%E2%8B%85(%E2%88%923%E2%88%92i).jpeg "(1+5i)⋅(−3−i)")
Multiplying Complex Numbers: (1 + 5i) ⋅ (-3 - i)
This article explores the process of multiplying two complex numbers: (1 + 5i) ⋅ (-3 - i).
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 (also known as FOIL method) similar to multiplying binomials:
- Multiply the first terms: (1) ⋅ (-3) = -3
- Multiply the outer terms: (1) ⋅ (-i) = -i
- Multiply the inner terms: (5i) ⋅ (-3) = -15i
- Multiply the last terms: (5i) ⋅ (-i) = -5i²
Now, we have: -3 - i - 15i - 5i²
Since i² = -1, we can substitute it: -3 - i - 15i - 5(-1)
Simplifying the expression: -3 - i - 15i + 5
Combining real and imaginary terms: (-3 + 5) + (-1 - 15)i
Therefore, the product of (1 + 5i) ⋅ (-3 - i) is 2 - 16i.
Conclusion
Multiplying complex numbers involves using the distributive property and simplifying the expression by substituting i² with -1. This process results in a new complex number, in this case, 2 - 16i.