Unlocking the Secrets of Complex Numbers: Properties and Inverses Explained!

Sure! Let's break down each concept using complex numbers. ### 1. Commutative Property of Multiplication: The commutative property states that the order in which you multiply two numbers does not affect the product. For complex numbers \( z_1 \) and \( z_2 \): \[ z_1 \cdot z_2 = z_2 \cdot z_1 \] **Example:** Let \( z_1 = 3 + 2i \) and \( z_2 = 1 - 4i \). Calculate both: \[ z_1 \cdot z_2 = (3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i \] \[ z_2 \cdot z_1 = (1 - 4i)(3 + 2i) = 3 + 2i - 12i - 8i^2 = 3 - 10i + 8 = 11 - 10i \] So, \( 3 + 2i \cdot 1 - 4i = 1 - 4i \cdot 3 + 2i \) (i.e., \( 11 - 10i = 11 - 10i \)). ### 2. Distributive Property: The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend and then adding the products. For complex numbers \( z_1 \), \( z_2 \), and \( z_3 \): \[ z_1 \cdot (z_2 + z_3) = z_1 \cdot z_2 + z_1 \cdot z_3 \] **Example:** Let \( z_1 = 2 + 3i \), \( z_2 = -1 + 4i \), and \( z_3 = 3 - i \). Calculate both sides: \[ z_1 \cdot (z_2 + z_3) = (2 + 3i)((-1 + 4i) + (3 - i)) \] \[ = (2 + 3i)(2 + 3i) = 2 + 3i = 4 + 6i = 10i \] \[ z_1 \cdot z_2 + z_1 \cdot z_3 = (2 + 3i)(-1 + 4i) + (2 + 3i)(3 - i) \] \[ = (-2 + 8i + (3i)^2(-1)) + (6 - 2i + 9i + i^2) \] \[ = 10 + 6 \] ### 3. Additive Inverse: The additive inverse of a complex number \( z \) is a number that, when added to \( z \), results in 0. For the number \( 5 + i \): \[ \text{Additive Inverse of } 5 + i = - (5 + i) = -5 - i \] So, \[ (5 + i) + (-5 - i) = 0 \] ### 4. Multiplicative Identity: The multiplicative identity is a number that, when multiplied by any complex number, gives the same complex number. The multiplicative identity for any complex number, including \( -12i \), is \( 1 \). \[ 1 \cdot -12i = -12i \] By keeping these properties in mind, you can handle multiplication, addition, and distribution of complex numbers effectively!