Solving the Square Root of i

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we solve for the square root of the imaginary number i. We use Euler's formula and some properties of exponentials to arrive at a very interesting result.

The Euler Formula

Euler's formula is remarkable! It relates the complex exponential, eiθ, to the sine and cosine functions.


We know sin π/2 = 1 and cos π/2 = 0. Let's see what happens in the Euler formula. Letting θ = π/2:


Now we have another way to write i.


Sine and cosine are periodic functions of period 2π. Thus,


Which is true for all integer values of n.

To be really general, we can write i as


Having an expression for i as a complex exponential, allows us to find the square root of i.

Finding the Square Root of i

The imaginary number, i, is equal to the square root of -1. But we can also find the square root of i. Symbolically, we can write the square root of i using a radical sign: √i or with i raised to the power 1/2.

The steps for finding the square root of i:

Step 1: Write i as a complex exponential.

We have already worked this out but it will be useful to organize the expression for what comes next. Since e(a + b) = ea eb,


We can also re-order the product of these exponents,


Step 2: Take the square root.

We take the square root by raising each side of the equation to the 1/2 power:


On the right-hand side, the 1/2 multiplies through the argument of each exponent:


The 2's canceled in the first exponential and in the second, π/2 divided by 2 is π/4.

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