Finding the Value of the Complex Expression

The problem asks us to find the value of the complex number expression \( \frac{1 - i}{1 + i} \), where \( i \) is the imaginary unit, defined such that \( i^2 = -1 \).

To simplify a fraction involving complex numbers like \( \frac{a+bi}{c+di} \), we use a standard technique. We multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number \( c+di \) is \( c-di \).

In our expression, the denominator is \( 1 + i \). Its conjugate is \( 1 - i \).

Let's multiply the numerator and the denominator by \( 1 - i \):

Now, we need to expand the numerator and the denominator separately.

Simplifying the Numerator

The numerator is \( (1 - i)(1 - i) \). We can expand this like a binomial square \( (a-b)^2 = a^2 - 2ab + b^2 \), where \( a=1 \) and \( b=i \):

Since \( i^2 = -1 \), we substitute this value:

So, the simplified numerator is \( -2i \).

Simplifying the Denominator

The denominator is \( (1 + i)(1 - i) \). This is in the form \( (a+b)(a-b) \), which expands to \( a^2 - b^2 \). Here, \( a=1 \) and \( b=i \):

Again, using \( i^2 = -1 \):

So, the simplified denominator is \( 2 \).

Combining the Simplified Parts

Now we put the simplified numerator and denominator back into the fraction:

Finally, we simplify the fraction by dividing the numerator by the denominator:

Thus, the value of the expression \( \frac{1 - i}{1 + i} \) is \( -i \).

Let's compare this result with the given options:

• Option 1: 1
• Option 2: i
• Option 3: -i
• Option 4: 1/i

Our calculated value \( -i \) matches Option 3.

Note: Option 4, \( 1/i \), can also be simplified. Multiplying the numerator and denominator by \( i \): \( \frac{1}{i} = \frac{1 \times i}{i \times i} = \frac{i}{i^2} = \frac{i}{-1} = -i \). So, \( 1/i \) is equivalent to \( -i \). However, the options list \( -i \) directly, which is the most simplified form.