Distribution of multlplication over addition

Starting from the construction of the complex numbers as ordered pairs of reals, prove that complex multiplication is distributive over complex addition.

Claim: (x,y)\cdot ((a,b)+(c,d))=(x,y)\cdot(a,b) + (x,y)\cdot(c,d)

Proof:

(x,y)\cdot ((a,b)+(c,d)) \\ = (x,y)\cdot(a+c,b+d) \text{ (by definition of complex addition)}\\ = (x\cdot(a+c)-y\cdot(b+d), y\cdot(a+c)+ x\cdot(b+d)) \text{ (by definition of complex multiplication)}\\ =(x\cdot a+ x\cdot c - y\cdot b - y\cdot d, y\cdot a + y\cdot c + x\cdot b + x\cdot d) \text{ (by distributivity of multiplication}\\ \text{over addition in } \mathbb{R}) \\ = (x\cdot a -y\cdot b, y\cdot a +x\cdot b) + (x\cdot c - y\cdot d, y\cdot c + x\cdot d) \text{ (by definition of complex addition)}\\ = (x,y)\cdot (a,b)+ (x,y)\cdot (c,d) \text{ (by definition of complex multiplication)}.

Thus, we have shown that which was to be demonstrated.

Let \alpha = (a, b), \beta = (c, d), and \gamma = (e, f), where \alpha, \beta, and \gamma are complex numbers being represented by ordered pairs of real numbers, a, b, c, d, e, and f.

The rules for complex addition and multiplication are as follows:

(a,b) + (c,d) = (a+c, b+d),

(a,b) \cdot (c,d) = (ac - bd, ad + bc).

For complex multiplication to be distributive over complex addition, it must be the case that

\alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma.

The former is expressed by

\alpha \cdot (\beta + \gamma) = (a,b) \cdot [(c,d) + (e,f)] = (a,b) \cdot (c+e, d+f).

(a,b) \cdot (c+e, d+f) = (a(c+e) - b(d+f), a(d+f) + b(c+e)).

Meanwhile, the latter is expressed by

\alpha \cdot \beta + \alpha \cdot \gamma = (a,b) \cdot (c,d) + (a,b) \cdot (e,f) = (ac - bd, ad + bc) + (ae - bf, af + be)

(ac - bd, ad + bc) + (ae - bf, af + be) = (ac + ae - bd - bf, ad + bc + af + be).

It is known that the multiplication of real numbers is distributive over the addition of real numbers, so we can rewrite \alpha \cdot \beta + \alpha \cdot \gamma as (a(c+e) - b(d+f), a(d+f) + b(c+e)).

Therefore, \alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma, so complex multiplication must be distributive over complex addition.