Complex Numbers
Each number system solves a problem from the one before. Natural numbers count \(1, 2, 3, \ldots\). Integers let you subtract larger quantities from smaller ones. Rationals let you divide \(\frac{a}{b}\) for \(b\neq 0\). Real numbers additionally capture quantities like \(\pi\) and complex numbers solve the next problem: square roots of negative numbers.
The issue shows up immediately in a simple quadratic:
\[ f(x) := x^2+1. \]
For every real number \(x\), the square \(x^2\) is non-negative. Therefore \(x^2+1\geq 1\), so the graph never touches the \(x\)-axis. The plot shows the problem geometrically: if \(x\) is restricted to the real numbers, the equation \(x^2+1=0\) has no solution.
Algebraically, the same equation asks for
\[ x^2+1=0 \quad\Longleftrightarrow\quad x^2=-1. \]
No real number squares to \(-1\). The trick is to invent a new symbol for it: the imaginary unit \(\mathbf{i}\).
\[ \mathbf{i}^2 := -1. \]
Successive powers of \(\mathbf{i}\) repeat in a cycle of length four:
\[ \mathbf{i}^0=1, \qquad \mathbf{i}^1=\mathbf{i}, \qquad \mathbf{i}^2=-1, \qquad \mathbf{i}^3=-\mathbf{i}, \qquad \mathbf{i}^4=1. \]
Consequently, every integer power is determined by the remainder modulo four:
\[ \mathbf{i}^n=\mathbf{i}^{\,n\bmod 4}, \qquad n\in\mathbb{Z}_{\geq 0}. \]
Once we have \(\mathbf{i}\), the equation has solutions:
\[ x=\mathbf{i} \qquad\text{and}\qquad x=-\mathbf{i}, \]
because
\[ \mathbf{i}^2=-1, \qquad (-\mathbf{i})^2=\mathbf{i}^2=-1. \]
More generally, the square roots of \(-k\) (for \(k\geq 0\)) are
\[ \pm\mathbf{i}\sqrt{k}, \]
since \((\mathbf{i}\sqrt{k})^2=-k\).
In particular, if \(\sqrt{\cdot}\) denotes the principal complex square root, then
\[ \sqrt{-a}=\mathbf{i}\sqrt{a}, \qquad a\geq 0. \]
The second square root is its negative, \(-\mathbf{i}\sqrt{a}\). Specifying the principal root matters because every nonzero complex number has two square roots.
Definition of Complex Numbers
A complex number is then a number of the form
\[ \mathbf{z} := a+\mathbf{i}b, \qquad a,b\in\mathbb{R}. \]
This representation is called Rectangular Form or Cartesian form. The real number \(a\) is the real part and the real number \(b\) is the imaginary part:
\[ \Re(\mathbf{z}) := a, \qquad \Im(\mathbf{z}) := b. \]
You can also think of a complex number as an ordered pair \((a,b)\) of real numbers:
\[ (a,b) := a+\mathbf{i}b. \]
In this pair notation, the imaginary unit and the real constants \(0\) and \(1\) are
\[ \mathbf{i}:=(0,1), \qquad \mathbf{0}:=(0,0), \qquad \mathbf{1}:=(1,0). \]
The set of all complex numbers is defined as
\[ \mathbb{C} := \{\,a+\mathbf{i}b \mid a,b\in\mathbb{R},\ \mathbf{i}^2:=-1\,\}. \]
The real numbers sit inside the complex numbers as the horizontal axis:
\[ \psi:\mathbb{R}\to\mathbb{C}, \qquad \psi(x):=x+\mathbf{i}0=(x,0). \]
Thus ordinary real arithmetic is still present inside complex arithmetic. If the imaginary part is zero, we simply write \(a\) instead of \(a+\mathbf{i}0\).
As a vector space over the reals, \(\mathbb{C}\) is two-dimensional with basis \(\{1, \mathbf{i}\}\). But what makes it interesting is not the vector structure but the multiplication that makes \(\mathbf{i}^2=-1\).
Gaussian Number Plane and Polar Form
A complex number \(\mathbf{z}:=a+\mathbf{i}b\) can be drawn as a point in a two-dimensional coordinate system. The horizontal axis represents \(\Re(\mathbf{z})\), and the vertical axis represents \(\Im(\mathbf{z})\). This plane is called the Gaussian number plane, or simply the complex plane.
This geometric view is already richer than the real number line. A real function \(y=f(x)\) can be drawn in two dimensions. Plotting a complex function would require to draw the graph in the third dimension if only one real-valued output such as \(|f(\mathbf{z})|\) is shown; the full graph of a function \(f:\mathbb{C}\to\mathbb{C}\) actually contains four real dimensions: two for the input and two for the output.
Drawing a line from the origin to \(\mathbf{z}\) produces a right triangle. The length of that line is the absolute value or modulus \(|\mathbf{z}|\). The angle with the positive real axis is the argument \(\arg\mathbf{z}\).
By the Pythagorean theorem, the magnitude is
\[ |\mathbf{z}| := \sqrt{a^2+b^2}. \]
The angle is described by
\[ \arg\mathbf{z} := \theta := \tan^{-1}\left(\frac{b}{a}\right), \]
as long as the correct quadrant is understood. In practical computation, one therefore uses \(\operatorname{atan2}(b,a)\), because it uses the signs of both \(a\) and \(b\) and returns the correct quadrant.
The argument is not unique. Adding a full turn does not change the point in the plane:
\[ \arg\mathbf{z} := \theta+2\pi k, \qquad k\in\mathbb{Z}. \]
For example,
\[ \arg(-1) := \pi+2\pi k, \qquad k\in\mathbb{Z}. \]
When one single angle is needed, the principal argument is often used and denoted by \(\operatorname{Arg}(\mathbf{z})\). A common convention is
\[ \operatorname{Arg}(\mathbf{z})\in(-\pi,\pi]. \]
The zero number has no argument: it has no direction in the complex plane. For nonzero \(\mathbf{z}_1\) and \(\mathbf{z}_2\), multiplication adds directions, division subtracts them, reciprocation reverses them, and conjugation reflects them across the real axis. Thus, modulo full turns,
\[ \begin{aligned} \arg(\mathbf{z}_1\mathbf{z}_2) &\equiv \arg\mathbf{z}_1+\arg\mathbf{z}_2 &&\pmod{2\pi},\\ \arg\!\left(\frac{\mathbf{z}_1}{\mathbf{z}_2}\right) &\equiv \arg\mathbf{z}_1-\arg\mathbf{z}_2 &&\pmod{2\pi},\\ \arg(\mathbf{z}^{-1}) &\equiv -\arg\mathbf{z} &&\pmod{2\pi},\\ \arg(\overline{\mathbf{z}}) &\equiv -\arg\mathbf{z} &&\pmod{2\pi}. \end{aligned} \]
Principal arguments require wrapping the result back into \(( -\pi,\pi]\). For example, two principal arguments may add to more than \(\pi\), in which case \(2\pi\) must be subtracted. Therefore \(\operatorname{Arg}(\mathbf{z}_1\mathbf{z}_2)\) need not equal the ordinary real sum \(\operatorname{Arg}(\mathbf{z}_1)+\operatorname{Arg}(\mathbf{z}_2)\), even though they always agree modulo \(2\pi\).
