Question

Prove that complex conjugation is an automorphism of the additive group of complex numbers.

A) Preserves addition
B) Inverts multiplication
C) Reverses exponentiation
D) Swaps real and imaginary parts

Answer 1

To prove that complex conjugation is an automorphism of the additive group of complex numbers, we need to show that it preserves addition.

Step-by-step explanation:

To prove that complex conjugation is an automorphism of the additive group of complex numbers, we need to show that it preserves addition.

Let z and w be two complex numbers, where z = a + bi and w = c + di. The sum of z and w is z + w = (a + bi) + (c + di) = (a + c) + (b + d)i. The complex conjugate of the sum is (z + w)* = (a + c) - (b + d)i, which is equal to the sum of the conjugates of z and w, z* + w* = (a - bi) + (c - di) = (a + c) - (b + d)i.

Therefore, complex conjugation preserves addition in the additive group of complex numbers.