Complex numbers — rectangular and polar form

Intuition Beginner

Some equations have no answer on the ordinary number line. The equation $x^2+1=0$ asks for a number whose square is $-1$. No real number works, because any real number squared is zero or positive. The real line is stuck.

The fix is to add one new number, called $i$, defined by the rule $i^2=-1$. With this single symbol added, and the usual rules of arithmetic kept, the complex numbers are born. Every complex number looks like $a+bi$, where $a$ and $b$ are real. The real part is $a$, and the imaginary part is $b$.

These numbers do not live on a line. They live in a plane. The number $a+bi$ is drawn as the point $(a,b)$: walk $a$ steps horizontally and $b$ steps vertically. The real numbers sit along the horizontal axis, and the numbers $bi$ sit along the vertical axis.

Why bother? Because closing this gap lets every polynomial equation have a solution, and because the geometry of this plane turns out to describe rotations, waves, and electrical circuits. The number $i$ is not a puzzle — it is a tool that makes algebra and geometry line up.

Visual Beginner

The complex plane has a horizontal axis for the real part and a vertical axis for the imaginary part. Each complex number $a+bi$ is one point $(a,b)$. The distance from the origin to $a+bi$ is $\sqrt{a^2+b^2}$, called the modulus. The angle from the positive real axis to that point is the argument, written $\theta$.

Every complex number carries two descriptions. The rectangular form $a+bi$ tells you where the point sits using a grid. The polar form tells you the same location using a distance $r$ and a direction $\theta$: walk out a distance $r$ along the angle $\theta$. Addition lives most naturally in rectangular form; multiplication lives most naturally in polar form.

Worked example Beginner

Multiply $(1+i)$ by $(1-i)$.

Use the distributive rule, the same one used for ordinary algebra:

$$(1+i)(1-i)=1\cdot 1+1\cdot (-i)+i\cdot 1+i\cdot (-i).$$

The last term is $i\cdot (-i)=-i^2$. Since $i^2=-1$, this becomes $-(-1)=1$. So the whole expression is $1-i+i+1$. The $-i$ and $+i$ cancel, leaving

$$(1+i)(1-i)=2.$$

Notice the answer is a pure real number. Multiplying a complex number by its mirror image $1-i$ (called its conjugate) always produces a real number equal to the square of the modulus: here $|1+i{|}^2=(\sqrt{2}{)}^2=2$.

Now convert $1+i$ to polar form. The modulus is $r=\sqrt{1^2+1^2}=\sqrt{2}$. The argument is the angle whose tangent is $1/1=1$, which is $45^{\circ }$ (or $\pi /4$ radians). So

$$1+i=\sqrt{2}(\cos 45^{\circ }+i\sin 45^{\circ }).$$

Two numbers that looked unrelated — a grid point and a distance-plus-angle — are the same point described two ways.

Check your understanding Beginner

Formal definition Intermediate+

Definition (complex numbers, rectangular form). A complex number is an expression $z=a+bi$ where $a,b\in \mathbb{R}$ and $i$ is a symbol satisfying $i^2=-1$. The set of all complex numbers is denoted $\mathbb{C}$. The real part is $\mathrm{Re}(z)=a$, and the imaginary part is $\mathrm{Im}(z)=b$. Two complex numbers are equal precisely when their real parts agree and their imaginary parts agree.

Equivalently, $\mathbb{C}$ is the set of ordered pairs $(a,b)$ with componentwise addition and the multiplication rule

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

Setting $i=(0,1)$ recovers $i^2=(0,1)(0,1)=(-1,0)=-1$. This ordered-pair model, made public by Gauss in 1831 ^{[Gauss 1831]}, shows that complex numbers need no metaphysical leap: they are pairs of real numbers with a particular multiplication.

Arithmetic. For $z=a+bi$ and $w=c+di$:

• Addition: $z+w=(a+c)+(b+d)i$.
• Subtraction: $z-w=(a-c)+(b-d)i$.
• Multiplication: $zw=(ac-bd)+(ad+bc)i$.
• Conjugate: $\bar{z}=a-bi$.
• Modulus: $|z|=\sqrt{a^2+b^2}$, so $z\bar{z}=a^2+b^2=|z{|}^2$.

Division uses the conjugate to clear the denominator:

$$\frac{z}{w}=\frac{z\bar{w}}{w\bar{w}}=\frac{z\bar{w}}{|w{|}^2}.$$

For example, $(1+2i)/(1-i)=\frac{(1+2i)(1+i)}{|1-i{|}^2}=\frac{-1+3i}{2}=-\frac{1}{2}+\frac{3}{2}i$.

Definition (complex plane). Identifying $z=a+bi$ with the point $(a,b)\in {\mathbb{R}}^2$ gives the complex plane (the Argand diagram). Addition of complex numbers is vector addition: $(a,b)+(c,d)=(a+c,b+d)$. Geometrically, two complex numbers add by the parallelogram rule.

