To show that $a + b = b + a$ for all $a, b \in \mathbb{C}$ (where $\mathbb{C}$ denotes the set of all complex numbers), we need to prove that addition of complex numbers is commutative.

Proof:

Let $a$ and $b$ be any two complex numbers. A complex number is generally represented as $z = x + iy$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit with the property $i^2 = -1$.

Thus, we can write: $a = a_1 + i a_2 \quad \text{and} \quad b = b_1 + i b_2,$ where $a_1, a_2, b_1, b_2$ are real numbers.

Now, consider the sum $a + b$: $a + b = (a_1 + i a_2) + (b_1 + i b_2).$ Using the distributive property of addition over the complex numbers, this becomes: $a + b = (a_1 + b_1) + i(a_2 + b_2).$

Next, consider the sum $b + a$: $b + a = (b_1 + i b_2) + (a_1 + i a_2).$ Again, using the distributive property of addition: $b + a = (b_1 + a_1) + i(b_2 + a_2).$

Since addition of real numbers is commutative, we have: $a_1 + b_1 = b_1 + a_1 \quad \text{and} \quad a_2 + b_2 = b_2 + a_2.$

Therefore: $(a_1 + b_1) + i(a_2 + b_2) = (b_1 + a_1) + i(b_2 + a_2).$

Thus, $a + b = b + a$, which shows that the addition of complex numbers is commutative.

Conclusion:

We have shown that for all complex numbers $a$ and $b$, the equality $a + b = b + a$ holds. Therefore, addition in $\mathbb{C}$ is commutative.

Would you like any further details or have any questions?

Here are 5 related questions:

1. How can the commutative property be extended to multiplication of complex numbers?
2. What is the difference between the commutative and associative properties?
3. Can you prove the commutative property for the subtraction of complex numbers?
4. How does the commutative property work in vector spaces?
5. How is the commutative property used in solving linear equations?

Tip: The commutative property is not only valid for addition of complex numbers, but also for other operations like multiplication in many algebraic structures, such as real numbers and matrices.