonometric functions cosine and sine were defined as its real part and imaginary part. From this, we derived the analytical expressions: \(\cos x=\sum_{n=0}^{+\infty} (-1)^n\dfrac{x^{2n}}{(2n)!}\) and \(\sin x=\sum_{n=0}^{+\infty} (-1)^n\dfrac{x^{2n+1}}{(2n+1)!}.\) We also rediscovered the essential properties \(\cos^2 t+\sin^2=1\), as well as \(\cos’t=-\sin t\) and \(\sin’t=\cos t\). Based solely on these properties, it is possible to define the number \(\pi,\) and to study its relationship with the circular exponential.

1.The cosine function has zeros on $\mathbb{R}_+$

Let us first show that the cosine function has zeros on $\mathbb{R}_+.$ By definition, we have $\cos 0 = 1$, and if the cosine did not have zeros on $\mathbb{R}_+,$ by continuity we would have $\cos x > 0$ for all $x \in \mathbb{R}_+.$ In this case, the sine function, whose derivative is $\cos x,$ would be (strictly) increasing: choosing a real number $a > 0,$ for any real number $x > a$ we would have $\sin x > \sin a > 0,$ since $\sin 0 = 0$.

Now consider the function $x \in I = ]a, +\infty[ \mapsto \cos x + x \sin a,$ whose derivative is $-\sin x + \sin a = \sin a – \sin x < 0.$ On the interval $I,$ this function would thus be strictly decreasing. However, since the cosine function is bounded and $\sin a > 0,$ this function has a limit of $+\infty$ as $x$ tends to $+\infty.$ These two properties are mutually contradictory, so we must conclude that the hypothesis is false: the cosine function has zeros on $\mathbb{R}_+.$

2.The smallest positive zero of the cosine function

Let us reformulate the conclusion of the previous proof: since $\cos 0 = 1,$ there exists at least one real number $x > 0$ such that $\cos x = 0$. By continuity of the cosine function, we can say a bit more: since $\cos 0 = 1 \neq 0$, the cosine does not vanish “in the neighborhood of $0$”, meaning that on an open interval $I$ containing $0$, the cosine does not vanish. In other words, the infimum of the set $X = \{x \in \mathbb{R}_+ : \cos x = 0\}$ is a strictly positive number: let us denote it by $a$. By definition, we then decree that the number $\p$ is the number $2 \times a$, that is, $a = \pi / 2$.

Now, the properties of the infimum and the continuity of the cosine function imply that $\cos a = 0$ (for instance, there exists a decreasing sequence $(x_n)$ of real numbers in $X$ that converges to $a$). In summary, the number $a = \pi / 2$ is the smallest positive zero of the cosine function. An analytic-geometric definition of the number $\pi$ is therefore that it is twice the smallest positive zero of the function $\cos x$.

3.The number $\pi$, the sine, and the circular exponential

If we have used circular functions to define the number $\pi$, this number has quite particular relationships with these functions. Since the cosine, the derivative of the sine, is $>0$ on the interval $[0, \pi/2[$, the sine function is strictly increasing on $[0, \pi/2]$. And since $\cos^2 \pi/2 + \sin^2 \pi/2 = 1$ and $\cos \pi/2 = 0$ by the definition of $\pi$, it follows that $\sin^2 \pi/2 = 1$, hence $\sin \pi/2 = 1$. It immediately follows that $e^{i\pi/2} = \exp(i.\pi/2) = \cos \pi/2 + i\sin \pi/2 = i$.

Due to the properties of the exponential, we can then deduce that $\exp(i\pi) = \exp(i \times 2 \times \pi/2) = \exp(i\pi/2)^2 = i^2 = -1$. Thus, we obtain the famous Euler formula:
$$e^{i\pi} + 1 = 0$$
(using the notation with an exponent for the circular exponential). From this, we deduce that $\cos \pi = -1$ and $\sin \pi = 0$ (which can also be derived from the formulas for $\sin(a+b)$ and $\cos(a+b)$).

Similarly, we can prove that $\exp(2i\pi) = 1$, and therefore for any complex number $z$, we have $\exp(z + 2i\pi) = \exp(z)$. In fact, the number $2\pi$ is the smallest real number $t > 0$ such that $\exp(it) = 0$, which is the period of this function.

Returning to the real trigonometric functions, we can deduce from the study of cosine and sine on the interval $[0, \pi/2]$ their behavior over the entire interval $[0, 2\pi]$, and thus over the whole set $\mathbb{R}$, using their elementary properties and periodicity. This study would precisely establish that the circular exponential defines a bijection from the interval $[0, 2\pi[$ onto the unit circle $S^1$, the set of complex numbers with modulus $1$.

Conclusion: Defining the number $\pi$

It might seem unnecessarily complicated to use the rigorous analytical definition of circular functions to define the number $\p$. There are other ways to define this number, or at least to provide expressions that could serve as definitions, but they all seem, in some way, to involve analysis, likely due to the transcendental nature of $\pi$ (it is not the solution of any algebraic equation). For example, one can define the number $\p$ from the formula $\sum_{n=1}^{+\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}$, the sum of a Riemann series. However, such a definition would need to characterize $\pi$ as the number defined here…

One could also consider defining $\pi$ as the smallest zero $>0$ of the sine function, using an approach similar to the one presented here. However, since $\sin 0 = 0$, another idea must be used instead of the one presented here; for example, one could consider the $\sin$ function as the solution to the differential equation $y^{(2)} + y = 0$ such that $y(0) = 0$ and $y'(0) = 1$, which is more advanced than the study conducted here. Finally, it is possible to define $\pi$ using the reciprocal trigonometric functions arctangent, arcsine, or arccosine and the theory of integration, which also introduces additional considerations. Thus, the present definition is one of the ‘best’ definitions of the number $\pi$, to our knowledge.