Warning: this question requires knowledge of complex numbers.
An Euler's identity is an identity, in which each of the following appears once and only once:
- the constant $0$: neutral element for addition;
- the constant $1$: neutral element for multiplication;
- the constant $e$: base of natural logarithm;
- the constant $i$: square root of $-1$;
- the constant $\pi$: ratio of a circle's circumference to its diameter;
- the operation $+$: addition;
- the operation $\times$: multiplication;
- the operation $^\wedge$: exponentiation;
- the equal sign $=$: symbol for equality.
How many different Euler's identities exist?
- The use of parentheses $()$ is unlimited.
- Other than the above mentioned symbols (including parentheses), no other symbol is allowed.
- Because of commutativity, $a + b$ and $b + a$ are considered the same. Same with $a \times b$ and $b \times a$.