√i = ?
Understand the Problem
The question is asking for the square root of the imaginary unit 'i'. This involves complex numbers and requires knowledge of how to manipulate them in mathematical terms.
Answer
The square root of $i$ is $\frac{\sqrt{2}}{2}(1 + i)$.
Answer for screen readers
The square root of the imaginary unit $i$ is:
$$ \sqrt{i} = \frac{\sqrt{2}}{2}(1 + i) $$
Steps to Solve
- Express the imaginary unit 'i' in exponential form
The imaginary unit can be expressed using Euler's formula. Recall that:
$$ i = e^{i \frac{\pi}{2}} $$
- Apply the square root formula in exponential form
To find the square root of 'i', we use the square root property for complex exponentials:
$$ \sqrt{z} = e^{\frac{1}{2} \log(z)} $$
So, we have:
$$ \sqrt{i} = \sqrt{e^{i \frac{\pi}{2}}} = e^{i \frac{\pi}{4}} $$
- Convert back to rectangular form
Now, we convert $e^{i \frac{\pi}{4}}$ back into rectangular form using Euler's formula:
$$ e^{i \theta} = \cos(\theta) + i \sin(\theta) $$
Thus:
$$ \sqrt{i} = \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) $$
- Calculate the trigonometric values
The values for cosine and sine at $\frac{\pi}{4}$ are both equal to $\frac{\sqrt{2}}{2}$:
$$ \sqrt{i} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} $$
- Final presentation
We can also write the answer in a more concise form for clarity:
$$ \sqrt{i} = \frac{\sqrt{2}}{2}(1 + i) $$
The square root of the imaginary unit $i$ is:
$$ \sqrt{i} = \frac{\sqrt{2}}{2}(1 + i) $$
More Information
The square root of the imaginary unit is a complex number. Understanding this concept is key in fields like electrical engineering and quantum mechanics, where complex numbers play a crucial role.
Tips
- Misunderstanding the definition of the square root in the context of complex numbers can lead to incorrect calculations.
- Failing to use the correct angle when converting between polar/exponential and rectangular forms.