algebra of complex numbers

Let us study algebra of complex numbers,
A complex number is an ordered pair of real numbers with addition defined
by
(a, b) + (c, d) = (a + c, b + d)

and multiplication defined by
(a, b) × (c, d) = (ac − bd, ad + bc),

where a, b, c, and d are any real numbers.
We will let i denote the complex number (0, 1). Then, by our definition of multiplication,
i2 = (0, 1) × (0, 1) = (0 − 1, 0 + 0) = (−1, 0).



Geometric representation of a complex number

If we identify the real number a with the complex number (a, 0), then we have
ai = (a, 0) × (0, 1) = (0 − 0, a + 0) = (0, a).
Then for any two real numbers, we have
(a, b) = (a, 0) + (0, b) = a + bi.

That is, a + bi is another way to write the complex number (a, b). In particular, with this
convention, becomes
i2 = −1,
that is,
i = p−1.
Moreover, we may write as
(a + bi) + (c + di) = (a + c) + (b + d)i
and (7.1.2) as
(a + bi) × (c + di) = (ac − bd) + (ad + bc)i.
In fact, we may view the latter as a consequence of the ordinary algebraic expansion of
the product
(a + bi)(c + di)
combined with the equality i2 = −1. That is,
(a + bi)(c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i.
It also follows from this formulation that if r is a real number, which we identify with
r + 0i, and z = a + bi is a complex number, then
rz = r(a + bi) = (r + 0i)(a + bi) = ra + rbi.
Hope the above explanation helped you.