The polar representation follows by factoring the magnitude out of the rectangular form:
\[ \mathbf{z} = a+\mathbf{i}b = \underbrace{\sqrt{a^2+b^2}}_{r} \left( \underbrace{\frac{a}{\sqrt{a^2+b^2}}}_{\cos\theta} + \mathbf{i}\underbrace{\frac{b}{\sqrt{a^2+b^2}}}_{\sin\theta} \right). \]
Hence, with \(r:=|\mathbf{z}|\), the polar form or phasor form is
\[ \mathbf{z} := |\mathbf{z}|(\cos\theta+\mathbf{i}\sin\theta) = |\mathbf{z}|\operatorname{cis}(\theta), \]
where
\[ \operatorname{cis}(\theta) := \cos\theta+\mathbf{i}\sin\theta. \]
Complex Numbers Form a Field
The complex numbers form a field. This means that addition and multiplication behave like the familiar operations on real numbers:
- addition is associative and commutative;
- there is an additive identity \(\mathbf{0}\);
- every complex number has an additive inverse \(-\mathbf{z}\);
- multiplication is associative and commutative;
- there is a multiplicative identity \(\mathbf{1}\);
- every nonzero complex number has a multiplicative inverse \(\mathbf{z}^{-1}\);
- multiplication distributes over addition.
The operator definitions show why these properties hold. They follow from the real-number laws applied to the two real components and from the rule \(\mathbf{i}^2:=-1\).
Operators of Complex Numbers
Complex arithmetic is defined so that the usual algebraic rules remain valid and the new relation \(\mathbf{i}^2:=-1\) is respected.
Complex Addition
\[ +:\mathbb{C}\times\mathbb{C}\to\mathbb{C} \]
For \(\mathbf{z}_1:=a_1+\mathbf{i}b_1\) and \(\mathbf{z}_2:=a_2+\mathbf{i}b_2\), complex addition is defined componentwise:
\[ \begin{aligned} \mathbf{z}_1+\mathbf{z}_2 &:= (a_1+\mathbf{i}b_1)+(a_2+\mathbf{i}b_2)\\ &:= (a_1+a_2)+\mathbf{i}(b_1+b_2). \end{aligned} \]
Geometrically, this is just vector addition in the plane. The plot shows the parallelogram law where real parts add and imaginary parts add.
The usual rules follow from real-number arithmetic:
\[ \begin{aligned} (\mathbf{z}_1+\mathbf{z}_2)+\mathbf{z}_3 &=((a_1+a_2)+a_3)+\mathbf{i}((b_1+b_2)+b_3)\\ &=(a_1+(a_2+a_3))+\mathbf{i}(b_1+(b_2+b_3))\\ &=\mathbf{z}_1+(\mathbf{z}_2+\mathbf{z}_3), \end{aligned} \]
and
\[ \mathbf{z}_1+\mathbf{z}_2 =(a_1+a_2)+\mathbf{i}(b_1+b_2) =(a_2+a_1)+\mathbf{i}(b_2+b_1) =\mathbf{z}_2+\mathbf{z}_1. \]
The additive identity is \(\mathbf{0}:=0+\mathbf{i}0\), because
\[ \mathbf{z}+\mathbf{0} =(a+0)+\mathbf{i}(b+0) =a+\mathbf{i}b =\mathbf{z}. \]
Addition is also consistent with ordinary real addition:
\[ (a_1+\mathbf{i}0)+(a_2+\mathbf{i}0) =(a_1+a_2)+\mathbf{i}0. \]
If the numbers are written in polar form, \(\mathbf{z}_1:=r_1e^{\mathbf{i}\theta_1}\) and \(\mathbf{z}_2:=r_2e^{\mathbf{i}\theta_2}\), then
\[ \mathbf{z}_1+\mathbf{z}_2 = (r_1\cos\theta_1+r_2\cos\theta_2) + \mathbf{i}(r_1\sin\theta_1+r_2\sin\theta_2). \]
Taking the squared magnitude gives the cosine law:
\[ |\mathbf{z}_1+\mathbf{z}_2|^2 = r_1^2+r_2^2+2r_1r_2\cos(\theta_1-\theta_2). \]
Complex Negation
\[ -:\mathbb{C}\to\mathbb{C} \]
The negation of \(\mathbf{z}:=a+\mathbf{i}b\) is defined componentwise:
\[ -\mathbf{z} := -a-\mathbf{i}b. \]
Geometrically, negation reflects the point through the origin.
