Raising a number $z$ to the power $p$ means multiplying $z$ by itself $p$ times, so multiplying two powers evidently adds the powers: $z^p z^q = z^{p+q}.$ This relationship uniquely defines the meaning of raising $z$ to the $0$ power or to a negative power: We must have $z^0 = 1$ and $z^{-p} = 1/z^p$ in order to preserve the rule that multiplying powers adds the powers. Thus we know the meaning of any integer power $p$, positive or negative.
Continuing in this way, raising a power to a power evidently multiplies the powers: $(z^p)^q = z^{pq},$ which we naturally hope to use to define any rational power by $(z^{p/q})^q = z^p.$ Unlike non-positive integer powers, however, this attempt hits a snag, which textbooks often gloss over with little comment. When $z$ is a real number (on a number line) the snag is subtle: For even denominators $q,$ the sign of $z^{p/q}$ is ambiguous (there is a plus and a minus square root for example), and the definition doesn't work at all when $z\lt 0.$ We can sweep away the whole problem by simply declaring that both $z$ and all its non-integer powers $z^p$ only exist for positive $z$. This step is imminently practical, but not as tidy as we like math to be.
Arguably, complex numbers are interesting precisely because they clean up the meaning of raising a number to a fractional power; when $z$ is complex, the restrictions on the values of $z^p$ disappear. Because complex multiplication rotates coordinates in a plane in addition to simply scaling them up and down (like ordinary multiplication), in the complex plane multiplication by a negative number becomes rotation by $180^\circ$, part of a smooth progression of rotations. However, in the complex plane, the ambiguity of a fractional power $z^p$ becomes impossible to sweep under the rug, and gives us multiple viable choices for what we mean by $z^p$ when $p$ is not an integer. (When $p$ is an integer all definitions agree on a unique result.) Here we embrace the ambiguity by showing how a non-standard choice can be useful.
The modulus of the product of any two complex numbers is the product of their moduli, and the argument of their product is the sum of their arguments. Geometrically, this means that when you raise a complex number $z$ to successive powers by squaring it, cubing it, raising it to the fourth power, and so on, what you are doing is constructing a series of triangles the same shape as the one formed by the points $0$, $1$, and $z$, stacked onto each other as shown in the figure. The orange dot is $z$, and the blue segment goes from $0$ to $1$. The gray dots are $z^2$, $z^3$, $z^4$, $z^5$, and $z^6$. You can drag the orange point $z$ around to observe the pattern its powers make on the plane: An arc that spirals outward or inward according to whether $z$ is outside or inside the unit circle (of radius one centered at the origin). Examine what happens when $z$ lies on the unit circle or on the real (horizontal) or imaginary (vertical) axis.
If the lovely arc of the powers of $z$ strikes your fancy, look up "logarithmic spiral" and you will find many people agree. We'll learn a lot more about these curves by the end of this page.
Let's take a closer look at the case when $z$ lies on the unit circle. All powers of any such $z$ also lie on the unit circle. In the figure below, you may select a power $p$ from two to six and explore how $z^p$ changes as you move $z$ around the unit circle.
When you drag the orange point $z$ once around the circle, the blue point $z^p$ to orbits $p$ times. Thus there are $p$ values of $z$ for which $z^p = 1$, marked by the orange ticks around the perimeter of the unit circle. Since the angles between successive powers are equal, these $p$ points are equally spaced around the unit circle, forming a regular $p$-gon with one corner at the point $1$. Each of these $p$ points is a $p$-th root of $1$, also called a $p$-th root of unity. We see that the fact that two real numbers ($+1$ and $-1$) whose squares are one is a special case of the general rule that there are $p$ complex numbers with a $p$-th power of one.
