Final answer:
To rewrite the complex number √2/2 - i√2/2, we calculate its magnitude, which is 1, and its angle, which is π/4 radians or 45 degrees. It can be rewritten in polar form as 1(cos π/4 + i sin π/4) or in exponential form as e^(iπ/4).
Step-by-step explanation:
The complex number in question is written as √2/2 - i√2/2. To rewrite this complex number in a different form, we can express it in either polar or exponential form.
First, let's find the magnitude (also known as the modulus) of the complex number:
- Magnitude = √((√2/2)^2 + (√2/2)^2) = √((2/4) + (2/4)) = √(1/2 + 1/2) = √1 = 1
Next, we calculate the angle θ (theta) using the arctan function, which gives us:
- θ = arctan(√2/2 / √2/2) = arctan(1) = π/4 radians or 45 degrees
Thus, the polar form of the complex number is:
r(cos θ + i sin θ) = 1(cos π/4 + i sin π/4)
The exponential form using Euler's formula, e^(iθ), is:
e^(iπ/4) = cos π/4 + i sin π/4
Therefore, the complex number √2/2 - i√2/2 can be rewritten in polar form as 1(cos π/4 + i sin π/4) or in exponential form as e^(iπ/4).