Engineering Procedure: Other ways to write 2 I am currently writing a college essay that is utilizing the engineering procedure. I am going to answer my question with the ways I know.  Are there any other ways to write mathematically write 2 besides the infinite series?
 A: $(1+1)^1$
Largest even prime
number of last names of the presidents with the same last name
A: $$2=\pi\prod_{i=1}^{\infty}\left(1-\frac{1}{4n^2}\right)$$
$$2=\sqrt{\frac{\pi}{\arctan(1)}}$$
It is also the smallest even prime.
A: $$2=\left(\sum_{k\ge 1}\frac{k}{\left(\left(\ln\left(\lim\limits_{n\to\infty}\left(1+\frac1n\right)^n\right)+\sin^2\theta+\cos^2\theta\right)\sum_{n\ge 0}\frac{\cosh y\sqrt{1-\tanh^2y}}{2^n}\right)^{\frac1{k+1}\binom{k+1}2}}\right)!$$
A: \begin{align}
2 &= \sqrt{\frac1\pi}\int_{-\infty}^\infty  e^{-x^2}\ \mathsf dx\\
&= \int_0^\infty x^2 e^{-x}\ \mathsf dx\\
&=\sum_{n=0}^\infty \frac{(ix)^n +(-ix)^n}{n!} + \frac1i\sum_{n=0}^\infty \frac{(ix)^n -(-ix)^n}{n!}\\
&= \lim_{n\to\infty}\left(1 + \int_1^2 \frac1{nx}\ \mathsf dx \right)^n\\
&= \frac4\pi \prod_{n=1}^\infty\left(\frac{4n^2}{4n^2-1}\right)
\end{align}
A: According to John Von Neumann this is actually the definition of $2$:
$\{\emptyset, \{\emptyset\} \}$
Granted, it is possible that the definition might have changed since then but the way I've understood that literally is two.
A: $$2=1+\left(\frac {1}{1\cdot 2}+\frac {1}{2\cdot 3}+\frac {1}{3\cdot 4}+...\right)$$ because of the telescoping series phenomenom:$$\sum_{j=1}^{\infty}\frac {1}{j(j+1)}=\lim_{n\to \infty}\sum_{j=1}^n\frac {1}{j(j+1)}=\lim_{n\to \infty}\sum_{j=1}^n\left(\frac {1}{j}-\frac {1}{j+1}\right)=$$ $$=\lim_{n\to \infty}\left(1-\frac {1}{n+1}\right)=1.$$
If you are a computer then $2$ is just a display character, but you know  a number $10$ in what humans call base $2.$
