Complex Numbers

Introduction

1.Complex numbers arise to overcome certain limitations of the set of real numbers $\mathbb{R}$, in particular, the impossibility of taking even-indexed roots of negative numbers. A consequence of this restriction is that some quadratic equations with a negative discriminant have no solution in $\mathbb{R}$.

2.In the set of real numbers $\mathbb{R}$, there is no number whose square is $-1$, because the square of any real number is always non-negative. Consequently, the equation $p(x) = x^2 + 1 = 0$ has no solutions in $\mathbb{R}$. Indeed, solving it would require $x^2 = -1$, which is never satisfied in the set of real numbers $\mathbb{R}$.

3.To address this, we introduce the symbol $i$, known as the imaginary unit, which is defined by the property:

$$ i^2 = -1 $$

4.With this definition, the equation $x^2 + 1 = 0$ now has two distinct complex roots, given by $\pm i$.

Construction of the complex numbers

5.The introduction of complex numbers is sometimes treated as a matter of convenient notation, as though the symbol $i$ were simply declared to satisfy $i^2 = -1$ and the matter were settled. This approach leaves an important question unanswered: does such an object actually exist, and if so, in what mathematical sense? Answering this requires a short excursion into the construction of $\mathbb{C}$ from the real numbers.

6.The starting point is the Cartesian product $\mathbb{R}^2$, the set of all ordered pairs of real numbers. Each element of this set is a pair of the form $(a, b)$ with $a, b \in \mathbb{R}$. This set is the familiar Euclidean plane, but here we want to equip it with an algebraic structure that makes it a field. To do so, we must define addition and multiplication on $\mathbb{R}^2$.

7.Addition is defined componentwise. Given two pairs $(a, b)$ and $(c, d)$, their sum is:

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

8.This is the natural extension of vector addition in the plane and presents no difficulty.

9.Multiplication is more subtle, and it is precisely here that the algebraic structure of the complex numbers diverges from that of $\mathbb{R}^2$ viewed merely as a vector space. The product of two pairs is defined by:

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

10.This rule is the unique multiplication that turns $\mathbb{R}^2$ into a field extending $\mathbb{R}$, as will become apparent once we make the connection with the standard algebraic notation explicit. The set $\mathbb{R}^2$ equipped with these two operations is denoted $\mathbb{C}$ and its elements are called complex numbers.

11.The real numbers embed into $\mathbb{C}$ via the identification $a \mapsto (a, 0)$. One can verify directly that this map preserves both addition and multiplication, so $\mathbb{R}$ sits inside $\mathbb{C}$ as a subfield in a precise algebraic sense. The element $(0, 1)$ has no counterpart in this embedded copy of $\mathbb{R}$. Computing its square using the multiplication rule yields:

$$ (0, 1) \cdot (0, 1) \ = \ (0 \cdot 0 - 1 \cdot 1,\ 0 \cdot 1 + 1 \cdot 0) \ =\ (-1,0) $$

12.Under the identification above, the pair $(-1, 0)$ corresponds to the real number $-1$. In other words, the element $(0, 1)$ of $\mathbb{C}$ satisfies exactly the relation that the symbol $i$ is traditionally required to satisfy. This element is called the imaginary unit and is denoted $i$, so that by definition $i = (0, 1)$ and consequently $i^2 = -1$. The property $i^2 = -1$ is therefore not a postulate imposed on an undefined symbol: it is a theorem that follows from the multiplication rule on $\mathbb{R}^2$.

13.With this notation in place, every complex number $(a, b)$ can be decomposed as a combination of the two basis elements $(1, 0)$ and $(0, 1)$, which correspond to $1$ and $i$ respectively. The decomposition takes the familiar form $a + bi$, since the following chain of equalities holds:

$$ \begin{align} (a, \ b) &= (a,\ 0) + (0,\ b) \\[6pt] &= a\cdot(1,\ 0) + b\cdot(0,\ 1) \\[6pt] &= a + bi \end{align} $$

14.The notation $z = a + bi$ is thus a compact encoding of the ordered pair $(a, b)$, with $a$ called the real part and $b$ the imaginary part of $z$. These are written as $\mathrm{Re}(z) = a$ and $\mathrm{Im}(z) = b$. Note that the imaginary part is the real number $b$, not the quantity $bi$.

15.It remains to verify that the algebraic properties required of a field actually hold. The verification is routine but worth summarising. Under addition, $\mathbb{C}$ forms an abelian group: commutativity and associativity are inherited from $\mathbb{R}$, the additive identity is $(0, 0)$, and the additive inverse of $(a, b)$ is $(-a, -b)$.

