Question 6 of 28beginner🔧 ApplyNumerical1 mark

Express the following complex number in the form a+iba + ib: (15+i25)−(4+i52)\left(\frac{1}{5} + i\frac{2}{5}\right) - \left(4 + i\frac{5}{2}\right)

Correct Answer

−195−i2110-\frac{19}{5} - i\frac{21}{10}

Exercise: EXERCISE 4.1 | Q: 6 | (Page 7)
For More Understanding

Explanation

This question tests the basic subtraction of complex numbers. According to the textbook, a complex number is expressed in the form a+iba + ib where aa is the real part and bb is the imaginary part. When subtracting two complex numbers, we subtract the real parts and imaginary parts separately.

Solution Steps

  1. Step 1: Write the given expression and distribute the negative sign.

    (15+i25)−(4+i52)=15+i25−4−i52\left(\frac{1}{5} + i\frac{2}{5}\right) - \left(4 + i\frac{5}{2}\right) = \frac{1}{5} + i\frac{2}{5} - 4 - i\frac{5}{2}

  2. Step 2: Group the real parts and imaginary parts together.

    =(15−4)+i(25−52)= \left(\frac{1}{5} - 4\right) + i\left(\frac{2}{5} - \frac{5}{2}\right)

  3. Step 3: Calculate the real part.

    15−4=15−205=1−205=−195\frac{1}{5} - 4 = \frac{1}{5} - \frac{20}{5} = \frac{1 - 20}{5} = -\frac{19}{5}

  4. Step 4: Calculate the imaginary part.

    25−52=410−2510=4−2510=−2110\frac{2}{5} - \frac{5}{2} = \frac{4}{10} - \frac{25}{10} = \frac{4 - 25}{10} = -\frac{21}{10}

  5. Step 5: Combine the results to express in the form a+iba + ib.

    =−195+i(−2110)=−195−i2110= -\frac{19}{5} + i\left(-\frac{21}{10}\right) = -\frac{19}{5} - i\frac{21}{10}
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