# i to the power i

Categories: complex numbers imaginary numbers

The imaginary unit *i* is defined to be the positive square root of -1. But what is *i* to the power *i*? Is it even possible to calculate, and what does it mean?

As we will see, it is possible to calculate *i* to the power *i*, and the result is quite surprising in a couple of different ways. But we will start with a quick recap on the real powers of complex numbers, in particular the real powers of *i*.

## Modulus-argument form for multiplication

We will be using the modulus-argument form for complex numbers, where a complex number *z* is represented as a radius *r* (called the *modulus*) and an angle *Θ* (called the *argument*):

The modulus of *z* is the distance from the origin to the point *z* on an Argand diagram. The argument of *z* is the angle *z* makes with the x-axis:

When we multiply two complex numbers *z1* and *z2* that are expressed in this form, the normal rules of the exponential function apply:

We multiply the moduli *r1* and *r2*. We add the arguments *Θ1* and *Θ2*. That is exactly the same as we would do if the exponents were real numbers.

## The value i in modulus-argument form

We will be using *i* quite a lot, so it is useful to know its modulus-argument form. Here is *i* on an Argand diagram:

*i* is 1 unit vertically above the origin. So the length *r* is 1, and the angle is *π/2* radians (which is 90 degrees of course). Here is the exponential form of *i*:

If we multiply any number *z* by *i*, then in modulus-argument form this is:

In other words, multiplying by *z* by *i* simply rotates *z* by *π/2* radians about the origin.

## Integer powers of i

Before calculating *i* to the power *i*, it is worth looking at *i* raised to a real power, as this will give us a couple of insights into the problem. We can calculate *i* squared like this:

This value has a unit length and an angle of *π* radians (half a full turn). This makes it equal to -1. But we already know that *i* squared is -1, by definition. So (as expected) the modulus-argument form of *i* squared gives the same result as simple complex number multiplication.

We can find *i* cubed in the same way. This time the angle is *3π/2* radians (three-quarters of a full turn), so the result is *-i*:

*i* to the fourth has an angle of *2π* radians a full turn), so the result is 1:

Here are *i* and its second, third and fourth powers plotted on an Argand diagram:

It is no great surprise that *i* to the fourth power is 1. *i* to the fourth is just *i* squared then squared again, and since *i* squared is -1 then we would expect *i* to the fourth to be 1.

We can generalise this and say that *i* to any integer power is equal to:

Using this we can find the fifth, sixth and seventh powers on the Argand diagram:

Higher integer powers of *i* continue rotating round and round the unit circle.

## Takeaways

There are two important takeaways from this. The first is that raising *i* to the power *n*, in modulus-argument form, works in the same way as raising any other exponential to a power *n*. We just multiply the exponent by *n*:

The second is that there are infinitely many ways to express *i* in modulus argument form. Since *i* to the fourth is equal to 1, it follows that:

In modulus-argument form:

In fact, for any complex number *z* with argument *Θ*, if we add an integer multiple of *2π* to *Θ*, we will get the same number. This follows from Euler's formula:

Adding a multiple of *2π* to the angle does not change the value of the sine or cosine functions, because those functions are periodic with period *2π*, so:

## Integer roots of i

So what is the square root of *i*? Well, the square root of a real number *x* is given by raising *x* to the power one-half. What happens if we try the same thing with *i*?

But remember that *i* can also be written as *i* to the power 5. If we take the square root of this alternate form we get a second square root:

We can draw these two roots on an Argand diagram:

We can do this again with *i* to the power 9 (which is also equal to *i*):

This gives a result that has an argument of *π/4* plus *2π*. Since adding *2π* has no effect on the value of a complex number, this result is identical to the original case where the argument was *π/4*. There are only two distinct square roots of *i*.

In fact, every complex number (except 0) has two distinct square roots, 3 distinct cube roots, and *n* distinct *nth* roots.

## Takeaway

*i* raised to a power *p* can sometimes have multiple values. Those values can be found by calculating the powers of the following equivalent numbers:

Not all of these roots are necessarily distinct.

## i to the power i

So now we are in a position to calculate the value of *i* to the power *i*. We will assume that we can raise *i* to the power *i* simply by setting *p* to the value *i* in the formula above. This can be shown to be true, but we won't prove it here.

Here is the result:

This is a very interesting result. The two *i* terms multiply to give -1, so the exponent is now a real number. This means that the power is a real number expression!

*i* to the power *i* is simply the exponential of *-π/2*. Which has a real value of approximately 0.207880.

But it gets a little weirder. We also have to consider the other possible results based on the alternate modulus-argument forms of *i*. For example when *n* equals 1, we add *2π* to the exponent:

This gives a value of approximately 0.000388203.

We can use negative values of *n* too, of course. When *n* equals -1, we subtract *2π* from the exponent:

This gives a value of approximately 111.318.

Since this formula is based on the exponential function of a real number, every different value of *n* will give a unique, real result.

So *i* to the power *i* has an infinite number of solutions, and they are all real numbers.

## See also

- Imaginary and complex numbers
- Complex number arithmetic
- Argand diagrams
- Why does complex number multiplication cause rotation?
- Modulus-argument form of complex numbers
- De Moivre's theorem
- Euler's formula - proof
- Complex powers and roots of complex numbers
- Semiprocal numbers - z to the power i
- Complex polynomials
- Complex number trigonometry functions

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