(3%2Bi).jpeg "(2-i)(3+i)")
Multiplying Complex Numbers: (2-i)(3+i)
This article will guide you through the process of multiplying two complex numbers: (2 - i)(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).
The Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL method):
(2 - i)(3 + i) = 2(3 + i) - i(3 + i)
Now, we distribute:
= 6 + 2i - 3i - i²
Remember that i² = -1, so we can substitute:
= 6 + 2i - 3i - (-1)
Combining real and imaginary terms:
= (6 + 1) + (2 - 3)i
Finally, we get:
(2 - i)(3 + i) = 7 - i
Conclusion
Therefore, the product of the complex numbers (2 - i) and (3 + i) is 7 - i.
By following the steps outlined above, you can confidently multiply any two complex numbers. Remember to distribute carefully and use the fact that i² = -1 to simplify the expression.