Can $\pi$ be represented exactly using a mixture of algebraic as well as exp/log functions, all real valued?
I know it can't be done using only algebra since its transcendental, but what if we introduced exp/log functions? Is there any theory about its possibility?
The short answer is "NO" unless you allow such inverse trigonometric expressions as $4\arctan(1)$ and similar. However, there are exceedingly good approximations using Heegner numbers (there are only nine: 1, 2, 3, 7, 11, 19, 43, 76, 163), the last of which produces: $$\frac{\ln(640320^3+744)} {\sqrt{163}}$$ which gives $\pi|$ to more than 30 decimal places and an error of $2.237\times 10^{-31}$
Even better approximations to more than 50 decimal places are also possible. Then there are the infinite series that can be generated from the series for expressions such as $$\frac{\pi}4=4 \arctan\frac1 5-\arctan\frac1 {239}$$ etc, which provide a means to calculate $\pi$ to any desired accuracy.