In standard mathematical notation, $1^{1/p} = 1$ for all $p$ -- the most boring choice is the most convenient and consistent for many purposes. But fractional powers are ambiguous, and in the case of roots of one, we can make a far more useful choice: On this page, let's write $1^{1/p}$ for the first interesting $p$-th root of unity - the nearest to $1$ as we go counterclockwise around the unit circle. The other $p$-th roots are then $1^{2/p}$, $1^{3/p}$, $1^{4/p}$, and so on up to $1^{p/p} = 1.$ Powers greater than $p$ hop counterclockwise around the circle again, looping back to $1$ at every multiple of $p$; negative powers loop clockwise.
Thus by defining $1^{1/p}$, we have in fact defined $1^\alpha$ for all rational numbers $\alpha=n/p$, and by continuity for all real numbers $\alpha$. Namely, $1^\alpha$ is the complex number which lies $\alpha$ counterclockwise revolutions around the unit circle starting from the point $1$, including fractions of a revolution. That is, an angle in units of revolutions (often called cycles) is a power of $1$ in the complex plane.
In trigonometry, the $x$ and $y$ coordinates of a point on the unit circle as functions of angle from the $x$-axis are called cosine and sine, respectively, so by our definition $$1^\alpha = \cos\alpha + i\sin\alpha.$$ Sine and cosine are available on many calculators and in any math library. However, note that our angle $\alpha$ is in units of revolutions, so you will usually need to multiply it by either $360$ to convert to degrees or by $2\pi$ to convert to radians before invoking a library function. Before computers, every scientist and engineer owned tables of sines (and several other functions) covering dozens of pages with numbers; they're important but not easy to calculate. But now we can.
The identity of angles with powers of $1$ makes it easy to derive all of the trigonometric identities from complex multiplication. The most important are the angle addition formulas, which we now recognize as nothing more than complex multiplication: $$\begin{align*} 1^{(\alpha+\beta)} &= 1^\alpha 1^\beta \\ \cos(\alpha+\beta) + i\sin(\alpha+\beta) &= (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta) \end{align*}$$ $$\quad\quad = (\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta)$$ The real parts are the cosine sum formula; the imaginary parts the sine sum formula. The formulas for the difference of angles $\alpha-\beta$ are the same but with the signs of $\sin\beta$ reversed: $$1^{-\beta} = 1/1^\beta = \overline{1^\beta}$$ where $\overline{z}$ means the complex conjugate of $z,$ that is, the number with equal and opposite imaginary part. Trigonometry teachers delight in torturing students with endless variants of this single idea, that complex multiplication marches around the unit circle.
Finally, if we want to build tables of values of the sine function, we need to be able to subdivide angles into smaller parts. To compute smaller angles, we can use the fact that the diagonal of the rhombus formed by the points $1^\alpha,$ $0,$ $1,$ and $1+1^\alpha$ bisects angle $\alpha$. To translate this geometric fact into algebra, first write $||z||=\sqrt{\overline{z}z}$ for the modulus of $z.$ Next, notice that $$1^\alpha||1+1^\alpha||^2 = 1^\alpha(1+1^{-\alpha})(1+1^\alpha) = (1+1^\alpha)^2,$$ which implies $$\begin{align*} 1^{\alpha/2} &= (1+1^\alpha) / ||1 + 1^\alpha|| \\ \cos(\alpha/2) + i\sin(\alpha/2) &= (1+\cos\alpha + i\sin\alpha) / \sqrt{2+2\cos\alpha}. \end{align*}$$
This half angle formula would suffice for building sine tables. However, if you want a challenge and know a little calculus, work out the equations you need to solve in order to subdivide an angle into three or five parts, by writing out the binomial expansion of $1^{3\alpha}$ and $1^{5\alpha}.$ The imaginary parts produce third and fifth degree polynomials, respectively, which you can very efficiently solve by Newton iteration (the guess-divide-average algorithm in the case of square roots) to get $\sin(\alpha/3)$ or $\sin(\alpha/5)$ given only $\sin\alpha$. Note that if you can divide an angle in half and into fifths, you can divide it into tenths, which is an interesting capability if you want to build a sine table. Dividing by three comes in handy if you work with angles in degrees, minutes, and seconds. There are also nice Newton iterations for versine, $\text{ver }\alpha=1-\cos\alpha$ for the half-angle or quarter angle, if you want to avoid the square root in the non-iterative half-angle formula. The real part of $1^{n\alpha}$ gives a nice iteration for $\text{ver}(\alpha/n)$ for even $n$, while the imaginary part produces a nice iteration for $\sin(\alpha/n)$ for odd $n$.
