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Multiplying Complex Numbers: (3 + i)(1 + 2i)
This article will guide you through the process of multiplying complex numbers, specifically focusing on the expression (3 + i)(1 + 2i).
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).
Multiplication of Complex Numbers
Multiplying complex numbers is similar to multiplying binomials. We use the distributive property (or FOIL method):
- Multiply the first terms: 3 * 1 = 3
- Multiply the outer terms: 3 * 2i = 6i
- Multiply the inner terms: i * 1 = i
- Multiply the last terms: i * 2i = 2i²
Now, we have: 3 + 6i + i + 2i²
Remember that i² = -1. Substituting this, we get:
3 + 6i + i + 2(-1)
Simplifying the expression:
3 + 6i + i - 2 = 1 + 7i
Conclusion
Therefore, the product of (3 + i) and (1 + 2i) is 1 + 7i.
This process demonstrates the fundamental steps involved in multiplying complex numbers. The key is to treat the imaginary unit 'i' as a variable, remembering its defining property i² = -1, and then simplify the resulting expression.