In the preceding section, the real numbers were considered from two perspectives: algebraic and geometric. It is customary to represent real numbers geometrically as points on a straight line, where a real number x corresponds to the point with abscissa x, measured according to a fixed scale and relative to a designated origin. This construction induces a total order on the set of real numbers, denoted by ℝ. At the same time, numerous algebraic properties of ℝ were identified, governing the arithmetic operations on real numbers.
The present section extends these concepts to the two-dimensional case, namely the Cartesian product ℝ² = ℝ × ℝ. As a natural consequence of this extension, the complex numbers are introduced. Their emergence resolves the longstanding problem of solving arbitrary quadratic equations, including x² + 1 = 0, a question that engaged mathematicians for centuries. The publication of Gerolamo Cardano’s Ars Magna (Latin for "Great Art") in 1545 is conventionally regarded as the formal inception of complex numbers.
Gerolamo Cardano Gerolamo (Girolamo) Cardano (1501–1576) was an Italian physician, engineer, and mathematician whose name is associated with several important innovations. He introduced complex numbers in the context of solving cubic equations and devised the Cardan shaft with universal joints, a mechanism that permits the transmission of rotary motion at variable angles and remains in use in modern vehicles. Cardano’s scientific output was extensive, comprising more than 200 works across diverse fields. The solution of cubic equations, however, was first discovered by Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1500–1557), whose methods were later publicized by Cardano.
Cardano’s personal life was marked by difficulties, and contemporary accounts often describe his eccentric and confrontational temperament. Biographical sources suggest that adverse circumstances during his early childhood may have contributed to these traits. He recorded that his mother had twice attempted to abort him, and the stigma of his birth combined with the devastation of plague within his family left indelible marks on his sense of precarity.
Based on his gambling experience for 25 years, Cardano analyzed dice and probability, making him one of the first to formalize probability theory in Liber de Ludo Aleae (“Book on Games of Chance”).
Complex numbers
As is well known, the Cartesian product ℝ² = ℝ × ℝ consists of all ordered pairs (x, y of real numbers. By definition, the Cartesian product itself carries no inherent algebraic structure; its elements are simply used to identify points in the plane. Operations such as addition and subtraction are not defined in ℝ² when viewed purely as a set of ordered pairs. In contrast, in physics one encounters vectors, which are entities characterized by both magnitude (length) and direction. A vector can be applied at any point, much like a force vector acting on a point mass.
As it is seen from Figure 1, a free vector can start at any point on the plane, so it is not clear how they can be identified in computer memory. Mathematicians found a way how to marry free vectors with points---they call this new object the affine space (see a linear algebra course). Every point P(x, y) on the plane ℝ² defines uniquely a vector [x, y] started from the origin and terminated at P, so P(x, y) = O(0, 0) + [x, y]. This formula establishes a one-to-one and onto correspondence between points on the plane (ℝ²) and the set of all plane vectors starting at the origin, which is denoted by ℂ. This relation is abbreviated as ℂ ≌ ℝ². The vector [x, y] ∈ ℂ is called the affix of the point P that represents it.
Elements of ℂ, the set of vectors on the plane starting at the origin, are called complex numbers (a rigorous definition will be given shortly); they are usually denoted by lower case letters written in italic style rather than bold font as a regular notation of vectors. This set is also called the complex plane. A geometrical representation of complex numbers as plane vectors is known as an Argand diagram. Though named after Jean-Robert Argand (1768–1822), such plots were first described by Norwegian–Danish land surveyor and mathematician Caspar Wessel (1745–1818). Every complex number z = [x, y], which is actually a vector starting at the origin, can be expanded as a linear combination
\[
z = \left[ x, y \right] = x\,\mathbf{i} + y\,\mathbf{j} ,
\]
where i = [1, 0] and j = [0, 1] are unit vectors, also commonly denoted by ⅈ and ⅉ, respectively. The direction of complex number z = [x, y] is identified by end point P(x, y) and its length is defined by Euclidean norm:
The historical development of complex numbers is marked by mystery, controversy, and conceptual difficulties. Even such eminent figures
as Descartes, Leibniz, Fermat, and the visionary Newton largely avoided or dismissed the subject. It is therefore unsurprising that the terminology surrounding complex numbers contains elements that appear strange or illogical. For example, the objects that function as vectors in the complex plane ℂ are conventionally referred to as “numbers.” The word "complex" means union or combination of two components (real numbers).
In 1777, the greatest mathematician of 18th century Leonhard Euler (1707--1783) introduced two notations: replace the unit vector along abscissa by 1 (or dropping ⅈ) and use i instead of ⅉ. This allows him to write a complex vector algebraicly as
\[
z = \left[ x, y \right] = x + y\,i .