It is impossible to overstate the importance of the relationship between the complex powers $1^\alpha$ and the trigonometric functions $\cos\alpha$ and $\sin\alpha$. Working with a power law is just far simpler than working with raw sines and cosines. Practical fields from signal processing to quantum mechanics rely heavily on the fact that the complex roots of unity and their powers orbit smoothly around the unit circle. We have approached this relationship from the opposite direction of current textbooks, and invented a notation $1^\alpha$ you won't see anywhere else. It's time to close the loop by demonstrating how our approach relates to the textbook approach.
We've worked our way from powers which are counting numbers, to zero or negative integer powers, to rational powers, until finally we have a useful definition of a complex value for $1^x$ for any real number $x$. So now the burning question becomes, how can we define $1^z$ for any complex number $z$ in a useful and consistent way? The general method for extending a partially defined function to cover the whole complex plane is called analytic continuation - a cornerstone of the theory of analytic functions (smooth functions of a complex number that return a value which is also a complex number).
The simplest case of analytic continuation is complex multiplication. Just as raising to a power begins with the idea that a power is repeated multiplication, the basic idea behind multiplication is repeated addition. For powers of a number $a$, we decided to preserve the property that $a^xa^y=a^{x+y}.$ For multiplication, the property we want to preserve is the distributive law, $ax+ay=a(x+y)$, and it leads us through the same progression - from counting numbers $x$ being repeated addition of $a$ to itself, to multiplying by zero or negative $x$, and finally to multiplying by any rational fraction $x$, all defined uniquely by the requirement that the distributive law continue to apply.
This whole chain of reasoning works for any complex value of $a$ (once we've defined complex addition), but we still have only defined how to multiply a complex number $a$ by a real number $x$. We now want to broaden this definition to cover the product $az$ for any complex $z=x+iy.$ Let's think of this product as a mapping from one complex number $z$ to a second complex number $w$, which is its product with $a$, $w=az.$ Our chain starting from repeated addition tells us how to map the real axis; it maps to the line in $w$ determined by the origin $0$ and the point $a$, corresponding to the $z$ values $0$ and $1$, respectively. When we move perpendicular to the real axis in $z$, the natural response is to move perpendicular to the line through $0$ and $a$ in $w$, with the same map scale factor as when we move along the real axis (namely the length of $a$). Geometrically, this means that we map a square grid in the $z$-plane with edges $0$, $1$, and $i$ into a square grid in the $w$-plane with edges $0$, $a$, and $ia$. The grid squares can be any size, for example, the grid based on $1/100$ and $i/100$ maps to the grid based on $a/100$ and $ia/100$.
Return now to our function $1^x$, which maps the real axis onto the unit circle periodically, revolving around the circle once every time $x$ increases by $1$. Let's assume this is a part of a function $w = 1^z$ which produces a complex value $w$ for every complex value $z$, that is, a mapping from a complex plane with coordinates $z$ to a second complex plane with coordinates $w$. We know the values $w = 1^z$ where $z = x + i0$, and we want to extend this definition to all values $z = x + iy.$ If we make a fine enough square grid in the $z$ plane, we can do the same thing for grid squares adjacent to the $y=0$ line as we did for multiplication, namely just rotate the squares in the $z$-grid so they fit adjacent to the unit circle in the $w$-plane. Because of the curvature, the squares won't quite fit in $w$, overlapping slightly inside the circle and leaving slight gaps outside the circle. However, the finer we make the $z$-grid, the smaller the overlaps or gaps in the $w$-grid become.