It is the additive inverse because
\[ \mathbf{z}+(-\mathbf{z}) = (a-a)+\mathbf{i}(b-b) =0+\mathbf{i}0 =\mathbf{0}. \]
Negation is additive, but not multiplicative in the sense of changing the sign of a product:
\[ (-\mathbf{z}_1)(-\mathbf{z}_2) = \mathbf{z}_1\mathbf{z}_2, \qquad \text{not }-\mathbf{z}_1\mathbf{z}_2. \]
Complex Subtraction
\[ -:\mathbb{C}\times\mathbb{C}\to\mathbb{C} \]
Subtraction is addition with the additive inverse:
\[ \begin{aligned} \mathbf{z}_1-\mathbf{z}_2 &:= \mathbf{z}_1+(-\mathbf{z}_2)\\ &=(a_1+\mathbf{i}b_1)+(-a_2-\mathbf{i}b_2)\\ &=(a_1-a_2)+\mathbf{i}(b_1-b_2). \end{aligned} \]
Complex Multiplication
\[ \times:\mathbb{C}\times\mathbb{C}\to\mathbb{C} \]
Complex multiplication is forced by two requirements: distributivity should still work, and \(\mathbf{i}^2:=-1\). Therefore
\[ \begin{aligned} \mathbf{z}_1\mathbf{z}_2 &:= (a_1+\mathbf{i}b_1)(a_2+\mathbf{i}b_2)\\ &=a_1a_2+\mathbf{i}a_1b_2+\mathbf{i}b_1a_2+\mathbf{i}^2b_1b_2\\ &=(a_1a_2-b_1b_2)+\mathbf{i}(a_1b_2+b_1a_2). \end{aligned} \]
In pair notation this is
\[ (a_1,b_1)(a_2,b_2) := (a_1a_2-b_1b_2,\ a_1b_2+b_1a_2). \]
Multiplication by \(\mathbf{i}\) has a direct geometric meaning:
\[ \mathbf{i}(a+\mathbf{i}b) = -b+\mathbf{i}a. \]
Thus the point \((a,b)\) becomes \((-b,a)\), which is a rotation by \(90^\circ\) counterclockwise.
To visualize any complex multiplication, think of your two numbers \(\mathbf{z}_1\) and \(\mathbf{z}_2\) as vectors:
Each number, together with the unit real segment, forms a triangle:
Rotate one triangle by the angle of the other and the angles add.
Scaling by the magnitude of the other number gives the final product at the reached point.
In polar form this becomes transparent. Let
\[ \mathbf{z}_1:=r_1(\cos\theta_1+\mathbf{i}\sin\theta_1), \qquad \mathbf{z}_2:=r_2(\cos\theta_2+\mathbf{i}\sin\theta_2). \]
Then
\[ \begin{aligned} \mathbf{z}_1\mathbf{z}_2 &=r_1r_2 (\cos\theta_1+\mathbf{i}\sin\theta_1) (\cos\theta_2+\mathbf{i}\sin\theta_2)\\ &=r_1r_2 \bigl( \cos\theta_1\cos\theta_2- \sin\theta_1\sin\theta_2 \bigr)\\ &\quad +\mathbf{i}r_1r_2 \bigl( \sin\theta_1\cos\theta_2+ \cos\theta_1\sin\theta_2 \bigr)\\ &=r_1r_2(\cos(\theta_1+\theta_2)+\mathbf{i}\sin(\theta_1+\theta_2)). \end{aligned} \]
So
\[ |\mathbf{z}_1\mathbf{z}_2| = |\mathbf{z}_1|\,|\mathbf{z}_2|, \qquad \arg(\mathbf{z}_1\mathbf{z}_2) = \arg\mathbf{z}_1+ \arg\mathbf{z}_2 \quad(\operatorname{mod} 2\pi). \]
If both have magnitude 1, you're just rotating.
Multiplication commutes because real multiplication commutes:
\[ \begin{aligned} \mathbf{z}_1\mathbf{z}_2 &=(a_1a_2-b_1b_2)+\mathbf{i}(a_1b_2+b_1a_2)\\ &=(a_2a_1-b_2b_1)+\mathbf{i}(a_2b_1+b_2a_1)\\ &=\mathbf{z}_2\mathbf{z}_1. \end{aligned} \]
It is associative because multiplication in polar form gives
\[ (\mathbf{z}_1\mathbf{z}_2)\mathbf{z}_3 = r_1r_2r_3e^{\mathbf{i}(\theta_1+\theta_2+\theta_3)} = \mathbf{z}_1(\mathbf{z}_2\mathbf{z}_3). \]
Distributivity follows by expanding:
\[ \mathbf{z}_1(\mathbf{z}_2+\mathbf{z}_3) = \mathbf{z}_1\mathbf{z}_2+ \mathbf{z}_1\mathbf{z}_3. \]
The multiplicative identity is \(\mathbf{1}:=1+\mathbf{i}0\), because
\[ \mathbf{1}\mathbf{z} = (1+\mathbf{i}0)(a+\mathbf{i}b) =a+\mathbf{i}b =\mathbf{z}. \]
Multiplication is consistent with ordinary real multiplication:
\[ (a_1+\mathbf{i}0)(a_2+\mathbf{i}0) = a_1a_2+\mathbf{i}0. \]
Complex Scalar Multiplication
\[ \cdot:\mathbb{R}\times\mathbb{C}\to\mathbb{C} \]
For a real scalar \(s\in\mathbb{R}\), scalar multiplication is defined by
\[ s\cdot\mathbf{z} := s(a+\mathbf{i}b) = sa+\mathbf{i}sb. \]
Geometrically, this moves \(\mathbf{z}\) along the line through the origin. If \(s>0\), the direction is preserved. If \(s<0\), the direction is reversed.
Complex Dot Product
\[ \cdot:\mathbb{C}\times\mathbb{C}\to\mathbb{R} \]
If complex numbers are treated as vectors in \(\mathbb{R}^2\), the ordinary Euclidean dot product is
This is the Euclidean dot product on \(\mathbb{R}^2\), not the Hermitian inner product on \(\mathbb{C}\).
\[ \mathbf{z}_1\cdot\mathbf{z}_2 := (a_1,b_1)\cdot(a_2,b_2) = a_1a_2+b_1b_2. \]
In complex notation this is
\[ \mathbf{z}_1\cdot\mathbf{z}_2 = \Re(\mathbf{z}_1\overline{\mathbf{z}_2}). \]
This operation is not the same as complex multiplication. Complex multiplication returns a complex number; the Euclidean dot product returns a real number measuring alignment of two directions.
