Understanding Exponents: Real, Irrational & Imaginary

Introduction

This is how both you and I learned exponents back in elementary school:

5³ = 5 × 5 × 5

Using repeated multiplication you can raise anything to a whole-number power. But what about fractions, negatives, irrationals, or imaginary exponents? Fractional and negative exponents are introduced in algebra: 5^-3 = 1 / 5³ and 5^3/2 = √(5³). Those follow from repeated multiplication and its inverses (roots). For irrational and imaginary exponents the “repeated multiplication” idea breaks down: what does it mean to multiply something by itself an irrational or imaginary number of times? To handle those cases we need a more general definition of exponentiation.

Definitions and basic properties

First, a few definitions and simple properties we will use:

• 0! = 1
• x⁰ = 1 (for x ≠ 0)
• x¹ = x
• e = 1/0! + 1/1! + 1/2! + 1/3! + … ≈ 2.718
• i² = -1, so i³ = -i, i⁴ = 1, and the powers of i repeat every 4 steps

If you understand these facts and the standard laws of exponents, you are ready for the broader definitions below. (See also: related background.)

e^x and the general definition of x^y

The exponential function is defined by its power series:

e^x = 1 + x/1! + x²/2! + x³/3! + … = ∑_n=0^∞ xⁿ/n!.

Using the exponential and the natural logarithm we define general exponentiation (for x > 0):

x^y := e^{y ln(x)},

where ln(x) is the natural logarithm, the inverse of e^x, so e^ln(x) = x.

This definition extends exponentiation beyond whole numbers and rationals: because e^x is defined for every real (and complex) x via its series, the expression e^{y ln(x)} makes sense even when y is irrational.

Consistency with familiar rules (short proofs)

The series definition makes common identities transparent. For example:

• e = e¹ = 1 + 1/1! + 1/2! + … (the series at x = 1)
• e⁰ = 1 + 0 + 0 + … = 1 (the series at x = 0)
• For x > 0, x^y = e^{y ln(x)} so identities like x^y+z = x^y × x^z follow from properties of the exponential: e^{(y+z)ln x} = e^{y ln x} × e^{z ln x}.

Thus the series/exponential definition is consistent with repeated multiplication where that interpretation applies, and it extends exponentiation to all real exponents in a consistent way.

Irrational exponents

Irrational exponents are covered by x^y = e^{y ln(x)} because ln(x) is a real number and y can be irrational. For example e^π means e raised to the irrational power π; the exponential series or the e^{y ln(x)} definition evaluates it unambiguously. You can also think of an irrational as a limiting or infinite sum (decimal expansion), but the cleaner approach is via the exponential and logarithm definitions.

Imaginary exponents and Euler’s formula

Imaginary exponents introduce rotation. A standard and deep fact is Euler’s formula:

e^ix = cos(x) + i sin(x).

Geometrically, consider the complex plane with the real axis horizontal and the imaginary axis vertical. The complex number 1 sits at (1, 0). Multiplying by e^ix corresponds to rotating that point counterclockwise by x radians around the origin. If x = π, the rotation sends 1 to -1 (half a turn); for other values of x you generally get a complex number with both real and imaginary parts, which is exactly cos(x) + i sin(x). (See also: background on real numbers.)

Combining real and imaginary exponents

A complex exponent a + ib behaves multiplicatively via e^a+ib = e^a e^ib = e^a (cos b + i sin b). The real part a scales the magnitude (a stretching or compression), and the imaginary part b rotates the point on the complex plane. This gives a clean geometric interpretation: real exponents change size, imaginary exponents change angle.

Operations on the complex plane

You can view the four basic operations on complex numbers as actions on the plane:

• Adding/subtracting a real number moves points left/right.
• Adding/subtracting an imaginary number moves points up/down.
• Multiplying/dividing by a real number stretches or compresses distances from the origin.
• Multiplying/dividing by a complex number with unit magnitude rotates points around the origin (and if the magnitude is not 1, it rotates and scales).

As you work with complex numbers more, these geometric interpretations make algebraic identities and transformations easier to understand and visualize. (See also: more on real numbers and views on complex numbers.)

Conclusion and thanks

In summary, defining exponentiation via the exponential series and the natural logarithm unifies whole-number, rational, irrational, and imaginary exponents into a single consistent framework. Repeated multiplication becomes a special-case consequence of these broader definitions when exponent values are whole numbers.

Thank you to Professor Tom Imbo for helping me learn this material.