In this way, we can define $1^{x+iy}$ for very small but non-zero values of $y$. Evidently, for a fixed value of $y$, $w$ will trace concentric circles, slightly inside the unit circle when $y\gt 0$ and slightly outside the unit circle when $y\lt 0.$ We can repeat this procedure for each new value of $y$ to work our way farther inward or outward. But as we move inward or outward a curious thing happens: The circumference of the circle at fixed $y$ shrinks or grows, while the number of grid steps in $x$ to go once arond remains fixed. Therefore, the size of the grid squares in the $w$-plane shrinks or grows in proportion to the radius of the circle corresponding to the value of $y$. We wind up with a $w$-grid like the one in this figure, consisting of squares (in the limit of a very fine grid) arranged in circles.
You can drag the marked set of four adjacent squares around on the $z$-plane or the $w$-plane to get a feeling for how our expanded mapping $w=1^z$ works. For this grid, you can still see the squares are slightly distorted in the $w$-plane, but you can also see that the distortion would be smaller for an even finer grid. Notice how moving along the blue line - the real axis in $z$ and the unit circle in $w$ - where we had originally defined $1^x$ defines adjacent concentric circles in $w$. Orbiting those adjacent circles defines the values on slightly larger circles, eventually mapping the whole $z$-plane to the whole the whole $w$-plane. In fact, because the mapping is periodic in $z$ ($z+1$ always maps to the same $w$ as $z$), our function $1^z$ covers the $w$-plane an infinite number of times.
The key principle of analytic continuation is that we expand our mapping in such a way that locally, over any small grid square, it looks like a simple complex multiplication and addition. Near any point $z_0$, our function $1^z = 1^{z_0} + c(z_0)(z-z_0)$ for some complex number $c$ (which may be different for each $z_0$). (If you know calculus, you'll recognize that the function $c$ is the derivative of the function $1^z$.) A function that is locally linear is called analytic. When you think about it, you realize that this is what makes it possible to compute functions like $1^z$: If we define a function so that we can compute it by addition and multiplication in a small region, and we have an appetite for doing lots of adding and multiplying, we can slowly march as far from our starting point as we please.
You will notice that according to our mapping, $1^{iy}$ is a point on the positive $w$-real axis, running from very near the origin $w=0$ for large positive $y$ through $w=1$ at $y=0,$ then out to infinity for large negative $y.$ In other words, $1^i$ is a real number less than $1$, and our mapping is simply $$1^z = 1^{x+iy} = 1^x(1^i)^y.$$ Since $1^i$ is a positive real number, we can raise it to a real power $y$ using the standard definition. The standard way to define $a^y$ for any positive real $a$ and any real $y$ goes through the same familiar steps we took to define $1^x$, beginning with counting numbers $y$, extending that to any integer $y$, then to any rational $y$ by defining reciprocal powers as roots.
So what is the value of $1^i$? If our grid has $n$ squares running from $z=0$ to $z=1$, then in the $w$-plane, we will go around the unit circle in $n$ steps. Hence near the unit circle, the side of the grid squares must be $2\pi/n$, since the perimeter of the unit circle is $2\pi.$ The radial edge of each square resting on the unit circle must also be $2\pi/n$, so the next circle inward has radius $1-2\pi/n$. This represents a step upward in the $z$-plane to the line $z=x+i/n.$ Exactly the same reasoning takes us to $z=x+i2/n,$ except that the perimeter of the circle was as factor of $1-2\pi/n$ smaller, so the corresponding circle radius drops by that factor to $(1-2\pi/n)^2$, after $n$ such steps, we will have reached the line $z=x+i$, which maps to the circle of radius $1^i$ in the $w$-plane. Therefore, $1^i \approx (1 - 2\pi/n)^n.$ Now for any finite $n$, this formula is not quite correct, because the grid in the $w$-plane is slightly distorted squares. (For example, $n=60$ in the figure.) So to get the exact answer we have to take the limit of an infinite $n$, an infinitely fine grid, written like this: $$1^i = \lim_{n\rightarrow\infty} (1 - 2\pi/n)^n.$$ Thus, $1^i$ is roughly $1/535,$ although you need to make $n$ very large, above ten thousand, to begin to see convergence.
