(2%2B3i)(4-3i).jpeg "(1+i)(2+3i)(4-3i)")
Multiplying Complex Numbers: (1+i)(2+3i)(4-3i)
This article explores the multiplication of complex numbers, specifically the expression (1+i)(2+3i)(4-3i). We'll break down the process step-by-step, providing a clear understanding of how to handle complex number multiplication.
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.
Multiplication Process
-
Multiply the first two factors:
(1+i)(2+3i) = 1(2+3i) + i(2+3i) = 2 + 3i + 2i + 3i² = 2 + 5i - 3 (since i² = -1) = -1 + 5i
-
Multiply the result by the third factor:
(-1 + 5i)(4-3i) = -1(4-3i) + 5i(4-3i) = -4 + 3i + 20i - 15i² = -4 + 23i + 15 (since i² = -1) = 11 + 23i
The Final Result
Therefore, the product of (1+i)(2+3i)(4-3i) is 11 + 23i.
Key Points
- Remember that the imaginary unit i is defined as the square root of -1. This means that i² = -1.
- Use the distributive property to multiply complex numbers, just like you would with real numbers.
- Complex numbers are often used in various fields like electrical engineering, quantum mechanics, and signal processing.
By understanding the process of multiplying complex numbers, you can tackle more complex calculations and apply them to real-world problems.