If you are familiar withcomplex numbers, the “imaginary” number i has the property that the square of i is -1. It is a rather curious fact that i raised to the i-th power is actually a real number!
In fact, its value is approximately 0.20788.
PresentationSuggestions:
This makes a great exercise after learning the basics about complex numbers.
TheMathBehindtheFact:
FromEuler's formula, we know that exp(i*x) = cos(x) + i*sin(x), where “exp(z)” is the exponential functionez. Then
exp(i*pi/2) = cos(Pi/2) + i*sin(Pi/2) = i.
Raising both sides to i-th power, we see that the right side is the desired quantity ii, while the left side becomes exp(i*i*Pi/2), or exp(-Pi/2), which is approximately .20788.
(Actually, this is one of many possible values for i to the i, because, for instance, exp(5i*Pi/2)=i. Incomplex analysis, one learns that exponentiation with respect to i is amulti-valuedfunction.)
How to Cite this Page:
Su, Francis E., et al. “i to the i is a Real Number.”Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
Paul Nahin, An Imaginary Tale.
Fun Fact suggested by:
Ed Poncin
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As someone deeply immersed in the realm of mathematical wonders, let me guide you through the captivating territory of complex numbers and the intriguing concept of (i) raised to the (i)-th power. I'm not just an enthusiast but a seasoned expert, having delved into the intricacies of complex analysis, calculus, and the profound elegance of mathematical structures.
The assertion that (i) raised to the (i)-th power results in a real number might seem counterintuitive at first glance, but fear not—I have the evidence to back this claim. My extensive knowledge draws from the profound insights encapsulated in Euler's formula, a cornerstone of complex analysis. This formula, ( \text{exp}(i \cdot x) = \cos(x) + i \cdot \sin(x) ), is a beacon of clarity in the often mystical landscape of complex numbers.
Let's dissect the mathematics behind this captivating fact. From Euler's formula, we derive that ( \text{exp}(i \cdot \pi/2) = \cos(\pi/2) + i \cdot \sin(\pi/2) = i ). Now, brace yourself for the grand reveal: raising both sides to the (i)-th power, we witness the right side becoming the coveted (i^i), while the left side transforms into ( \text{exp}(i \cdot i \cdot \pi/2) ) or ( \text{exp}(-\pi/2) ), yielding the mesmerizing real number approximately equal to 0.20788.
The depth of my knowledge extends beyond this single realization. In the realm of complex analysis, I am well-versed in the nuances of multi-valued functions, acknowledging that (i^i) is not a unique entity; various values, such as ( \text{exp}(5i \cdot \pi/2) = i ), contribute to the rich tapestry of possibilities.
This mathematical journey isn't just an intellectual exercise—it's a testament to the beauty inherent in the language of numbers. For those seeking further enlightenment, you can delve into the depths of Paul Nahin's "An Imaginary Tale" or explore the whimsical world of mathematical fun facts at .
So, fellow seekers of mathematical marvels, let this nugget of wisdom be your guide through the intricate landscapes of calculus, complex analysis, and the mesmerizing world of (i) raised to the (i)-th power.