Euler studied the function he defined by $$f(x) = \lim_{n\rightarrow\infty} (1 + x/n)^n.$$ We have just demonstrated that $1^i = f(-2\pi),$ so we'll follow Euler's reasoning about this more general form. You can replace $n$ on the right hand side by any multiple of $n$ since all grow without limit. If you replace $n$ by $n|x|$ and work through some technical details, you find that Euler's limit is just a disguised form of an exponential function: $$f(x) = f(1)^x = f(-1)^{-x} = e^x.$$ Here, Euler writes $e$ for the number $f(1)$, which is roughly $2.72$, and his choice has become the mathematical standard.
Thus, our notation here is related to standard notation by $1^i = e^{-2\pi},$ where the left side is non-standard and the right side is standard. Now, Euler had to work out how to analytically continue the function $e^x$ from the real axis into the complex plane. We've already worked this out in reverse. Since $1^{iy} = e^{-2\pi y}$ for all real $y$, we know $$e^z = 1^{-iz/(2\pi)}$$ for any complex $z.$ This gives the standard notation in terms of ours, which means that our notation would look like this in textbooks: $$1^z = e^{i2\pi z}.$$ In other words, to get from our notation on this page to textbook notation, just substitute $e^{i2\pi}$ for $1$ wherever $1$ appears as the base of an exponent. In the standard notation, $1$ is useless as the base of an exponent, so no one uses it and there is little chance of confusion.
The relation to the trigonometric functions that appeared naturally from our discussion of the complex roots of unity, $$1^\alpha = \cos\alpha + i\sin\alpha,$$ looks like this in standard notation $$e^{i\theta} = \cos\theta + i\sin\theta.$$ This is the celebrated Euler formula. The major difference is that the power of the number $e$ in the Euler formula is the pure imaginary number $i\theta,$ while in our non-standard notation, the exponent of $1$ is the ordinary real number $\alpha$. The secondary difference is that the units of the angle $\theta$ are implicitly radians, while the units of the angle $\alpha$ are revolutions or cycles. (So $\theta = 2\pi\alpha.$)
To understand the meaning of an imaginary exponent, you need to work through the analytic continuation argument. In the geometric approach on this page, the key element of that argument was that we want our extended function to look like a linear function, a complex multiplication and a complex add, on small scales, so that a grid of fine squares in $z$ maps to a grid of fine squares in $w$. In Euler's purely algebraic approach, this analytic continuation step hardly needs to be mentioned. Since his definition of $e^x$ in terms of $(1+x/n)^n$ begins with just addition and multiplication operations, all he needs to to is plug in complex numbers for $x$ and he automatically has definition of $e^x$ for complex values that will look like complex multiplication on small scales.
In our roots of unity approach, angles were the primary players from the beginning; we didn't need to introduce radian angle measure or the constants $e$ or $\pi$ at all to reach the important relations between complex multiplication and the trigonometric functions. It was only when we analytically continued our function $1^z$ to pure imaginary numbers that we discovered its relation to ordinary real exponential functions. The standard textbook approach, Euler's approach, is to begin with ordinary real exponentials and discover their connection to angles by analytic continuation to pure imaginaries.
Our notation $1^\alpha$ simplifies many important mathematical formulas, especially in Fourier analysis. However, it makes many others more complicated, notably anything involving rates of change of angles. Similarly, measuring angles in revolutions or cycles is often the best choice in engineering - witness the units Hertz (cycles per second) or RPM (revolutions per minute) - while measuring angles in radians is often simplest in problems involving rates of change.