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Given z = (2−i)/(1+i).

Mathematics

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Answer

a

z=12−32iz = \frac{1}{2} - \frac{3}{2}i

Step-by-step solution

3 steps

Here are the solutions to your questions:

1. Given z=2−i1+iz = \frac{2-i}{1+i}.

Part a· 3 steps

Express zz in the form a+bia+bi.

  1. Multiply the numerator and denominator by the conjugate of the denominator.

    z=2−i1+i×1−i1−iz = \frac{2-i}{1+i} \times \frac{1-i}{1-i}

  2. Perform the multiplication.

    Numerator: (2−i)(1−i)=2(1)+2(−i)−i(1)−i(−i)=2−2i−i+i2=2−3i−1=1−3i(2-i)(1-i) = 2(1) + 2(-i) - i(1) - i(-i) = 2 - 2i - i + i^2 = 2 - 3i - 1 = 1 - 3i.
    Denominator: (1+i)(1−i)=12−i2=1−(−1)=1+1=2(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1+1 = 2.

  3. Write zz in the form a+bia+bi.

    z=1−3i2=12−32iz = \frac{1-3i}{2} = \frac{1}{2} - \frac{3}{2}i

    The final answer is z=12−32i\boxed{z = \frac{1}{2} - \frac{3}{2}i}.

Part b

Calculate $|

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