\]
Mathematicians still follow this genius, but in engineering, computer science, and physics, the symbol j (or ⅉ) is often used instead of i. We follow the latter and write
\begin{equation} \label{EqComplex.1}
z = x + y\,{\bf j} .
\end{equation}
Using the above convention, we can write symbolically: ℂ = ℝ + ⅉℝ.
The complex numbers [0, 0], [1, 0], and [0, 1] are called zero, unit, and
imaginary unit and are identified with 0, 1, and ⅉ or j, respectively.
Cardano (1545) and Bombelli (1572) worked with square roots of negative numbers but did not give them a dedicated symbol, treated ⅉ as a kind of “fiction." When y = 0,a complex number x [1, 0] = xⅈ can be identifies with x ∈ ℝ, making real numbers as a subset of ℂ. Note that complex numbers possess other algebraic properties that include all arithmetic operations.
The number x is called the real part and y the imaginary part of the complex number z = [x, y] = x + yⅉ, often denoted as Re(z) = x or ℜ(z) = x and Im(z) = y or ℑ(z) = y.
René Descartes (1596–1650) coined the term imaginary (“imaginaires”) in 1637 in La Géométrie.
He acknowledged their existence but dismissed them as “impossible” numbers.
Let xOy be a right-handed orthogonal axes system on a plane, which means that the abscissa is rotated into the ordinate in counterclockwise direction. Then the axis x will be called the real axis, and y will be called the imaginary axis.
The complex conjugatez of a given complex number z = x + ⅉy is defined by
\[
\overline{z} = x - {\bf j}\,y \qquad \mbox{or} \qquad z^{\ast} = x - {\bf j}\,y .
\]
There is no universal notation for complex conjugate numbers, so we utilize both of them
Now we define ℂ algebraically, posponing its rigorous definition till next subsection. Since the set of real numbers is a vector space, we define ℝ² as the direct product of two copies of ℝ that inherites the algebraic structure of ℝ as a vector space.
The direct product ℝ² = ℝ × ℝ of two copies of vector space ℝ is the set
\[
\mathbb{R}^2 = \left\{ (x, y) \ : \quad \forall \ x, y \in \mathbb{R} \right\} ,
\]
which is equipped with a binary operation
\[
\left( x_1 , y_1 \right) + \left( x_2 , y_2 \right) = \left( x_1 + x_2 , y_1 + y_2 \right)
\]
and a binary function (scalar multiplication)
\[
\lambda \left( x , y \right) = \left( \lambda\,x , \lambda\,y \right) , \quad \forall \lambda \in \mathbb{R} .
\]
In order to emphasize that ℝ² is a direct product of vector spaces, we will denote its elements either as [x, y] or x + yⅉ.
Theorem 1:
The direct product ℝ² is a vector space over ℝ, that is, it has the binary operation of addition and the binary function of scalar multiplication that satisfy the following axioms:
Additive axioms. For every z₁, z₂, z₃ in ℝ², we have
z₁ + z₂ = z₂ + z₁,
(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃),
0 + z = z + 0 = z,
(−z) + z = z + (−z) = 0.
Multiplicative axioms. For every z in ℝ² and real numbers α, β, we have
0·z = z·0 = 0,
1·z = z·1 = z,
(αβ)·z = α·(β·z).
Distributive axioms. For every z₁, z₂ in ℝ² and real numbers α, β, we have
α·(z₁ + z₂) = α·z₁ + α·z₂,
(α + β)·z = α·z + β·z.
For two arbitrary complex numbers z = [x, y] = x + ⅉy and w = [𝑎, b] = 𝑎 + ⅉb. we have
\[
\left[ x, y \right] + \left[ a, b \right] = \left[ a+x, b+y \right] = \left[ a, b \right] + \left[ x, y \right]
\]
because sums of reals are real and they commute.
Also, for any real number λ,
\[
\lambda\left[ x, y \right] = \left[ \lambda\,x, \lambda\,y \right] .
\]
For any complex number z = [x, y], we have
\[
\left[ x, y \right] + \left[ 0, 0 \right] = \left[ 0, 0 \right] + \left[ x, y \right] = \left[ x, y \right] .
\]
For any complex number z = [x, y], its additive inverse is −z = [−x, −y] because
\[
z + (-z) = \left[ x , y \right] + \left[ -x , -y \right] = \left[ 0, 0 \right] .
\]
Conclusion:
All vector space axioms hold under these operations, so ℝ² is a vector space over ℝ.
Example 1:
■
End of Example 1
The operations of additing or scalar multiplication by real number could now be given equally definite meaning as geometric operations on the two corresponding vectors in the plane.
