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Multiplying Complex Numbers: (1 - 2i)(1 + 2i)
This article will walk you through the process of multiplying the complex numbers (1 - 2i) and (1 + 2i).
Understanding Complex Numbers
A complex number is a number 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.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just like we would with any other binomial multiplication.
Let's multiply (1 - 2i) and (1 + 2i):
(1 - 2i)(1 + 2i) = 1(1 + 2i) - 2i(1 + 2i)
Now, distribute:
= 1 + 2i - 2i - 4i²
Remember that i² = -1. Substitute this in:
= 1 + 2i - 2i - 4(-1)
Simplify:
= 1 + 4
= 5
Conclusion
The product of (1 - 2i) and (1 + 2i) is 5. Notice that the result is a real number. This is because (1 - 2i) and (1 + 2i) are complex conjugates of each other.
Complex conjugates have the interesting property that their product always results in a real number.