Definition (polar form). The modulus of $z=a+bi$ is $r=|z|=\sqrt{a^2+b^2}$, and the argument $\arg (z)=\theta$ is the angle from the positive real axis to the point $(a,b)$, measured counter-clockwise, with $\tan \theta =b/a$. Then

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

This is the polar form. The angle $\theta$ is determined up to addition of $2\pi$, since sine and cosine have period $2\pi$ (a fact inherited from the unit circle of 00.07.01).

Definition (exponential form). Defining $e^{i\theta }=\cos \theta +i\sin \theta$ (Euler's formula, justified from power series in 00.05.03), every complex number admits the compact form

$$z=r{e}^{i\theta }.$$

The modulus $r$ is the distance from the origin; the argument $\theta$ is the direction. Numbers with $r=1$ lie on the unit circle and have the form $e^{i\theta }$.

Why polar form: multiplication is angle addition

The decisive reason polar form matters is that multiplication becomes geometric. If $z_1=r_1{e}^{i{\theta }_1}$ and $z_2=r_2{e}^{i{\theta }_2}$, then

$$z_1{z}_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}.$$

Multiply the moduli, add the arguments. Multiplying by $e^{i\theta }$ rotates the whole plane by angle $\theta$ and leaves lengths unchanged; multiplying by a general $z$ rotates by $\arg (z)$ and scales by $|z|$. Division subtracts arguments: $z_1/z_2=(r_1/r_2)e^{i({\theta }_1-{\theta }_2)}$.

Counterexamples to common slips

• $\arg (z)$ is not unique. The argument is defined only up to $2\pi$: $e^{i\theta }=e^{i(\theta +2\pi )}$. A single-valued principal argument is usually fixed in $(-\pi ,\pi ]$.
• Multiplication is not componentwise. In rectangular form, $(a,b)(c,d)\ne (ac,bd)$; the correct rule is $(ac-bd,ad+bc)$. Componentwise multiplication does not respect $i^2=-1$.
• $|z_1{z}_2|=|z_1||z_2|$ holds, but $|z_1+z_2|\ne |z_1|+|z_2|$ in general. The triangle inequality $|z_1+z_2|\le |z_1|+|z_2|$ is the correct relation; equality holds only when $z_1$ and $z_2$ point in the same direction.

Key theorem with proof Intermediate+

Theorem (De Moivre's theorem). For every integer $n$ and every real $\theta$,

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta ).$$

Equivalently, $(e^{i\theta }{)}^n=e^{in\theta }$.

Proof. We induct on $n$ for positive integers. The base case $n=1$ reads $(\cos \theta +i\sin \theta {)}^1=\cos \theta +i\sin \theta$, which matches $\cos (1\cdot \theta )+i\sin (1\cdot \theta )$, so the formula holds.

For the inductive step, assume the formula holds for some $n\ge 1$:

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta ).$$

Multiply both sides by $\cos \theta +i\sin \theta$:

$$(\cos \theta +i\sin \theta {)}^{n+1}=(\cos (n\theta )+i\sin (n\theta )\left)\right(\cos \theta +i\sin \theta ).$$

Expand the right side, using $i^2=-1$:

$$=[\cos (n\theta )\cos \theta -\sin (n\theta )\sin \theta ]+i[\sin (n\theta )\cos \theta +\cos (n\theta )\sin \theta ].$$

The addition formulas of 00.08.01 — $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$ and $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$ — with $\alpha =n\theta$ and $\beta =\theta$ collapse the brackets:

$$=\cos ((n+1)\theta )+i\sin ((n+1)\theta ).$$

That is exactly the formula with $n$ replaced by $n+1$. By induction the identity holds for every positive integer $n$.

For $n=0$, both sides equal $1$. For negative integers, write $n=-m$ with $m>0$ and use that the conjugate of $e^{i\theta }$ is $e^{-i\theta }$: $(e^{i\theta }{)}^{-m}=(e^{-i\theta }{)}^m=e^{-im\theta }=e^{in\theta }$. So the formula holds for all integers $n$. $\square$

Corollary (powers in polar form). If $z=r(\cos \theta +i\sin \theta )$, then for any integer $n$,

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

To raise a complex number to the $n$th power, raise the modulus to the $n$th power and multiply the argument by $n$. A worked instance: $1+i=\sqrt{2}{e}^{i\pi /4}$, so $(1+i{)}^8=(\sqrt{2}{)}^8{e}^{i\cdot 8\pi /4}=2^4{e}^{i2\pi }=16\cdot 1=16$.

Bridge. The foundational reason De Moivre's theorem is true is that complex multiplication adds angles, and this is exactly the content of the trigonometric addition formulas of 00.08.01 rewritten in exponential form. The central insight is that polar form converts exponentiation into a single scaling and rotation: $r$ becomes $r^n$ and $\theta$ becomes $n\theta$, so the pattern is dual to the angle-addition identities and generalises from integer $n$ to rational and real $n$, which produces the $n$th roots below. Putting these together, the bridge is between the algebra of powers and the geometry of rotations: exponentiation in $\mathbb{C}$ is repeated rotation. This builds toward the $n$th roots of unity forming regular polygons, and appears again in 00.05.03 where Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$ is derived from power series, in 00.07.01 where the modulus-1 complex numbers are exactly the points of the unit circle, and in the analysis strand 02.01.01 where the same exponential law underpins Fourier decomposition and the complex-valued functions that model waves.

