%E2%8B%85(1%E2%88%92i).jpeg "(−2−4i)⋅(1−i)")
Multiplying Complex Numbers: (−2−4i)⋅(1−i)
This article explores the multiplication of two complex numbers: (−2−4i)⋅(1−i). We will demonstrate the process and provide the solution in both rectangular and polar forms.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplication Process
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
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Distribute: (−2−4i)⋅(1−i) = (−2)⋅(1) + (−2)⋅(−i) + (−4i)⋅(1) + (−4i)⋅(−i)
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Simplify: = −2 + 2i − 4i + 4i²
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Substitute i² with -1: = −2 + 2i − 4i + 4(-1)
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Combine real and imaginary terms: = (−2 − 4) + (2 − 4)i
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Solution in Rectangular Form: = −6 − 2i
Polar Form
Complex numbers can also be represented in polar form, using magnitude (r) and angle (θ):
- Magnitude (r): √(a² + b²)
- Angle (θ): arctan(b/a)
For (−6 − 2i):
- r: √((-6)² + (-2)²) = √40 = 2√10
- θ: arctan(-2/-6) = arctan(1/3) ≈ 18.43° (Note: adjust angle based on quadrant)
Solution in Polar Form: 2√10 * cis(18.43°)
Conclusion
By applying the distributive property and substituting i² with -1, we successfully multiplied the complex numbers (−2−4i) and (1−i), obtaining the result −6 − 2i in rectangular form and 2√10 * cis(18.43°) in polar form.