Imaginary unit
In math, imaginary units, or [math]\displaystyle{ i }[/math], are numbers that can be represented by equations but refer to values that could not physically exist in real life. The mathematical definition of an imaginary unit is [math]\displaystyle{ i = \sqrt{-1} }[/math], which has the property [math]\displaystyle{ i \times i = i^2 = -1 }[/math].
The reason [math]\displaystyle{ i }[/math] was created was to answer a polynomial equation, [math]\displaystyle{ x^2 + 1 = 0 }[/math], which normally has no solution as the value of x^2 would have to equal -1. Though the problem is solvable, the square root of -1 could not be represented by a physical quantity of any objects in real life.
Square root of i
It is sometimes assumed that one must create another number to show the square root of [math]\displaystyle{ i }[/math], but that is not needed. The square root of [math]\displaystyle{ i }[/math] can be written as: [math]\displaystyle{ \sqrt{i} = \pm \frac{\sqrt{2}}{2} (1 + i) }[/math].
This can be shown as:
[math]\displaystyle{ \left( \pm \frac{\sqrt{2}}{2} (1 + i) \right)^2 \ }[/math] [math]\displaystyle{ = \left( \pm \frac{\sqrt{2}}{2} \right)^2 (1 + i)^2 \ }[/math] [math]\displaystyle{ = (\pm 1)^2 \frac{2}{4} (1 + i)(1 + i) \ }[/math] [math]\displaystyle{ = 1 \times \frac{1}{2} (1 + 2i + i^2) \quad \quad (i^2 = -1) \ }[/math] [math]\displaystyle{ = \frac{1}{2} (2i) \ }[/math] [math]\displaystyle{ = i \ }[/math]
Powers of i
The powers of [math]\displaystyle{ i }[/math] follow a predictable pattern:
- [math]\displaystyle{ i^{-3} = i }[/math]
- [math]\displaystyle{ i^{-2} = -1 }[/math]
- [math]\displaystyle{ i^{-1} = -i }[/math]
- [math]\displaystyle{ i^0 = 1 }[/math]
- [math]\displaystyle{ i^1 = i }[/math]
- [math]\displaystyle{ i^2 = -1 }[/math]
- [math]\displaystyle{ i^3 = -i }[/math]
- [math]\displaystyle{ i^4 = 1 }[/math]
- [math]\displaystyle{ i^5 = i }[/math]
- [math]\displaystyle{ i^6 = -1 }[/math]
This can be shown with the following pattern where n is any integer:
- [math]\displaystyle{ i^{4n} = 1 }[/math]
- [math]\displaystyle{ i^{4n+1} = i }[/math]
- [math]\displaystyle{ i^{4n+2} = -1 }[/math]
- [math]\displaystyle{ i^{4n+3} = -i }[/math]