(4%2B2i)-in-standard-form.jpeg "(4-2i)(4+2i) In Standard Form")
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.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just as we would with real numbers. However, we must remember that i² = -1.
(4 - 2i)(4 + 2i) in Standard Form
Let's multiply the expression (4 - 2i)(4 + 2i) using the distributive property (also known as FOIL):
- First: (4)(4) = 16
- Outer: (4)(2i) = 8i
- Inner: (-2i)(4) = -8i
- Last: (-2i)(2i) = -4i²
Now, combining the terms and substituting i² = -1:
16 + 8i - 8i - 4(-1) = 16 + 4
Therefore, (4 - 2i)(4 + 2i) simplified to 20.
Observations
The result, 20, is a real number. This is because we multiplied a complex number by its complex conjugate. The complex conjugate of a complex number a + bi is a - bi. When we multiply a complex number by its conjugate, the imaginary terms cancel out, resulting in a real number.