Complex Conjugate
\[ \overline{\cdot}:\mathbb{C}\to\mathbb{C} \]
The complex conjugate of \(\mathbf{z}:=a+\mathbf{i}b\) is defined as
\[ \overline{\mathbf{z}} := a-\mathbf{i}b. \]
Geometrically, conjugation reflects \(\mathbf{z}\) across the real axis. In polar form, it negates the angle:
\[ \overline{\mathbf{z}} = r(\cos(-\theta)+\mathbf{i}\sin(-\theta)) = r\operatorname{cis}(-\theta). \]
Conjugation is compatible with addition, subtraction and multiplication:
\[ \overline{\mathbf{z}_1\pm\mathbf{z}_2} = \overline{\mathbf{z}_1}\pm\overline{\mathbf{z}_2}, \qquad \overline{\mathbf{z}_1\mathbf{z}_2} = \overline{\mathbf{z}_1}\,\overline{\mathbf{z}_2}. \]
These identities follow directly from the definition. For multiplication:
\[ \begin{aligned} \overline{\mathbf{z}_1}\,\overline{\mathbf{z}_2} &=(a_1-\mathbf{i}b_1)(a_2-\mathbf{i}b_2)\\ &=(a_1a_2-b_1b_2)-\mathbf{i}(a_1b_2+a_2b_1)\\ &=\overline{\mathbf{z}_1\mathbf{z}_2}. \end{aligned} \]
The real and imaginary parts can be extracted with conjugation:
\[ \Re(\mathbf{z}) := \frac{\mathbf{z}+\overline{\mathbf{z}}}{2}, \qquad \Im(\mathbf{z}) := \frac{\mathbf{z}-\overline{\mathbf{z}}}{2\mathbf{i}}. \]
Multiplying a number by its conjugate gives the squared magnitude:
\[ \mathbf{z}\overline{\mathbf{z}} = (a+\mathbf{i}b)(a-\mathbf{i}b) = a^2+b^2 = |\mathbf{z}|^2. \]
Key conjugation facts:
\[ \overline{\overline{\mathbf{z}}}=\mathbf{z}, \qquad |\overline{\mathbf{z}}|=|\mathbf{z}|, \qquad \overline{\mathbf{z}}=\mathbf{z} \Longleftrightarrow \Im(\mathbf{z})=0, \]
and
\[ \overline{\mathbf{z}}=-\mathbf{z} \Longleftrightarrow \Re(\mathbf{z})=0. \]
For \(\mathbf{z}\neq\mathbf{0}\), also
\[ (\overline{\mathbf{z}})^{-1} = \overline{\mathbf{z}^{-1}}, \qquad \overline{\left(\frac{\mathbf{z}_1}{\mathbf{z}_2}\right)} = \frac{\overline{\mathbf{z}_1}}{\overline{\mathbf{z}_2}} \quad(\mathbf{z}_2\neq\mathbf{0}). \]
The exponential also respects conjugation:
\[ \exp(\overline{\mathbf{z}}) = \overline{\exp(\mathbf{z})}. \]
For the multi-valued logarithm introduced below, conjugation preserves the complete set of values:
\[ \log(\overline{\mathbf{z}}) = \left\{\overline{\mathbf{w}}:\mathbf{w}\in\log(\mathbf{z})\right\}. \]
Complex Multiplicative Inverse
\[ (\cdot)^{-1}:\mathbb{C}\setminus\{\mathbf{0}\}\to\mathbb{C} \]
For a nonzero complex number, the multiplicative inverse is the normalized conjugate:
\[ \begin{aligned} \mathbf{z}^{-1} &:= \frac{\mathbf{1}}{\mathbf{z}}\\ &=\frac{\overline{\mathbf{z}}}{|\mathbf{z}|^2}\\ &=\frac{a-\mathbf{i}b}{a^2+b^2}. \end{aligned} \]
This works because
\[ \mathbf{z}\cdot\frac{\overline{\mathbf{z}}}{|\mathbf{z}|^2} = \frac{\mathbf{z}\overline{\mathbf{z}}}{|\mathbf{z}|^2} = \frac{|\mathbf{z}|^2}{|\mathbf{z}|^2} = \mathbf{1}. \]
So
\[ \mathbf{i}^{-1}=-\mathbf{i}, \qquad \mathbf{1}^{-1}=\mathbf{1}, \qquad |\mathbf{z}^{-1}|=|\mathbf{z}|^{-1}. \]
Complex Division
\[ /:\mathbb{C}\times(\mathbb{C}\setminus\{\mathbf{0}\})\to\mathbb{C} \]
Division is multiplication by the inverse:
\[ \frac{\mathbf{z}_1}{\mathbf{z}_2} := \mathbf{z}_1\mathbf{z}_2^{-1} = \frac{\mathbf{z}_1\overline{\mathbf{z}_2}}{|\mathbf{z}_2|^2}, \qquad \mathbf{z}_2\neq\mathbf{0}. \]
For \(\mathbf{z}_1:=a_1+\mathbf{i}b_1\) and \(\mathbf{z}_2:=a_2+\mathbf{i}b_2\), this gives
\[ \begin{aligned} \frac{\mathbf{z}_1}{\mathbf{z}_2} &= \frac{a_1+\mathbf{i}b_1}{a_2+\mathbf{i}b_2} \cdot \frac{a_2-\mathbf{i}b_2}{a_2-\mathbf{i}b_2}\\ &= \frac{(a_1a_2+b_1b_2)+\mathbf{i}(b_1a_2-a_1b_2)}{a_2^2+b_2^2}\\ &= \frac{a_1a_2+b_1b_2}{a_2^2+b_2^2} + \mathbf{i}\frac{b_1a_2-a_1b_2}{a_2^2+b_2^2}. \end{aligned} \]
Multiplying numerator and denominator by the conjugate of the denominator is really just the inverse formula in a different form:
\[ \frac{\mathbf{z}_1}{\mathbf{z}_2} = \frac{\mathbf{z}_1\overline{\mathbf{z}_2}}{\mathbf{z}_2\overline{\mathbf{z}_2}} = \frac{\mathbf{z}_1\overline{\mathbf{z}_2}}{|\mathbf{z}_2|^2}. \]
Multiplication by \(\mathbf{i}\) rotates by 90°, so division by \(\mathbf{i}\) rotates by -90°:
\[ \frac{1}{\mathbf{i}} = \frac{\overline{\mathbf{i}}}{|\mathbf{i}|^2} = -\mathbf{i}. \]
Complex Exponential Form
\[ \exp:\mathbb{C}\to\mathbb{C} \]
Euler's formula connects the trigonometric and exponential views:
\[ e^{\mathbf{i}\theta} := \cos\theta+\mathbf{i}\sin\theta. \]
For a general complex number \(\mathbf{z}:=x+\mathbf{i}y\), the complex exponential is
\[ e^{\mathbf{z}} := e^x e^{\mathbf{i}y} = e^x(\cos y+\mathbf{i}\sin y). \]
Its modulus and argument are therefore
\[ |e^{\mathbf{z}}|=e^x, \qquad \arg(e^{\mathbf{z}})\equiv y\pmod{2\pi}. \]
In particular, \(e^{\mathbf{z}}\neq0\) and the exponential is periodic in the imaginary direction:
\[ e^{\mathbf{z}+2\pi\mathbf{i}k}=e^{\mathbf{z}}, \qquad k\in\mathbb{Z}. \]
The familiar exponential law remains valid for all \(\mathbf{z}_1,\mathbf{z}_2\in\mathbb{C}\):
\[ e^{\mathbf{z}_1+\mathbf{z}_2} = e^{\mathbf{z}_1}e^{\mathbf{z}_2}. \]
This follows by multiplying the positive real scale factors and applying the addition formulas to the sine and cosine terms. Consequently, \(e^{-\mathbf{z}}=1/e^{\mathbf{z}}\).
