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.


Euler_formula


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


theta=pi/2


Now we have another way to write i.


e^i_pi/2=i


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


periodic


Which is true for all integer values of n.

To be really general, we can write i as


i_as_periodic


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,


expanding_exponent


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


re-order_the_product


Step 2: Take the square root.

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


both_sides_to_1/2_power


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


multiplying_through_by_1/2


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

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