16.Multiplication is likewise commutative and associative, as can be confirmed by direct computation, and the multiplicative identity is $(1, 0)$. The distributive law also holds. The only property requiring attention is the existence of multiplicative inverses for nonzero elements. Given $(a, b) \neq (0, 0)$, one checks that its multiplicative inverse is the following pair:

$$ (a, \ b)^{-1} \ = \ \left(\frac{a}{a^2 + b^2}, \ \frac{-b}{a^2 + b^2}\right) $$

17.The denominator $a^2 + b^2$ is strictly positive when $(a, b) \neq (0, 0)$, which ensures the formula is well defined for every nonzero complex number. The conclusion is that $\mathbb{C}$, as constructed, is a field. Moreover, since $\mathbb{R}$ embeds into it as a subfield, $\mathbb{C}$ is an extension field of $\mathbb{R}$. This is the precise mathematical sense in which the complex numbers extend the real number system.

18.One may also observe that, as a vector space over $\mathbb{R}$, the field $\mathbb{C}$ has dimension two, with basis $\{1, i\}$. This two-dimensionality underlies the natural geometric interpretation in the complex plane. The real and imaginary parts of a complex number serve as coordinates with respect to this basis.

19.The construction just described also generalises: replacing $\mathbb{R}$ with an arbitrary field $F$ and seeking an extension in which a chosen irreducible polynomial has a root leads to the broader theory of field extensions. The example $\mathbb{C} \cong \mathbb{R}[x]/(x^2 + 1)$ is the simplest and one of the most important instances of this general construction.

Definition

20.A complex number $z$ is a number of the form $z = a + bi$, where $a$ and $b$ are real numbers. The set of complex numbers is denoted by $\mathbb{C}$ and is formally defined by: $$ \mathbb{C} := \{ z = a + ib \mid a, b \in \mathbb{R}\} $$

21.Let $z$ be any complex number. The quantity $a$ is the real part of $z$ and is denoted by $\mathrm{Re}(z)$, while $b$ is the imaginary part of $z$ and is denoted by $\mathrm{Im}(z)$:

$$ z = a + ib \quad \Longrightarrow \quad \begin{cases} \mathrm{Re}(z) = a \\[6pt] \mathrm{Im}(z) = b \\ \end{cases} $$

23.While the algebraic form is the most familiar representation of complex numbers, an alternative and often more convenient way to express them is through their polar trigonometric form:

$$z = r(\cos\theta + i\sin\theta)$$

24.Another widely used representation is the exponential form:

$$z = re^{i\theta}$$

Complex plane

25.Because the set $\mathbb{C}$ has the structure of a Cartesian product, complex numbers can be represented geometrically in the complex plane (also known as the Gaussian or Argand plane). In this representation, the real part corresponds to the $x$-coordinate and the imaginary part corresponds to the $y$-coordinate.

26.Thus, the complex number $z = x + iy$ can be represented as the point $(x, y)$ in the complex plane.

27.A purely imaginary number is represented, for example, by the ordered pair $i = (0,1)$.

Conjugate and modulus

28.Given a complex number $z = a + bi$ its conjugate is defined as the complex number:

$$ \overline{z} = a - bi $$

29.Geometrically, $\overline{z}$ is the reflection of $z$ across the real axis in the complex plane.

30.Given the complex number $z = a + bi$, the modulus of $z$ is defined as:

$$ |z| = \sqrt{a^2 + b^2} $$

31.It is the distance from the origin to the point $(a, b)$ in the complex plane. This follows directly from the Pythagorean theorem, since the modulus is the length of the hypotenuse of a right triangle whose legs have lengths $|a|$ and $|b|$:

$$ |z|^2 = a^2 + b^2 $$

32.For example, consider the complex number $z = 3 + 2i$. Substituting $a = 3$ and $b = 2$ into the formula for the modulus, we get:

$$|z| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$$

33.Thus, the modulus of $z = 3 + 2i$ is:

$$|z| = \sqrt{13} \approx 3.61 $$

34.

This value is the distance of $z = 3 + 2i$ from the origin in the complex plane.

Argument

35.The argument of a complex number $z = a + bi$ is the angle $\theta$ between the positive real axis and the line segment from the origin to the point $(a, b)$ in the complex plane. It is measured in radians, counterclockwise from the positive real axis, and is denoted by $\arg(z)$.

