Math Problem Statement

(c) Berechnen Sie das Produkt der beiden komplexen Zahlen z_1 = 2 + 2j und z_2 = -1 + j. Berechnen Sie ferner die Beträge von z_1, z_2 und dem Produkt z_1 * z_2 und kontrollieren Sie, dass gilt |z_1 * z_2| = |z_1| * |z_2|.

Solution

The exercise involves two complex numbers, z1=2+2jz_1 = 2 + 2j and z2=1+jz_2 = -1 + j, and you are asked to perform the following:

Steps to solve:

  1. Calculate the product z1z2z_1 \cdot z_2: z1z2=(2+2j)(1+j)z_1 \cdot z_2 = (2 + 2j) \cdot (-1 + j) Use the distributive property (FOIL): z1z2=2(1)+2j+2j(1)+2jjz_1 \cdot z_2 = 2 \cdot (-1) + 2 \cdot j + 2j \cdot (-1) + 2j \cdot j Simplifying each term: z1z2=2+2j2j+2j2z_1 \cdot z_2 = -2 + 2j - 2j + 2j^2 Since j2=1j^2 = -1, this becomes: z1z2=2+2(1)=22=4z_1 \cdot z_2 = -2 + 2(-1) = -2 - 2 = -4 So, the product is: z1z2=4z_1 \cdot z_2 = -4

  2. Calculate the magnitudes z1|z_1|, z2|z_2|, and z1z2|z_1 \cdot z_2|:

    The magnitude of a complex number z=a+bjz = a + bj is given by z=a2+b2|z| = \sqrt{a^2 + b^2}.

    • For z1=2+2jz_1 = 2 + 2j: z1=22+22=4+4=8=22|z_1| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
    • For z2=1+jz_2 = -1 + j: z2=(1)2+12=1+1=2|z_2| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}
    • For z1z2=4z_1 \cdot z_2 = -4: Since it's a real number, the magnitude is just its absolute value: z1z2=4=4|z_1 \cdot z_2| = | -4 | = 4
  3. Check if z1z2=z1z2|z_1 \cdot z_2| = |z_1| \cdot |z_2|:

    Now, verify the relation: z1z2=z1z2|z_1 \cdot z_2| = |z_1| \cdot |z_2| Substituting the values: 4=(22)2=222=22=44 = (2\sqrt{2}) \cdot \sqrt{2} = 2\sqrt{2} \cdot \sqrt{2} = 2 \cdot 2 = 4 The equation holds true.

Summary of answers:

  • z1z2=4z_1 \cdot z_2 = -4
  • z1=22|z_1| = 2\sqrt{2}
  • z2=2|z_2| = \sqrt{2}
  • z1z2=4|z_1 \cdot z_2| = 4

Would you like more details or have any questions?

Related Questions:

  1. How do you compute the product of two complex numbers geometrically?
  2. What is the significance of the magnitudes of complex numbers in the context of multiplication?
  3. How do we represent complex numbers in polar form, and how does multiplication work there?
  4. What is the relationship between complex numbers and their conjugates in multiplication?
  5. Can the product of two complex numbers ever be a real number? Under what conditions?

Tip:

When multiplying complex numbers, always keep in mind that j2=1j^2 = -1, which is key to simplifying the result.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Magnitude of Complex Numbers
Multiplication of Complex Numbers

Formulas

Product of two complex numbers: (a + bj) * (c + dj) = (ac - bd) + (ad + bc)j
Magnitude of a complex number: |z| = sqrt(a^2 + b^2)

Theorems

Theorem: |z_1 * z_2| = |z_1| * |z_2| (Product of magnitudes theorem for complex numbers)

Suitable Grade Level

Grade 10-12 / Introductory University