The one-unit imaginary number with itself as an exponent, ii, seems strictly unreal. But its looks are quite deceiving. Here’s how Euler’s formula can be used to show that it’s a real number:
First, notice that when equal numbers expressed in different ways are raised to the same power, the resulting numbers are also equal. Example: Since 4/2 = 2, we know that (4/2)2 = 22. Next, recall from Chapter 11 that eiπ/2 = i. (It can be derived by plugging π/2 into eiθ = cos θ + i sin θ.) If we raise both of these equal numbers to the ith power, we should get two numbers that are also equal. That is, (eiπ/2)i = ii. This equation means that evaluating (eiπ/2)i will reveal the numerical identity of ii.
Now, to evaluate (eiπ/2)i, consider a similar expression, 22 raised to the third power, or (22)3, which can be written (2 × 2)3, or (2 × 2) × (2 × 2) × (2 × 2), which equals 26, or 22 × 3. This example illustrates a general rule that can be stated with variables as (xa)b = xa × b. Applying the rule to (eiπ/2)i gives us (eiπ/2)i = eiπ/2 × i = ei × i × π/2 (by rearranging multiplied terms in the exponent) = e−π/2 (since, i × i = i2 = −1). Thus, ii = e−π/2, and although e−π/2 has a funny-looking negative exponent, it is a real number—it’s actually about 0.208. (In fact, ii is equal to an infinity of real numbers related to this decimal, which is termed the principal value of ii, but that’s another story.) When Euler discovered that ii is real, he exclaimed in a letter to a friend that this “seems extraordinary to me”—part of his genius, as well as of his charm, was an inexhaustible capacity to be surprised and delighted by his discoveries.