sqrt(i)

TL;DR

The square root of i can be expressed as two complex numbers: 1/√(2) + (1/√(2))i and -1/√(2) - (1/√(2))i.

Transcript

Okay, as we all know, the square root of negative 1 is equal to i, but have we ever thought about what is the square root of i? Oh well. As you know, i is the imaginary unit, and when you put the i inside the square root, do we still end up with a complex number? Most likely, it isn't? So the way to deal with this is, well, we're just going to set ... Read More

Key Insights

• 😑 The square root of i can be expressed as two complex numbers, each with real and imaginary parts.
• ❎ The process of finding the square root of i involves squaring both sides of the equation and solving a system of equations.
• 🥳 The solutions to the system of equations result in two possible values for the real and imaginary parts of the square root of i.

Questions & Answers

Q: How can the square root of i be expressed as a complex number?

The square root of i can be expressed as two complex numbers: 1/√(2) + (1/√(2))i and -1/√(2) - (1/√(2))i. These numbers have real and imaginary parts.

Q: How is the system of equations derived?

By squaring both sides of the equation √(i) = a + bi, the i term cancels out on the left-hand side. This results in a system of equations with the real and imaginary parts separated.

Q: What are the solutions to the system of equations?

The system of equations yields two possible solutions for a and b: a = 1/√(2), b = 1/√(2) and a = -1/√(2), b = -1/√(2). These values represent the real and imaginary parts of the square root of i.

Q: Are there any other ways to represent the square root of i?

No, the square root of i can only be represented as the two complex numbers mentioned earlier: 1/√(2) + (1/√(2))i and -1/√(2) - (1/√(2))i. These are the standard forms for the square root of i.

Summary & Key Takeaways

• The video discusses finding the square root of i and expresses it as a complex number in standard form.

• By squaring both sides of the equation, the i term cancels out, resulting in a system of equations.

• Solving the system of equations leads to two possible values for a and b, which are then used to represent the square root of i in standard form.