You can derive this by comparing power series:
\[ e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}, \qquad \cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}, \qquad \sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}. \]
Substituting \(x=\mathbf{i}\theta\) into the exponential series separates the even and odd powers:
\[ \begin{aligned} e^{\mathbf{i}\theta} &= 1+\mathbf{i}\theta+\frac{(\mathbf{i}\theta)^2}{2!} +\frac{(\mathbf{i}\theta)^3}{3!} +\frac{(\mathbf{i}\theta)^4}{4!}+\cdots\\ &= \left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right) + \mathbf{i}\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots\right)\\ &= \cos\theta+\mathbf{i}\sin\theta. \end{aligned} \]
Setting \(\theta:=\pi\) gives \(\cos\pi=-1\) and \(\sin\pi=0\), hence Euler's identity
\[ e^{\mathbf{i}\pi}+1=0. \]
It places the constants \(0\), \(1\), \(\mathbf{i}\), \(e\), and \(\pi\) in one equation and is also the special case of Euler's formula describing a half-turn around the unit circle.
Since \(|e^{\mathbf{i}\theta}|=1\) for every real \(\theta\) (a rotation never scales), squaring the real and imaginary parts of Euler's formula recovers the Pythagorean identity
\[ \cos^2\theta+\sin^2\theta=1. \]
Multiplying two unit exponentials, \(e^{\mathbf{i}(\theta_1+\theta_2)}=e^{\mathbf{i}\theta_1}e^{\mathbf{i}\theta_2}\), and comparing real and imaginary parts on both sides recovers the addition theorems
\[ \cos(\theta_1+\theta_2)=\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2, \qquad \sin(\theta_1+\theta_2)=\sin\theta_1\cos\theta_2+\cos\theta_1\sin\theta_2, \]
without ever expanding a trigonometric identity by hand. The exponential form also bounds how fast \(\theta\mapsto e^{\mathbf{i}\theta}\) can move:
\[ \bigl|e^{\mathbf{i}\theta_1}-e^{\mathbf{i}\theta_2}\bigr| = \bigl|e^{\mathbf{i}(\theta_1-\theta_2)}-1\bigr| = 2\left|\sin\frac{\theta_1-\theta_2}{2}\right| \leq |\theta_1-\theta_2|, \]
using \(|\sin t|\leq|t|\) for every real \(t\). So \(\theta\mapsto e^{\mathbf{i}\theta}\) (and hence \(\cos\) and \(\sin\) separately) is Lipschitz with constant \(1\), and therefore continuous everywhere.
So the polar form is
\[ \mathbf{z} := re^{\mathbf{i}\theta}, \qquad r:=|\mathbf{z}|, \quad \theta:=\arg\mathbf{z}. \]
With this notation, multiplication becomes more compact:
\[ (r_1e^{\mathbf{i}\theta_1})(r_2e^{\mathbf{i}\theta_2}) = r_1r_2e^{\mathbf{i}(\theta_1+\theta_2)}. \]
Complex Logarithm and Complex Powers
The exponential is not one-to-one because adding \(2\pi\mathbf{i}k\) does not change its value. Its inverse must therefore return all possible values. For \(\mathbf{z}\neq0\), write
\[ \mathbf{z}=|\mathbf{z}|e^{\mathbf{i}(\theta+2\pi k)}, \qquad k\in\mathbb{Z}. \]
The complex logarithm is the set
\[ \log(\mathbf{z}) := \left\{ \ln|\mathbf{z}|+\mathbf{i}(\operatorname{Arg}\mathbf{z}+2\pi k) :k\in\mathbb{Z} \right\}. \]
There is no logarithm of zero because the complex exponential never vanishes. Choosing only the principal argument gives the principal logarithm
\[ \operatorname{Log}(\mathbf{z}) := \ln|\mathbf{z}|+\mathbf{i}\operatorname{Arg}(\mathbf{z}), \qquad \mathbf{z}\neq0. \]
With the convention \(\operatorname{Arg}(\mathbf{z})\in(-\pi,\pi]\), this principal value jumps across the negative real axis. It satisfies \(e^{\operatorname{Log}\mathbf{z}}=\mathbf{z}\), but the reverse composition wraps the imaginary part into the principal interval:
The negative real axis is therefore called the branch cut of the principal logarithm: approaching it from above gives arguments near \(\pi\), whereas approaching it from below gives arguments near \(-\pi\).
\[ \operatorname{Log}(e^{\mathbf{z}}) = \mathbf{z}-2\pi\mathbf{i}k \]
for the unique integer \(k\) that makes the imaginary part lie in \(( -\pi,\pi]\). The familiar real logarithm laws consequently need branch corrections. For example,
\[ \operatorname{Log}(\mathbf{z}_1\mathbf{z}_2) = \operatorname{Log}(\mathbf{z}_1) + \operatorname{Log}(\mathbf{z}_2) +2\pi\mathbf{i}k \]
for some \(k\in\mathbb{Z}\), not necessarily \(k=0\). Indeed, \(\operatorname{Log}(-1)=\mathbf{i}\pi\), while \(\operatorname{Log}((-1)(-1))=\operatorname{Log}(1)=0\).
For \(\mathbf{z}\neq0\) and \(\mathbf{w}\in\mathbb{C}\), a general complex power is defined through the logarithm:
\[ \mathbf{z}^{\mathbf{w}} := \left\{ \exp\!\left( \mathbf{w} \left(\ln|\mathbf{z}|+\mathbf{i}(\operatorname{Arg}\mathbf{z}+2\pi k)\right) \right) :k\in\mathbb{Z} \right\}. \]
This set may contain infinitely many, finitely many, or only one distinct value. Selecting \(k=0\) defines the principal value \(\exp(\mathbf{w}\operatorname{Log}\mathbf{z})\). For integer exponents all choices coincide, recovering ordinary repeated multiplication; for non-integer exponents the branch choice is part of the definition.