36.The argument is not uniquely determined. Any two angles differing by an integer multiple of $2\pi$ represent the same geometric direction. To remove this ambiguity, one usually works with the principal argument, denoted $\mathrm{Arg}(z)$, which is the unique value of $\theta$ satisfying the following condition.

$$ -\pi < \mathrm{Arg}(z) \leq \pi $$

37.Computing the argument requires care, because the naive formula $\theta = \arctan(b/a)$ is inadequate. The arctangent function returns values only in the interval $(-\pi/2, \pi/2)$, which covers only the right half of the complex plane and is undefined when $a = 0$. The correct value of $\theta$ depends on the quadrant containing $(a, b)$ and must be determined case by case.

38.When $a > 0$, the point lies in the right half-plane and the principal argument is given by the arctangent:

$$ \mathrm{Arg}(z) = \arctan \left(\frac{b}{a}\right) $$

39.When $a < 0$ and $b \geq 0$, the point lies in the second quadrant, and a correction of $\pi$ must be added to bring the angle into the correct range:

$$ \mathrm{Arg}(z) = \arctan \left(\frac{b}{a}\right) + \pi $$

40.When $a < 0$ and $b < 0$, the point lies in the third quadrant, and the correction is $-\pi$:

$$ \mathrm{Arg}(z) = \arctan \left(\frac{b}{a}\right) - \pi $$

41.When $a = 0$, the point lies on the imaginary axis and the arctangent is undefined. In this case the argument is determined directly from the sign of $b$: if $b > 0$ then $\mathrm{Arg}(z) = \pi/2$, and if $b < 0$ then $\mathrm{Arg}(z) = -\pi/2$. The case $z = 0$ is excluded, since the argument of the origin is undefined.

42.As an illustration, consider the complex number $z = -1 + i$. Its real part is negative and its imaginary part is positive, so the point lies in the second quadrant. Applying the arctangent to the ratio $b/a = 1/(-1) = -1$ gives $\arctan(-1) = -\pi/4$, which lies in the fourth quadrant and therefore does not represent the correct argument. Since $a < 0$ and $b \geq 0$, we must apply the correction of $+\pi$:

$$ \mathrm{Arg}(z) = -\frac{\pi}{4} + \pi = \frac{3\pi}{4} $$

43.

This value agrees with the geometric position of $z = -1 + i$: the point is equidistant from both axes in the second quadrant, forming an angle of $135°$ with the positive real axis.

Properties of $\mathbb{C}$

44.The sum and product of complex numbers satisfy the associative, commutative, and distributive properties, just as the real numbers do.

45.Associative property for sum and product: when adding or multiplying complex numbers, the way in which the numbers are grouped does not affect the result.

$$(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) $$

$$(z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3) $$

46.Commutative property: the order in which two complex numbers are added or multiplied does not change the result.

$$z_1 + z_2 = z_2 + z_1 $$

$$z_1 \cdot z_2 = z_2 \cdot z_1 $$

47.Distributive property: multiplying a number by a sum gives the same result as multiplying each addend individually and then adding the products.

$$z_1 \cdot (z_2 + z_3) = z_1 \cdot z_2 + z_1 \cdot z_3 $$

48.The complex number $0 + 0i$ is the additive identity in $\mathbb{C}$, since for every complex number $z = a + bi$, we have:

$$ \begin{align} z + (0 + 0i) &= (a + bi) + (0 + 0i) \\[6pt] &= (a + 0) + (b + 0)i \\[6pt] &= a + bi \\[6pt] &= z \end{align} $$

49.The complex number $1 + 0i$ is the multiplicative identity in $\mathbb{C}$, since for every complex number $z = a + bi$, we have:

$$ \begin{align} z \cdot (1 + 0i) &= (a + bi) \cdot (1 + 0i) \\[6pt] &= a \cdot 1 + a \cdot 0i + bi \cdot 1 + bi \cdot 0i \\[6pt] &= a + bi \\[6pt] &= z \end{align} $$

50.The opposite of $a + bi$ is the complex number:

$$-(a + bi) = -a - bi $$

51.The reciprocal of a nonzero complex number $z = a + bi$ is the complex number:

$$\frac{1}{z} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} i $$

52.Complex numbers of the form $z = a + 0i$, where the imaginary part is zero, are precisely the real numbers.

53.The set of complex numbers $\mathbb{C}$ cannot be ordered in a way that is compatible with addition and multiplication. Suppose there existed a total order $\leq$ on $\mathbb{C}$ that behaved well with these operations. Then we should be able to compare $i$ with $0$. There are two possible cases:

55.Since neither case is possible, no total order on $\mathbb{C}$ can be defined that is compatible with the field operations.