Question 7 of 28beginner🔧 ApplyNumerical1 mark

Express the following complex number in the form a+iba + ib: [(13+i73)+(4+i13)]−(−43+i)\left[\left(\frac{1}{3} + i\frac{7}{3}\right) + \left(4 + i\frac{1}{3}\right)\right] - \left(-\frac{4}{3} + i\right)

Correct Answer

173\frac{17}{3} + i53\frac{5}{3}

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

Explanation

This question tests the addition and subtraction of complex numbers. According to the context, for complex numbers z₁ = a + ib and z₂ = c + id, the sum is z₁ + z₂ = (a + c) + i(b + d). We apply this rule to add the first two complex numbers, then subtract the third complex number by changing signs appropriately.

Solution Steps

  1. Step 1: Add the two complex numbers inside the first bracket: (13+i73)+(4+i13)=(13+4)\left(\frac{1}{3} + i\frac{7}{3}\right) + \left(4 + i\frac{1}{3}\right) = \left(\frac{1}{3} + 4\right) + i(73+13)=133\left(\frac{7}{3} + \frac{1}{3}\right) = \frac{13}{3} + i83\frac{8}{3}

  2. Step 2: Subtract the third complex number: (133+i83)−(−43+i)=(133+43)\left(\frac{13}{3} + i\frac{8}{3}\right) - \left(-\frac{4}{3} + i\right) = \left(\frac{13}{3} + \frac{4}{3}\right) + i(83−1)\left(\frac{8}{3} - 1\right)

  3. Step 3: Simplify real and imaginary parts: = 173\frac{17}{3} + i(83−33)=173\left(\frac{8}{3} - \frac{3}{3}\right) = \frac{17}{3} + i53\frac{5}{3}

  4. Step 4: Final answer in the form a + ib: 173\frac{17}{3} + i53\frac{5}{3}, where a = 173\frac{17}{3} and b = 53\frac{5}{3}

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