(3-2i).jpeg "(1+i)(3-2i)")
Multiplying Complex Numbers: (1+i)(3-2i)
This article will walk you through the process of multiplying two complex numbers: (1+i) and (3-2i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers
- i is the imaginary unit, defined as the square root of -1 (i² = -1)
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
Step 1: Expand the Expression
(1 + i)(3 - 2i) = 1(3 - 2i) + i(3 - 2i)
Step 2: Simplify
= 3 - 2i + 3i - 2i²
Step 3: Substitute i² with -1
= 3 - 2i + 3i - 2(-1)
Step 4: Combine Real and Imaginary Terms
= (3 + 2) + (-2 + 3)i
Step 5: Final Result
= 5 + i
Therefore, the product of (1 + i) and (3 - 2i) is 5 + i.
Visual Representation
Complex numbers can be represented graphically on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Multiplying complex numbers can be visualized as a rotation and scaling of the points representing the numbers on the complex plane.
Conclusion
Multiplying complex numbers involves applying the distributive property and simplifying the resulting expression. Understanding this process is essential for working with complex numbers in various fields like mathematics, physics, and engineering.