Complex numbers (introductory)

Intuition [Beginner]

Some equations have no real solution. The equation $x^2+1=0$ asks for a number whose square is $-1$. No real number works, because every real square is zero or positive. The number line has no answer.

The solution is to invent a new number, called $i$, with the property $i^2=-1$. This single addition, plus the rule that arithmetic works as usual, produces the complex numbers. Every complex number has the form $a+bi$ where $a$ and $b$ are real. The letter $i$ is not a variable — it is a fixed symbol meaning "the number whose square is $-1$."

Complex numbers live in a plane, not on a line. The real part $a$ gives the horizontal coordinate, and the imaginary part $b$ gives the vertical coordinate. The number $3+2i$ sits three units right and two units up from the origin. This picture turns every algebraic operation into geometry: adding shifts the point, and multiplying rotates and stretches it.

Visual [Beginner]

The complex plane has a horizontal axis for the real part and a vertical axis for the imaginary part. Each complex number $a+bi$ corresponds to exactly one point $(a,b)$ in this plane.

The horizontal axis holds the real numbers (where $b=0$). The vertical axis holds the purely imaginary numbers (where $a=0$). The distance from the origin to the point $a+bi$ is $\sqrt{a^2+b^2}$, called the modulus, measured by the same distance formula used in coordinate geometry.

Worked example [Beginner]

Add and multiply $(2+3i)$ and $(1-4i)$.

Addition. $(2+3i)+(1-4i)=(2+1)+(3-4)i=3-i$. Add real parts together, add imaginary parts together.

Multiplication. $(2+3i)(1-4i)=2(1)+2(-4i)+3i(1)+3i(-4i)=2-8i+3i-12i^2$. Since $i^2=-1$, the last term becomes $-12(-1)=12$. So the result is $(2+12)+(-8+3)i=14-5i$.

What this tells us: complex arithmetic follows the same distributive rules as real arithmetic, with the single extra rule $i^2=-1$ handling the new symbol.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The complex numbers $\mathbb{C}$ are the set of ordered pairs $(a,b)$ of real numbers, equipped with addition and multiplication:

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

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

The notation $a+bi$ abbreviates $(a,b)$, where $i$ denotes the pair $(0,1)$. The multiplication rule is engineered so that $i^2=(0,1)\cdot (0,1)=(-1,0)=-1$.

Definition. For $z=a+bi\in \mathbb{C}$: the real part $\mathrm{Re}(z)=a$, the imaginary part $\mathrm{Im}(z)=b$, the modulus $|z|=\sqrt{a^2+b^2}$, and the complex conjugate $\bar{z}=a-bi$.

The set $\mathbb{C}$ with these operations is a field. Additive identity is $0=0+0i$; multiplicative identity is $1=1+0i$. The multiplicative inverse of $z=a+bi\ne 0$ is:

$$z^{-1}=\frac{\bar{z}}{|z{|}^2}=\frac{a-bi}{a^2+b^2}=\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i.$$

The key identities are: $z\bar{z}=|z{|}^2$, $\overline{z+w}=\bar{z}+\bar{w}$, and $\overline{zw}=\bar{z}\bar{w}$.

Counterexamples to common slips

• $i$ is not "imaginary" in the colloquial sense. It is a well-defined algebraic object: the ordered pair $(0,1)$ satisfying $(0,1{)}^2=(-1,0)$.
• The imaginary part of $a+bi$ is $b$, not $bi$. The imaginary part is a real number.
• Complex numbers are not ordered. There is no relation $<$ on $\mathbb{C}$ compatible with the field operations, because $i^2=-1$ contradicts the requirement that squares be non-negative.

Key theorem with proof [Intermediate+]

Theorem (fundamental theorem of algebra, statement). Every non-constant polynomial with complex coefficients has at least one root in $\mathbb{C}$. Equivalently, every polynomial of degree $n\ge 1$ with complex coefficients factors as $a_n(z-r_1)(z-r_2)\cdots (z-r_n)$ for some $r_1,\dots ,r_n\in \mathbb{C}$.

The proof (due to Gauss, who gave four different proofs between 1799 and 1849) uses topological properties of $\mathbb{C}$ that are beyond this unit's scope. The algebraic content is that $\mathbb{C}$ is algebraically closed: the process of solving polynomial equations terminates inside $\mathbb{C}$.