Complex Integer Power: De Moivre's Formula
\[ (\cdot)^n:\mathbb{C}\to\mathbb{C}, \qquad n\in\mathbb{Z} \]
Let
\[ \mathbf{z}:=|\mathbf{z}|(\cos\theta+\mathbf{i}\sin\theta) = |\mathbf{z}|e^{\mathbf{i}\theta}. \]
For an integer \(n\), repeated multiplication gives
\[ \begin{aligned} \mathbf{z}^n &=\left(|\mathbf{z}|e^{\mathbf{i}\theta}\right)^n\\ &=|\mathbf{z}|^n e^{\mathbf{i}n\theta}\\ &=|\mathbf{z}|^n(\cos(n\theta)+\mathbf{i}\sin(n\theta)). \end{aligned} \]
De Moivre's formula:
\[ \bigl(\cos\theta+\mathbf{i}\sin\theta\bigr)^n = \cos(n\theta)+\mathbf{i}\sin(n\theta). \]
It says that powers repeatedly scale the radius and rotate the angle:
\[ |\mathbf{z}^n|=|\mathbf{z}|^n, \qquad \arg(\mathbf{z}^n)=n\arg(\mathbf{z}) \quad(\text{mod }2\pi). \]
Complex Roots
\[ \sqrt[n]{\cdot}:\mathbb{C}\to\mathcal{P}(\mathbb{C}) \]
Complex roots are multi-valued. Take
\[ \mathbf{z}:=re^{\mathbf{i}\theta}, \qquad r>0. \]
We want all \(\mathbf{w}\) such that \(\mathbf{w}^n=\mathbf{z}\). Writing \(\mathbf{w}:=\rho e^{\mathbf{i}\varphi}\):
\[ \rho^n e^{\mathbf{i}n\varphi} = re^{\mathbf{i}\theta}. \]
So \(\rho:=r^{1/n}\) and the angles satisfy
\[ n\varphi = \theta+2\pi k, \qquad k\in\mathbb{Z}. \]
The \(n\) distinct \(n\)-th roots are therefore
\[ \mathbf{w}_k := r^{1/n}e^{\mathbf{i}(\theta+2\pi k)/n}, \qquad k=0,1,\dots,n-1. \]
Choosing \(\theta:=\operatorname{Arg}(\mathbf{z})\) and \(k:=0\) gives the principal \(n\)-th root
\[ \sqrt[n]{\mathbf{z}}_{\mathrm{principal}} := |\mathbf{z}|^{1/n} e^{\mathbf{i}\operatorname{Arg}(\mathbf{z})/n}. \]
They lie evenly spaced on a circle of radius \(r^{1/n}\). When \(\mathbf{z}=\mathbf{1}\), these are the roots of unity:
\[ \omega_k := e^{2\pi\mathbf{i}k/n}, \qquad k=0,1,\dots,n-1. \]
If \(\mathbf{w}_0\) is any one \(n\)-th root of \(\mathbf{z}\) and \(\omega:=e^{2\pi\mathbf{i}/n}\), then every root is obtained by rotating \(\mathbf{w}_0\) through successive multiples of \(2\pi/n\):
\[ \mathbf{w}_k=\mathbf{w}_0\omega^k, \qquad k=0,1,\dots,n-1. \]
This is why complex numbers matter so much for algebra. The fundamental theorem of algebra says every non-constant polynomial with complex coefficients has a root. Equivalently, every degree-\(n\) polynomial factors into exactly \(n\) linear factors (counting multiplicity). The real numbers don't have this property yet; the complex numbers do.
Real coefficients impose an additional symmetry. If
\[ p(\mathbf{z}) := a_0+a_1\mathbf{z}+\cdots+a_n\mathbf{z}^n, \qquad a_0,\dots,a_n\in\mathbb{R}, \]
then conjugation distributes through every sum and product, while \(\overline{a_k}=a_k\). Therefore
\[ p(\overline{\mathbf{z}}) = \overline{p(\mathbf{z})}. \]
If \(p(\mathbf{z}_0)=0\), it follows that \(p(\overline{\mathbf{z}_0})=\overline{0}=0\). Hence every nonreal root of a polynomial with real coefficients occurs together with its conjugate, with the same multiplicity. For \(\mathbf{z}_0:=a+\mathbf{i}b\), the paired linear factors combine into a real quadratic:
\[ (x-\mathbf{z}_0)(x-\overline{\mathbf{z}_0}) = (x-a)^2+b^2 = x^2-2ax+(a^2+b^2). \]
Consequently, every polynomial with real coefficients factors over \(\mathbb{R}\) into real linear factors and irreducible real quadratic factors. Over \(\mathbb{C}\), those quadratic factors split into their two conjugate linear factors.
Proving this needs one more tool than complex arithmetic alone: the extreme value theorem for continuous functions on compact sets, which is where the full proof is given.
