Find the modulus and argument of the complex number: 1+i/1-i

To find the modulus and argument of the complex number (1 + i) / (1 - i), let's break the problem down step by step:

Step 1: Express the complex number in standard form
We are given the complex number:

(1 + i) / (1 - i)

To simplify, multiply both the numerator and the denominator by the conjugate of the denominator, which is (1 + i).

So, we multiply by (1 + i) / (1 + i):

[(1 + i) / (1 - i)] * [(1 + i) / (1 + i)]

This results in:

= [(1 + i)(1 + i)] / [(1 - i)(1 + i)]

Step 2: Simplify the numerator
(1 + i)(1 + i) = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i

So, the numerator is 2i.

Step 3: Simplify the denominator
(1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2

So, the denominator is 2.

Step 4: Final simplified expression
Now, the complex number simplifies to:

(2i) / 2 = i

Step 5: Find the modulus of i
The modulus of a complex number a + bi is given by:

|z| = √(a² + b²)

For i, a = 0 and b = 1, so the modulus is:

|i| = √(0² + 1²) = √1 = 1

Step 6: Find the argument of i
The argument of a complex number is the angle θ it makes with the positive real axis in the complex plane. For i, it lies along the positive imaginary axis, so its argument is:

θ = π/2

Final Answer:
Modulus: 1
Argument: π/2