Theorem (de Moivre's formula). If $z=r(\cos \theta +i\sin \theta )$, then for any integer $n$:

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

Proof. For $n=0$: $z^0=1=r^0(\cos 0+i\sin 0)$. For the inductive step, assume $z^n=r^n(\cos n\theta +i\sin n\theta )$. Then:

$$z^{n+1}=z^n\cdot z=r^n(\cos n\theta +i\sin n\theta )\cdot r(\cos \theta +i\sin \theta ).$$

Multiplying out:

$$=r^{n+1}[(\cos n\theta \cos \theta -\sin n\theta \sin \theta )+i(\cos n\theta \sin \theta +\sin n\theta \cos \theta )]$$

$$=r^{n+1}[\cos (n\theta +\theta )+i\sin (n\theta +\theta )]=r^{n+1}(\cos (n+1)\theta +i\sin (n+1)\theta )$$

by the angle-addition formulas. The case $n<0$ follows from $z^{-n}=1/z^n$. $\square$

Bridge. The foundational reason complex numbers are indispensable in mathematics is that they close the algebraic gap left by the reals: where $x^2+1=0$ has no solution in $\mathbb{R}$, the number $i$ provides one, and the fundamental theorem of algebra guarantees that no further extension is needed for polynomial equations. This is exactly the content that identifies $\mathbb{C}$ with the algebraic closure of $\mathbb{R}$. The central insight is that the two-dimensional geometry of the complex plane — modulus, argument, rotation — is the same structure as the algebra of $\mathbb{C}$, and de Moivre's formula puts these together by showing that raising to the $n$-th power is rotation by $n\theta$ and scaling by $r^n$. The bridge is between algebra and geometry: multiplication of complex numbers is geometrically a rotation and scaling, and this pattern builds toward 00.05.01 where the exponential function $e^x$ is reinterpreted as $e^{i\theta }=\cos \theta +i\sin \theta$ (the Euler identity), and appears again in 00.03.02 where the quadratic formula produces complex roots when the discriminant is negative.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Euler identity). For any real $\theta$:

$$e^{i\theta }=\cos \theta +i\sin \theta .$$

This is proved by substituting $z=i\theta$ into the power series $e^z=1+z+z^2/2!+z^3/3!+\cdots$ and separating real and imaginary parts: $e^{i\theta }=(1-{\theta }^2/2!+{\theta }^4/4!-\cdots )+i(\theta -{\theta }^3/3!+{\theta }^5/5!-\cdots )=\cos \theta +i\sin \theta$. The special case $\theta =\pi$ gives $e^{i\pi }=-1$, the celebrated equation linking $e$, $i$, $\pi$, $1$, and $0$.

Theorem 2 (polar form). Every nonzero complex number $z$ has a unique representation $z=r(\cos \theta +i\sin \theta )$ with $r=|z|>0$ and $\theta \in [0,2\pi )$. Equivalently, $z=r{e}^{i\theta }$.

The angle $\theta$ is the argument $\arg (z)$. Multiplication in polar form is $z_1{z}_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}$: moduli multiply, arguments add. This is the geometric content of complex multiplication — rotation by the sum of angles and scaling by the product of lengths.

Theorem 3 (roots of unity). The $n$-th roots of unity are $e^{2\pi ik/n}$ for $k=0,1,\dots ,n-1$. They form a regular $n$-gon inscribed in the unit circle.

The $n$-th roots of any nonzero complex number $w=\rho e^{i\varphi }$ are ${\rho }^{1/n}{e}^{i(\varphi +2\pi k)/n}$ for $k=0,1,\dots ,n-1$, evenly spaced on a circle of radius ${\rho }^{1/n}$.

Theorem 4 (conjugate root theorem). If $p(z)=a_n{z}^n+\cdots +a_0$ has real coefficients and $p(r)=0$, then $p(\bar{r})=0$. Non-real roots of real polynomials occur in conjugate pairs.

The proof uses $\overline{p(r)}=\overline{a_n{r}^n+\cdots +a_0}=a_n{\bar{r}}^n+\cdots +a_0=p(\bar{r})$ because the coefficients are real, so conjugation passes through.

Theorem 5 (geometric interpretation of inversion). The map $z\mapsto 1/z$ on $\mathbb{C}\setminus \{0\}$ is the composition of reflection in the unit circle ($z\mapsto z/|z{|}^2$) and complex conjugation ($z\mapsto \bar{z}$). Geometrically, inversion takes a point at distance $r$ from the origin to a point at distance $1/r$ along the same ray, then reflects across the real axis.