Complex Trigonometric and Hyperbolic Functions
The exponential definitions extend sine and cosine from real to complex arguments. For every \(\mathbf{z}\in\mathbb{C}\), define
\[ \cos\mathbf{z} := \frac{e^{\mathbf{i}\mathbf{z}}+e^{-\mathbf{i}\mathbf{z}}}{2}, \qquad \sin\mathbf{z} := \frac{e^{\mathbf{i}\mathbf{z}}-e^{-\mathbf{i}\mathbf{z}}}{2\mathbf{i}}. \]
For real arguments these definitions agree with the usual trigonometric functions. They also make Euler's formula valid for every complex argument:
\[ e^{\mathbf{i}\mathbf{z}} = \cos\mathbf{z}+\mathbf{i}\sin\mathbf{z}. \]
The remaining trigonometric functions are defined by quotients wherever the corresponding denominator is nonzero; for example,
\[ \operatorname{tan}\mathbf{z}:=\frac{\sin\mathbf{z}}{\cos\mathbf{z}}, \qquad \cot\mathbf{z}:=\frac{\cos\mathbf{z}}{\sin\mathbf{z}}. \]
The complex hyperbolic functions are the even and odd parts of the exponential:
\[ \cosh\mathbf{z}:=\frac{e^{\mathbf{z}}+e^{-\mathbf{z}}}{2}, \qquad \sinh\mathbf{z}:=\frac{e^{\mathbf{z}}-e^{-\mathbf{z}}}{2}. \]
Substituting \(\mathbf{i}\mathbf{z}\) into these definitions links the two families directly:
\[ \begin{aligned} \cosh(\mathbf{i}\mathbf{z})&=\cos\mathbf{z}, &\qquad \sinh(\mathbf{i}\mathbf{z})&=\mathbf{i}\sin\mathbf{z},\\ \cos(\mathbf{i}\mathbf{z})&=\cosh\mathbf{z}, & \sin(\mathbf{i}\mathbf{z})&=\mathbf{i}\sinh\mathbf{z}. \end{aligned} \]
Restricting the hyperbolic functions to a real argument \(x\) gives
\[ \cosh x:=\frac{e^x+e^{-x}}{2}, \qquad \sinh x:=\frac{e^x-e^{-x}}{2}. \]
\(\cosh\) is the even part of \(e^x\) and \(\sinh\) is the odd part, so \(e^x=\cosh x+\sinh x\) by construction — mirroring \(e^{\mathbf{i}\theta}=\cos\theta+\mathbf{i}\sin\theta\), with the roles of "even/odd part" and "real/imaginary part" swapped. The Pythagorean identity above has a hyperbolic counterpart with a sign flip instead of complex numbers:
\[ \cosh^2x-\sinh^2x=\frac{(e^x+e^{-x})^2-(e^x-e^{-x})^2}{4}=\frac{4e^xe^{-x}}{4}=1, \]
which is why the pair \((\cosh t,\sinh t)\) traces the hyperbola \(u^2-v^2=1\) the way \((\cos t,\sin t)\) traces the unit circle. The addition theorems carry over the same way, directly from \(e^{x+y}=e^xe^y\):
\[ \cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y, \qquad \sinh(x+y)=\sinh x\cosh y+\cosh x\sinh y. \]
Two more hyperbolic functions are defined as ratios, in analogy with tangent and cotangent:
\[ \tanh x:=\frac{\sinh x}{\cosh x}, \qquad \coth x:=\frac{\cosh x}{\sinh x}\ \ (x\neq0). \]
Since \(\cosh x\geq1>0\) for every real \(x\), \(\sinh\) is a strictly increasing bijection \(\mathbb{R}\to\mathbb{R}\) and \(\cosh\) is a strictly increasing bijection \([0,\infty)\to[1,\infty)\); both have inverse functions, found in closed form the same way complex \(n\)-th roots were found above — undo the exponential with a logarithm. Solving \(y=\sinh x=\frac{e^x-e^{-x}}{2}\) for \(x\) as a quadratic in \(e^x\) gives
\[ \operatorname{arsinh}y=\ln\!\left(y+\sqrt{y^2+1}\right), \qquad \operatorname{arcosh}y=\ln\!\left(y+\sqrt{y^2-1}\right)\ \ (y\geq1). \]
Extending a familiar real function to the complex numbers this way is not limited to roots and inverse trigonometric functions: the ordinary factorial, defined only for non-negative integers, extends the same way to a function of a complex variable via an integral that reproduces every factorial exactly and stays finite everywhere except at the non-positive integers.
Absolute Value of a Complex Number
\[ |\cdot|:\mathbb{C}\to\mathbb{R}_{\geq 0} \]
The absolute value of a real number is its distance from zero. For complex numbers, the same idea becomes the Euclidean distance from the origin in the Gaussian number plane:
\[ |\mathbf{z}| := |a+\mathbf{i}b| := \sqrt{a^2+b^2}. \]
Equivalently,
\[ |\mathbf{z}|^2 := \mathbf{z}\overline{\mathbf{z}} = a^2+b^2. \]
The absolute value is always real and non-negative. Its basic properties are
\[ |\mathbf{z}|\geq 0, \qquad |\mathbf{z}|=0\Longleftrightarrow\mathbf{z}=\mathbf{0}, \]
\[ |\Re(\mathbf{z})|\leq|\mathbf{z}|, \qquad |\Im(\mathbf{z})|\leq|\mathbf{z}|, \]
\[ |\overline{\mathbf{z}}|=|\mathbf{z}|, \qquad |\mathbf{z}^{-1}|=|\mathbf{z}|^{-1} \quad(\mathbf{z}\neq\mathbf{0}), \]
and
\[ |\mathbf{z}_1\mathbf{z}_2| = |\mathbf{z}_1|\,|\mathbf{z}_2|. \]
The multiplicative property follows from conjugation:
\[ \begin{aligned} |\mathbf{z}_1\mathbf{z}_2|^2 &=(\mathbf{z}_1\mathbf{z}_2)\overline{(\mathbf{z}_1\mathbf{z}_2)}\\ &=\mathbf{z}_1\mathbf{z}_2\overline{\mathbf{z}_1}\,\overline{\mathbf{z}_2}\\ &=(\mathbf{z}_1\overline{\mathbf{z}_1})(\mathbf{z}_2\overline{\mathbf{z}_2})\\ &=|\mathbf{z}_1|^2|\mathbf{z}_2|^2. \end{aligned} \]
The triangle inequality:
\[ |\mathbf{z}_1+\mathbf{z}_2| \leq |\mathbf{z}_1|+|\mathbf{z}_2|. \]
It follows from
\[ \begin{aligned} |\mathbf{z}_1+\mathbf{z}_2|^2 &=(\mathbf{z}_1+\mathbf{z}_2) (\overline{\mathbf{z}_1}+\overline{\mathbf{z}_2})\\ &=|\mathbf{z}_1|^2+|\mathbf{z}_2|^2 +\mathbf{z}_1\overline{\mathbf{z}_2} +\overline{\mathbf{z}_1}\mathbf{z}_2\\ &=|\mathbf{z}_1|^2+|\mathbf{z}_2|^2 +2\Re(\mathbf{z}_1\overline{\mathbf{z}_2})\\ &\leq |\mathbf{z}_1|^2+|\mathbf{z}_2|^2 +2|\mathbf{z}_1\overline{\mathbf{z}_2}|\\ &=(|\mathbf{z}_1|+|\mathbf{z}_2|)^2. \end{aligned} \]
Equality holds precisely when the two numbers point in the same direction, including the case where either number is zero. Indeed, the only estimate in the proof is
\[ \Re(\mathbf{z}_1\overline{\mathbf{z}_2}) \leq |\mathbf{z}_1\overline{\mathbf{z}_2}|. \]
Equality therefore requires \(\mathbf{z}_1\overline{\mathbf{z}_2}\) to be real and non-negative. If neither number is zero, this is equivalent to
\[ \mathbf{z}_1=t\mathbf{z}_2 \qquad\text{for some }t>0. \]
Geometrically, the two vectors are parallel and point the same way, so the triangle formed by their sum degenerates into a line segment. Consequently,
\[ |\mathbf{z}_1+\mathbf{z}_2| =|\mathbf{z}_1|+|\mathbf{z}_2| \]
if and only if one number is zero or one is a positive real multiple of the other.
