Definition 73 (Landau 57). A complex number is two real numbers Ξ₁, Ξ₂ taken in a definite order, written [Ξ₁, Ξ₂]. [Ξ₁, Ξ₂] and [Θ₁, Θ₂] are the same when Ξ₁ = Θ₁ and Ξ₂ = Θ₂. Otherwise they are different.
Small letters x, y, z, u, w stand for complex numbers. The letters k, m, n, p, q are kept for whole numbers.
What a complex number is. Landau gives only the pair. But a number that is not a multitude must be a specific operation (Definition 3), so the pair needs an operand and a sense.
The operand is twofold: two real numbers (a, b) held in order, as a quantity reckoned in two kinds at once.
As addend, [Ξ₁, Ξ₂] adds Ξ₁ to the first and Ξ₂ to the second.
As multiplier, [Ξ, 0] takes each of the two Ξ-fold. [0, 1] is the operation that puts the first in the second place and the contrary of the second in the first place: (a, b) becomes (−b, a). Done twice, it gives (−a, −b), which is what −1 does. So its square is −1 because of what it does (Theorem 432). Nothing is posited. And [Ξ₁, Ξ₂] as multiplier does Ξ₁-fold the operand together with Ξ₂-fold the exchanged operand: (a, b) becomes (Ξ₁ • a − Ξ₂ • b, Ξ₁ • b + Ξ₂ • a).
On this reading Definitions 76 and 79 below are consequences, as Definitions 68 and 71 were for real numbers.
Why i is admitted. i is not a number. Neither are +n, −n, 0 or 1/n. All of these are derived numbers: specific numeric operations. i is one more, further removed from the unit than the others, since its operand is twofold. Its justification is not that it works. It is that it completes the numeric operations. Adding is undone by removing, and −n makes removing always possible. Multiplying is undone by dividing, and 1/n makes dividing always possible. Squaring is undone by taking the side, and the bounds make that possible for what is positive (Theorem 255). What is still lacking is the side of what is negative, and i supplies exactly that. Theorem 434 below proves the first part of this claim: with i, every one of these derived numbers has a side. The whole claim is larger: that every operation made by adding and multiplying can be undone, that is, that every polynomial has a root. That is not in Landau. It needs Theorem 339, and it is owed. Until it is proved, "completes" is a promise and should be called one.