Theorem 6 (the complex numbers as a real algebra). The field $\mathbb{C}$ is a two-dimensional vector space over $\mathbb{R}$ with basis $\{1,i\}$. Multiplication by a fixed complex number $a+bi$ is a linear map ${\mathbb{R}}^2\to {\mathbb{R}}^2$ represented by the matrix $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$, which is a rotation through angle $\theta$ and scaling by $r=\sqrt{a^2+b^2}$.

Synthesis. The foundational reason complex numbers unify algebra and geometry is that the algebraic structure of $\mathbb{C}$ as a field coincides with the geometric structure of the plane as a space of rotations and scalings. The central insight is that multiplication by $a+bi$ is the linear map $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$, and this is exactly the rotation-by-$\theta$-and-scale-by-$r$ transformation where $r$ and $\theta$ are the modulus and argument. Putting these together with the Euler identity, the bridge is between the exponential function and the rotation group: $e^{i\theta }$ parametrises the unit circle, de Moivre's formula generalises to $n$-th powers as $n$-fold rotations, and the $n$-th roots of unity generalise into the cyclic group of order $n$. This pattern identifies $\mathbb{C}$ with the algebraic closure of $\mathbb{R}$ and identifies the unit circle with the group of rotations in the plane. The complex plane appears again in 00.05.01 through the exponential function, in 00.03.02 through the quadratic formula's discriminant, and throughout the complex-analysis strand beginning with 06.01.01.

Full proof set [Master]

Proposition 1 ($\mathbb{C}$ is a field). With the addition $(a,b)+(c,d)=(a+c,b+d)$ and multiplication $(a,b)(c,d)=(ac-bd,ad+bc)$, the set $\mathbb{C}=\{(a,b):a,b\in \mathbb{R}\}$ satisfies all field axioms.

Proof. Addition is commutative and associative because it is component-wise real addition. The additive identity is $(0,0)$ and additive inverses are $-(a,b)=(-a,-b)$. Multiplication is commutative because $(a,b)(c,d)=(ac-bd,ad+bc)=(ca-db,da+cb)=(c,d)(a,b)$. Associativity: $[(a,b)(c,d)](e,f)=(ac-bd,ad+bc)(e,f)=((ac-bd)e-(ad+bc)f,(ac-bd)f+(ad+bc)e)=(ace-bde-adf-bcf,acf-bdf+ade+bce)$. Direct computation of $(a,b)[(c,d)(e,f)]$ yields $(a,b)(ce-df,cf+de)=(a(ce-df)-b(cf+de),a(cf+de)+b(ce-df))=(ace-adf-bcf-bde,acf+ade+bce-bdf)$, which agrees. The multiplicative identity is $(1,0)$. For $(a,b)\ne (0,0)$, the inverse is $(a/(a^2+b^2),-b/(a^2+b^2))$ because $(a,b)\cdot (a/(a^2+b^2),-b/(a^2+b^2))=((a^2+b^2)/(a^2+b^2),(-ab+ba)/(a^2+b^2))=(1,0)$. Distributivity is verified by expanding both sides and comparing. $\square$

Proposition 2 (conjugate root theorem). If $p(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_0$ with $a_k\in \mathbb{R}$ for all $k$, and if $p(r)=0$ for some $r\in \mathbb{C}$, then $p(\bar{r})=0$.

Proof. For real $a$ and complex $z$: $\overline{az}=a\bar{z}$ because $a=\bar{a}$. By induction: $\overline{z^n}={\bar{z}}^n$. So $\overline{p(r)}=\overline{a_n{r}^n+\cdots +a_0}={\bar{a}}_n\overline{r^n}+\cdots +{\bar{a}}_0=a_n{\bar{r}}^n+\cdots +a_0=p(\bar{r})$. Since $p(r)=0$, $\overline{p(r)}=\bar{0}=0$, giving $p(\bar{r})=0$. $\square$

Proposition 3 (Euler identity from power series). The power series $e^z={\sum }_{k=0}^{\infty }{z}^k/k!$ converges for all $z\in \mathbb{C}$ and satisfies $e^{i\theta }=\cos \theta +i\sin \theta$.