Generalized Triangle Inequalities
The reverse inequality is just as useful. It holds for both a sum and a difference:
\[ \bigl||\mathbf{z}_1|-|\mathbf{z}_2|\bigr| \leq |\mathbf{z}_1\pm\mathbf{z}_2|. \]
It follows from applying the triangle inequality to \(\mathbf{z}_1=(\mathbf{z}_1-\mathbf{z}_2)+\mathbf{z}_2\) and by symmetry to \(\mathbf{z}_2\). Replacing \(\mathbf{z}_2\) by \(-\mathbf{z}_2\) gives the sum case because \(|-\mathbf{z}_2|=|\mathbf{z}_2|\). Together with the triangle inequality, this gives
\[ \bigl||\mathbf{z}_1|-|\mathbf{z}_2|\bigr| \leq |\mathbf{z}_1\pm\mathbf{z}_2| \leq |\mathbf{z}_1|+|\mathbf{z}_2|. \]
More generally, for every \(\mathbf{u}\in\mathbb{C}\),
\[ |\mathbf{z}_1-\mathbf{z}_2| \leq |\mathbf{z}_1-\mathbf{u}|+|\mathbf{u}-\mathbf{z}_2|. \]
Distance Measure
\[ d:\mathbb{C}\times\mathbb{C}\to\mathbb{R}_{\geq 0} \]
The distance between two complex numbers is the magnitude of their difference:
\[ d(\mathbf{z}_1,\mathbf{z}_2) := |\mathbf{z}_1-\mathbf{z}_2|. \]
In rectangular coordinates:
\[ d(a_1+\mathbf{i}b_1,\ a_2+\mathbf{i}b_2) = \sqrt{(a_1-a_2)^2+(b_1-b_2)^2}. \]
The absolute value is therefore the special case
\[ |\mathbf{z}| = d(\mathbf{0},\mathbf{z}). \]
This distance is a metric on \(\mathbb{C}\). Definiteness follows from \(|\mathbf{z}_1-\mathbf{z}_2|=0\) exactly when \(\mathbf{z}_1=\mathbf{z}_2\), symmetry follows from \(|-\mathbf{z}|=|\mathbf{z}|\), and the triangle inequality gives
\[ \begin{aligned} d(\mathbf{z}_1,\mathbf{z}_3) &=|\mathbf{z}_1-\mathbf{z}_3|\\ &=|(\mathbf{z}_1-\mathbf{z}_2)+(\mathbf{z}_2-\mathbf{z}_3)|\\ &\leq d(\mathbf{z}_1,\mathbf{z}_2)+d(\mathbf{z}_2,\mathbf{z}_3). \end{aligned} \]
Convergence in this metric is exactly componentwise convergence. For \(\mathbf{z}_n:=a_n+\mathbf{i}b_n\) and \(\mathbf{z}:=a+\mathbf{i}b\),
\[ \mathbf{z}_n\to\mathbf{z} \quad\Longleftrightarrow\quad a_n\to a\ \text{ and }\ b_n\to b. \]
The forward implication follows from \(|a_n-a|\leq|\mathbf{z}_n-\mathbf{z}|\) and \(|b_n-b|\leq|\mathbf{z}_n-\mathbf{z}|\). Conversely,
\[ |\mathbf{z}_n-\mathbf{z}| =\sqrt{(a_n-a)^2+(b_n-b)^2} \longrightarrow 0 \]
whenever both real component sequences converge. The same estimates show that a sequence in \(\mathbb{C}\) is a Cauchy sequence exactly when its real and imaginary parts are Cauchy sequences in \(\mathbb{R}\). Since \(\mathbb{R}\) is complete, both components then have real limits, and those limits combine into a limit in \(\mathbb{C}\). Thus the complex plane is a complete metric space.
Complex Equality
\[ =:\mathbb{C}\times\mathbb{C}\to\{\mathrm{true},\mathrm{false}\} \]
Complex numbers can't be ordered in a way compatible with their algebraic structure. But you can compare them for equality: they're equal when both real and imaginary parts are equal.
\[ \mathbf{z}_1 = \mathbf{z}_2 \Longleftrightarrow \Re(\mathbf{z}_1)=\Re(\mathbf{z}_2) \;\text{and}\; \Im(\mathbf{z}_1)=\Im(\mathbf{z}_2) \]
So for \(\mathbf{z}_1=a_1+\mathbf{i}b_1\) and \(\mathbf{z}_2=a_2+\mathbf{i}b_2\), equality means
\[ a_1+\mathbf{i}b_1 = a_2+\mathbf{i}b_2 \Longleftrightarrow a_1=a_2 \;\text{and}\; b_1=b_2. \]
Complex Numbers Have No Order
Complex numbers can be checked for equality, but they cannot be sorted like real numbers in a way that respects addition and multiplication. The problem is not that we cannot invent some ordering of points in the plane. We could sort them lexicographically, by magnitude, or by angle. The problem is that none of these orders behaves like the usual order on real numbers.
For an order compatible with multiplication, every nonzero square must be positive. That's true for reals: if \(x\neq 0\), then \(x^2>0\). But for complex numbers we have the imaginary unit \(\mathbf{i}\) with
\[ \mathbf{i}^2 := -1. \]
If complex numbers had such an order, then \(\mathbf{i}^2=-1\) would be the square of a nonzero number, so it'd have to be positive. That means \(-1>0\). But also \(1=1^2>0\). Both 1 and -1 are positive, which is impossible in an ordered field.
So complex numbers have no natural order compatible with their field structure. You can measure size with the absolute value \(|\mathbf{z}|\) and test equality, but there's no \(\leq\) that works like it does for reals.