The addition of complex numbers coincides with that of ordered pairs of
real numbers in the real linear space ℝ² (Fig. 1.2); it corresponds to the usual
parallelogram rule for the sum of vectors. The multiplication of real numbers by
complex numbers coincides with the multiplication of scalars by vectors in the real
linear space; it corresponds to the expansion or contraction of the distance to 0,
according to the real number having a modulus greater or less than 1, keeping or
inverting the direction if the real number is positive or negative.
What distinguishes the complex linear space ℂ from the real linear space ℝ² is the substraction operation where the difference of two vectors is the vector which must be moved to start at the origin.
Since the set of real numbers is a field, we remind its definition.
A field is a set 𝔽 equipped with two operations (addition and multiplication) that satisfy several rules: Closure: Results of addition and multiplication stay inside 𝔽, commutativity, associativity, distributivity, rxistence of identity elements, and existence of inverses except multiplicative zero.
There are many equivalent ways to think about a complex number, each of which is useful in its own right. In this subsection, we begin with a formal definition of a complex number.
The complex numbers are pairs of real numbers,
\[
\mathbb{C} = \left\{ [x, y]\ : \quad x, y \in \mathbb{R} \right\} ,
\]
equipped with addition
\[
[x, y] + [a, b] = [x+a , y+b] ,
\]
and multiplication
\[
[x, y] \cdot [a, b] = \left[ xa - yb, xb + ya \right] .
\]
What we are stating here can be compressed in the language of algebra: equations: Equations a -- e say that (ℂ, +) is an Abelian group with unit element [0, 0]; equations f -- j say that (ℂ ∖ {[0, 0]}, •) is a commutative monoid with unit element [1, 0]. The last property is known as the distributive one.
With closure, commutativity, associativity, identities, inverses (for nonzero elements under multiplication), and distributivity all holding for the usual operations on ℝ, the reals form a commutative field.
All these properties are inherited from the properties of ℚ and preserved by the real construction, with proofs done via equivalence classes (Cauchy sequences) or set operations (Dedekind cuts).
Example 2:
■
End of Example 2
Theorem 3:
Any complex number z = [x, y] may be written in the form x + ⅉ y (also x + jy), where ⅉ = [0, 1] is the imaginary unit so ⅉ² = −1.
A simple calculation shows that [0, 1] • [0, 1] = [−1, 0] = −[1, 0], and hence ⅉ² = −1. For arbitrary complex number z, we have
\[
z = \left[ x, y \right] = \left[ x, 0 \right] + \left[ 0, 1 \right] \cdot \left[ y, 0 \right] = x + {\bf j}\, y .
\]
The form z = x + ⅉy or x + jy of the complex number z = [x, y], is called the algebraic form of the given complex number.
Example 3:
■
End of Example 3
The quantity 𝑎 + ⅉb should be viewed simply as the point in the xy-plane having Cartesian coordinates (𝑎, b), or equivalently as the affix vector connecting the origin to that point.
Observation 1:
The subset ℝ × {0} = { (x, 0) | x ∈ ℝ } ⊂ ℂ is a subfield of the ℂ with
respect to the field operations.
The function ϕ : ℝ → ℝ × {0} ⊂ ℂ defined by ϕ(x) = (x, 0) is an isomorphism between the above fields.
The division is defined as the operation, which is inverse to multiplication. Namely, if z₂ = x₂ + ⅉy₂ ≠ 0 (i.e., x₂ ≠ 0 or y₂ ≠ 0, so \( \displaystyle \quad x_2^2 + y_2^2 > 0), \quad \) then
\( \displaystyle \quad \frac{1}{z} = \frac{\overline{z}}{| z |^2} \quad \) for z ≠ 0,
−|z| ≤ Re(z) ≤ |z|, \quad −|z| ≤ Im(z) ≤ |z|,
|z₁ + z₂| ≤ |z₁| + |z₂| .
where \( \displaystyle \quad \overline{z} = \overline{a + {\bf j}\,b} = a - {\bf j}\,b = \left( a + {\bf j}\,b \right)^{\ast} = z^{\ast} \quad \) is complex conjugate of complex number z.
Show that the equation of a line in complex plane
\[
z = a + b\,t, \qquad \mbox{where} \quad a \in \mathbb{C}, \ b\in \mathbb{C} \setminus \{ 0 \} , \ t \in \mathbb{R},
\]
can be written as
\[
\mbox{Im} \frac{z-a}{b} = 0 , \qquad b\ne 0.
\]
Apostol, T.M., Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability, Wiley; 2nd edition, 1991; ISBN-13: 978-0471000075.
