Find the modulus of the following complex numbers
`(1+i)/(1-i) - (1-i)/(1+i)`

Answer

To find the modulus of the complex number \(\frac{1+i}{1-i} - \frac{1-i}{1+i}\), we will follow these steps:

### Step 1: Simplify the Expression
We start with the expression:
\[ Z = \frac{1+i}{1-i} - \frac{1-i}{1+i} \]
To simplify this, we will take a common denominator. The common denominator is \((1-i)(1+i)\).

### Step 2: Rewrite with Common Denominator
Rewriting the expression with the common denominator:
\[ Z = \frac{(1+i)(1+i) - (1-i)(1-i)}{(1-i)(1+i)} \]

### Step 3: Expand the Numerator
Now, we will expand the numerator:
\[ (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \]
\[ (1-i)(1-i) = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \]
So, the numerator becomes:
\[ 2i - (-2i) = 2i + 2i = 4i \]

### Step 4: Simplify the Denominator
Next, we simplify the denominator:
\[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 2 \]

### Step 5: Combine the Results
Now we can combine the results:
\[ Z = \frac{4i}{2} = 2i \]

### Step 6: Find the Modulus
To find the modulus of \(Z\), we use the formula for the modulus of a complex number \(x + iy\):
\[ |Z| = \sqrt{x^2 + y^2} \]
In our case, \(Z = 0 + 2i\), so \(x = 0\) and \(y = 2\):
\[ |Z| = \sqrt{0^2 + 2^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]

### Final Answer
Thus, the modulus of the complex number is:
\[ \boxed{2} \]