Proof. For $z=i\theta$ the series is $1+i\theta +(i\theta {)}^2/2!+(i\theta {)}^3/3!+\cdots =1+i\theta -{\theta }^2/2!-i{\theta }^3/3!+{\theta }^4/4!+i{\theta }^5/5!-\cdots$. Separating real and imaginary parts: $\mathrm{Re}(e^{i\theta })=1-{\theta }^2/2!+{\theta }^4/4!-\cdots =\cos \theta$ and $\mathrm{Im}(e^{i\theta })=\theta -{\theta }^3/3!+{\theta }^5/5!-\cdots =\sin \theta$. Both series converge absolutely for all $\theta$, so the rearrangement is valid. $\square$

Connections [Master]

• Quadratic equations and the discriminant 00.03.02. When the discriminant $b^2-4ac$ of a quadratic equation $a{x}^2+bx+c=0$ is negative, the quadratic formula $x=(-b\pm \sqrt{b^2-4ac})/(2a)$ involves the square root of a negative number. Complex numbers make sense of this: $\sqrt{-16}=4i$, and the two roots $-1+2i$ and $-1-2i$ are complex conjugates (by Proposition 2). The quadratic formula's failure over the reals is the historical motivation for complex numbers.

• Real numbers and the completeness of $\mathbb{R}$ 00.01.01. The complex numbers $\mathbb{C}$ are built from ordered pairs of real numbers, inheriting the completeness of $\mathbb{R}$ through the identification $\mathbb{C}\cong {\mathbb{R}}^2$. The modulus $|z|=\sqrt{a^2+b^2}$ uses the real square root whose existence depends on completeness, and the convergence of the power series $e^z=\sum z^k/k!$ depends on completeness of $\mathbb{R}$ to guarantee absolute convergence.

• Real exponents and the exponential function 00.05.01. The Euler identity $e^{i\theta }=\cos \theta +i\sin \theta$ extends the real exponential function $e^x$ to complex arguments. Where the real exponential maps $\mathbb{R}\to {\mathbb{R}}_{>0}$ along the positive real axis, the complex exponential maps $\mathbb{C}\to \mathbb{C}\setminus \{0\}$, and its restriction to the imaginary axis traces out the unit circle. The exponential law $e^{z+w}=e^z{e}^w$ holds for complex $z,w$ exactly as it does for real exponents.

Historical & philosophical context [Master]

Bombelli 1572 L'Algebra ^{[Bombelli1572]} was the first text to treat square roots of negative numbers as legitimate arithmetic objects. Bombelli encountered expressions like $\sqrt{-1}$ while applying Cardano's cubic formula to the equation $x^3=15x+4$, and showed that the imaginary parts cancel to produce the real root $x=4$. Wessel 1797 Om directionens analytiske betegning (read before the Royal Danish Academy) introduced the geometric representation of complex numbers as points in the plane, defining addition and multiplication geometrically. Argand 1806 Essai sur une maniere de representer les quantites imaginaires dans les constructions geometriques independently rediscovered the same plane representation. Hamilton 1837 Theory of Conjugate Functions or Algebraic Couples (Trans. Royal Irish Academy 17) gave the formal definition of $\mathbb{C}$ as ordered pairs $(a,b)$ with the multiplication rule $(a,b)(c,d)=(ac-bd,ad+bc)$, removing all mystery by grounding complex arithmetic in real-pair arithmetic with no reference to "imaginary" quantities. Gauss 1831 Theoria residuorum biquadraticorum, Commentatio secunda (Comment. Soc. Regiae Sci. Gottingensis 7) popularised the complex plane and coined the term "complex number," establishing the geometric viewpoint as standard.

Bibliography [Master]

@book{Bombelli1572,
  author = {Bombelli, Rafael},
  title = {L'Algebra},
  publisher = {Giovanni Rossi},
  address = {Bologna},
  year = {1572}
}

@article{Wessel1797,
  author = {Wessel, Caspar},
  title = {Om directionens analytiske betegning},
  journal = {Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter},
  volume = {5},
  pages = {469--518},
  year = {1797}
}

@article{Argand1806,
  author = {Argand, Jean-Robert},
  title = {Essai sur une mani{\`e}re de repr{\'e}senter les quantit{\'e}s imaginaires dans les constructions g{\'e}om{\'e}triques},
  publisher = {privately printed},
  address = {Paris},
  year = {1806}
}

@article{Hamilton1837,
  author = {Hamilton, William Rowan},
  title = {Theory of Conjugate Functions, or Algebraic Couples},
  journal = {Transactions of the Royal Irish Academy},
  volume = {17},
  pages = {293--422},
  year = {1837}
}

@book{Lang1988,
  author = {Lang, Serge},
  title = {Basic Mathematics},
  publisher = {Springer},
  year = {1988}
}