(4-i).jpeg "(2+3i)(4-i)")
Multiplying Complex Numbers: (2 + 3i)(4 - i)
This article will guide you through the process of multiplying two complex numbers, specifically (2 + 3i)(4 - i).
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 (i² = -1).
Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last) just like we do with binomials.
-
Expand: (2 + 3i)(4 - i) = 2(4) + 2(-i) + 3i(4) + 3i(-i)
-
Simplify: = 8 - 2i + 12i - 3i²
-
Substitute i² with -1: = 8 - 2i + 12i - 3(-1)
-
Combine Real and Imaginary Terms: = (8 + 3) + (-2 + 12)i
-
Final Result: = 11 + 10i
Conclusion
Therefore, the product of (2 + 3i) and (4 - i) is 11 + 10i. This process illustrates how complex numbers are manipulated through multiplication, maintaining the form a + bi.