Tiger Algebra Calculator

Imaginary numbers, almost always written as i, are unique in that they equal a negative number when multiplied by themselves. You might be wondering how this is possible since even negative numbers multiplied by themselves equal a positive number. The trick is that $$i=\surd -1$$, which, when multiplied by itself, removes the radical symbol but does not change the symbol of the number inside the radical symbol.

What is even more interesting about imaginary numbers, is that raising them by increasing powers results in a predictable, repetitive cycle that helps us quickly solve problems that might otherwise be unruly. For example, we can use this cycle to quickly solve $$i^{3473}$$, which might otherwise require a lot of extra work. Here is how it works: i, when raised to the powers of 0 through 3, results in different outcomes. After this, however, the outcomes start to repeat themselves every four digits, forever. So, $$i^0=i^4=i^8=1$$ and $$i^3=i^7=i^{11}=-i$$ and so on.


This means that, instead of manually calculating i raised to any power higher than 4, we can find a number close to that power and use the pattern described above, as well as properties of exponents, to simplify it.

For example, let's calculate $$i^{23}$$