Questions: Example 1.8 Write the following complex number as a single complex number. a) (2+5j)/(3+j) c) 3/(1+j) b) 1/j + 1/(1-j) d) 2/(1+j)^4

Solution Steps

To simplify each complex number expression, we need to multiply the numerator and the denominator by the conjugate of the denominator. This will eliminate the imaginary part in the denominator, allowing us to express the result as a single complex number.

To simplify the expression \( \frac{2+5j}{3+j} \), we multiply the numerator and denominator by the conjugate of the denominator, \( 3-j \):

\[ \frac{(2+5j)(3-j)}{(3+j)(3-j)} = \frac{6 - 2j + 15j - 5j^2}{9 + 1} = \frac{6 + 13j + 5}{10} = \frac{11 + 13j}{10} \]

Thus, the simplified form is:

\[ \frac{11}{10} + \frac{13}{10}j \approx 1.1 + 1.3j \]

For the expression \( \frac{1}{j} + \frac{1}{1-j} \), we first simplify each term. The first term can be rewritten as \( -j \). The second term is simplified by multiplying the numerator and denominator by the conjugate \( 1+j \):

\[ \frac{1}{1-j} = \frac{1(1+j)}{(1-j)(1+j)} = \frac{1+j}{1 + 1} = \frac{1}{2} + \frac{1}{2}j \]

Now, combining both terms:

\[ -j + \left(\frac{1}{2} + \frac{1}{2}j\right) = -\frac{1}{2}j + \frac{1}{2}j = -\frac{1}{2}j \]

For the expression \( \frac{3}{1+j} \), we multiply the numerator and denominator by the conjugate \( 1-j \):

\[ \frac{3(1-j)}{(1+j)(1-j)} = \frac{3 - 3j}{1 + 1} = \frac{3 - 3j}{2} = \frac{3}{2} - \frac{3}{2}j \]

Thus, the simplified form is:

\[ \frac{3}{2} - \frac{3}{2}j \approx 1.5 - 1.5j \]

Final Answer