%E2%8B%85(3%2B2i).jpeg "(−4+4i)⋅(3+2i)")
Multiplying Complex Numbers: (-4 + 4i) * (3 + 2i)
This article will guide you through multiplying the complex numbers (-4 + 4i) and (3 + 2i).
Understanding Complex Numbers
Before diving into the multiplication, let's quickly recap complex numbers. Complex numbers are expressed in the form a + bi, where:
- a is the real part.
- b is the imaginary part.
- i is the imaginary unit, where i² = -1.
Multiplication Process
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
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Expand the product: (-4 + 4i) * (3 + 2i) = (-4 * 3) + (-4 * 2i) + (4i * 3) + (4i * 2i)
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Simplify the terms: = -12 - 8i + 12i + 8i²
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Substitute i² with -1: = -12 - 8i + 12i + 8(-1)
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Combine real and imaginary terms: = (-12 - 8) + (-8 + 12)i
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Final Result: = -20 + 4i
Therefore, the product of (-4 + 4i) and (3 + 2i) is -20 + 4i.
Visual Representation
You can visualize this multiplication on the complex plane. Each complex number corresponds to a point on the plane. Multiplying complex numbers can be thought of as a rotation and scaling operation on the complex plane.
Applications
Complex numbers have numerous applications in various fields, including:
- Engineering: Electrical circuits, signal processing, control systems
- Physics: Quantum mechanics, wave phenomena
- Mathematics: Solving equations, representing geometric transformations
Understanding complex number multiplication is essential for working with these applications.