Find the modulus and argument of the complex number `z =(i^2 +i^3)/(i^4 +i^5)`

Answer

To find the modulus and argument of the complex number \( z = \frac{i^2 + i^3}{i^4 + i^5} \), we will follow these steps:

### Step 1: Simplify the numerator and denominator
First, we need to calculate \( i^2 \), \( i^3 \), \( i^4 \), and \( i^5 \):
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \)

Now substituting these values into the expression for \( z \):
\[ z = \frac{i^2 + i^3}{i^4 + i^5} = \frac{-1 - i}{1 + i} \]

### Step 2: Multiply numerator and denominator by the conjugate of the denominator
To simplify \( z \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 1 - i \):
\[ z = \frac{(-1 - i)(1 - i)}{(1 + i)(1 - i)} \]

### Step 3: Calculate the denominator
The denominator simplifies as follows:
\[ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 4: Calculate the numerator
Now, calculate the numerator:
\[ (-1 - i)(1 - i) = -1 + i + i - i^2 = -1 + 2i + 1 = 2i \]

### Step 5: Combine the results
Now we can write \( z \):
\[ z = \frac{2i}{2} = i \]

### Step 6: Find the modulus of \( z \)
The modulus of a complex number \( z = x + iy \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
In our case, \( z = 0 + 1i \), so \( x = 0 \) and \( y = 1 \):
\[ |z| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \]

### Step 7: Find the argument of \( z \)
The argument \( \theta \) of a complex number is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Here, since \( x = 0 \) and \( y = 1 \):
\[ \theta = \tan^{-1}\left(\frac{1}{0}\right) \]
The argument is \( \frac{\pi}{2} \) (or \( 90^\circ \)), since the point lies on the positive imaginary axis.

### Final Answer
Thus, the modulus and argument of the complex number \( z \) are:
- Modulus: \( 1 \)
- Argument: \( \frac{\pi}{2} \)