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

Of course! Let's go through each concept one by one using complex numbers. ### Cumulative Property of Multiplication The cumulative property of multiplication states that changing the order of the factors does not change the product. Mathematically, it's represented as: \[ a \times b = b \times a \] For complex numbers, let \( z_1 = 1 + 2i \) and \( z_2 = 3 + 4i \): \[ z_1 \times z_2 = (1 + 2i) \times (3 + 4i) \] Calculate each term separately: \[ (1 \times 3) + (1 \times 4i) + (2i \times 3) + (2i \times 4i) \] \[ = 3 + 4i + 6i + 8i^2 \] Since \( i^2 = -1 \), we get: \[ = 3 + 10i - 8 \] \[ = -5 + 10i \] Now, multiply in the reverse order: \[ z_2 \times z_1 = (3 + 4i) \times (1 + 2i) \] Using the same approach: \[ (3 \times 1) + (3 \times 2i) + (4i \times 1) + (4i \times 2i) \] \[ = 3 + 6i + 4i + 8i^2 \] \[ = 3 + 10i - 8 \] \[ = -5 + 10i \] Thus: \[ (1 + 2i) \times (3 + 4i) = (3 + 4i) \times (1 + 2i) \] This demonstrates the cumulative property of multiplication for complex numbers. ### Distributive Property The distributive property states that \[a(b + c) = ab + ac\]. Let's use complex numbers \( z_1 = 2 + 3i \), \( z_2 = 1 + 4i \), and \( z_3 = 2 + i \): \[ z_1 (z_2 + z_3) = (2 + 3i) \left((1 + 4i) + (2 + i)\right) \] First, sum \( z_2 \) and \( z_3 \): \[ z_2 + z_3 = (1 + 4i) + (2 + i) = 3 + 5i \] Then, distribute \( z_1 \): \[ (2 + 3i) \times (3 + 5i) \] Calculate the product: \[ (2 \times 3) + (2 \times 5i) + (3i \times 3) + (3i \times 5i) \] \[ = 6 + 10i + 9i + 15i^2 \] \[ = 6 + 19i - 15 \] \[ = -9 + 19i \] Now let's verify the right side: \[ z_1 z_2 + z_1 z_3 = (2 + 3i) \times (1 + 4i) + (2 + 3i) \times (2 + i) \] Calculate each product separately: \[ (2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i^2 = 2 + 11i - 12 = -10 + 11i \] \[ (2 + 3i)(2 + i) = 4 + 2i + 6i + 3i^2 = 4 + 8i - 3 = 1 + 8i \] Sum the two results: \[ (-10 + 11i) + (1 + 8i) = -9 + 19i \] Thus: \[ (2 + 3i) ((1 + 4i) + (2 + i)) = (2 + 3i) (1 + 4i) + (2 + 3i) (2 + i) \] This demonstrates the distributive property for complex numbers. ### Additive Inverse of \( 5 + i \) The additive inverse of a complex number \( z \) is a number that, when added to \( z \), results in 0. For \( z = 5 + i \), the additive inverse will be a number that, when added to 5 + i, gives 0. Thus, the additive inverse of \( 5 + i \) is: \[ - (5 + i) = -5 - i \] ### Multiplicative Identity of \( -12i \) The multiplicative identity is a number that, when multiplied by any complex number, leaves the number unchanged. For complex numbers, the multiplicative identity is \( 1 \). Thus, regardless of the complex number: \[ -12i \times 1 = -12i \] So the multiplicative identity of \( -12i \) is 1. ### Summary: - **Cumulative property of multiplication**: \( (1 + 2i) \times (3 + 4i) = (3 + 4i) \times (1 + 2i) \) - **Distributive property**: \( (2 + 3i) \times \left((1 + 4i) + (2 + i)\right) = (2 + 3i) \times (1 + 4i) + (2 + 3i) \times (2 + i) \) - **Additive inverse of \( 5 + i \)**: \( -5 - i \) - **Multiplicative identity of \( -12i \)**: \( 1 \)