Exercises Intermediate+

Advanced results Master

Theorem 1 (Euler's identity). $e^{i\pi }+1=0$. Setting $\theta =\pi$ in Euler's formula gives $e^{i\pi }=\cos \pi +i\sin \pi =-1$. This single equation links the five central constants $e,i,\pi ,1,0$.

Theorem 2 (roots of a complex number). For any non-zero $w=R{e}^{i\varphi }$ and any positive integer $n$, the equation $z^n=w$ has exactly $n$ distinct solutions,

$$z_k=R^{1/n}{e}^{i(\varphi +2\pi k)/n},k=0,1,\dots ,n-1.$$

Each has modulus $R^{1/n}$, and their arguments are equally spaced by $2\pi /n$. Geometrically they form the vertices of a regular $n$-gon centred at the origin. This is De Moivre's theorem inverted: roots are powers with rational exponent.

Theorem 3 (roots of unity and the cyclic group). The $n$th roots of unity ${\omega }_k=e^{2\pi ik/n}$ form a cyclic group of order $n$ under multiplication, isomorphic to $\mathbb{Z}/n\mathbb{Z}$ via $k\mapsto {\omega }_k$. Their sum vanishes:

$$\sum _{k=0}^{n-1}{\omega }_k=0(n\ge 2).$$

This is the algebraic echo of the geometric fact that the vertices of a regular $n$-gon balance at the origin.

Theorem 4 (fundamental theorem of algebra). Every non-constant polynomial with complex coefficients has at least one complex root. Equivalently, a degree-$n$ polynomial in $\mathbb{C}[x]$ factors completely as $c{\prod }_{j=1}^n(x-r_j)$. Gauss gave the first accepted proof in his 1799 doctoral thesis. The proof uses the topology of the plane (or, in modern proofs, Liouville's theorem that a bounded entire function is constant) and is beyond precalc; what matters here is that complex numbers are algebraically closed, which is the precise sense in which they complete the number system.

Theorem 5 (conjugate-root theorem). If a polynomial has real coefficients and $z$ is one of its roots, then $\bar{z}$ is also a root. Real polynomials therefore factor into real linear factors and real irreducible quadratic factors $(x^2+bx+c)$ with negative discriminant — exactly the quadratic pairs that first forced complex numbers into existence.

Theorem 6 (multiplication is a similarity of the plane). Multiplication by a fixed $w=R{e}^{i\varphi }$ is the map $z\mapsto wz$, which rotates every point about the origin by angle $\varphi$ and scales every distance by factor $R$. It preserves angles between curves, so it is a conformal map. This is the geometric fact that makes complex multiplication indispensable for describing rotations in two dimensions and is the gateway to complex analysis.

Theorem 7 (preview: complex functions and analysis). A function $f:\mathbb{C}\to \mathbb{C}$ that is complex-differentiable is called holomorphic. Holomorphic functions are exactly those that preserve angles locally (conformal), and they admit power-series representations $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ around every point of their domain. The integral of a holomorphic function around a closed loop depends only on the loop's winding around singularities (Cauchy's theorem). This is the entry point to the analysis strand and to the wave-based models of quantum mechanics.

Synthesis. The foundational reason complex numbers unify algebra, geometry, and analysis is that the single equation $i^2=-1$ turns the number line into a plane on which multiplication is rotation plus scaling, and this is exactly the content encoded by Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$. The central insight is that polar form makes exponentiation into repeated rotation, so the angle-addition identities of 00.08.01 become the exponential law $e^{i(\alpha +\beta )}=e^{i\alpha }{e}^{i\beta }$, and the bridge is between the algebraic operation of raising powers and the geometric operation of rotating the plane. Putting these together, De Moivre's theorem is dual to the angle-addition formulas, it generalises from integer exponents to rational exponents (producing the regular $n$-gon of roots of unity), and it generalises further to real and complex exponents in the holomorphic functions of analysis. The pattern is that $\mathbb{C}$ is algebraically closed (the fundamental theorem of algebra), conformally rich (multiplication is a similarity), and analytically rigid (holomorphic functions admit power series), and these three faces — algebraic, geometric, analytic — are three descriptions of the same object.

Full proof set Master

Proposition 1 (De Moivre's theorem for all integers). For every $n\in \mathbb{Z}$ and every $\theta \in \mathbb{R}$, $(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$.

Proof. The case $n\ge 1$ is proved by induction in the Key theorem above. For $n=0$ both sides equal $1$. For $n<0$, write $n=-m$ with $m\ge 1$. Since $e^{-i\theta }=\cos \theta -i\sin \theta =\overline{e^{i\theta }}$ is also $e^{i(-\theta )}$, and $(e^{i\theta }{)}^{-1}=e^{-i\theta }$, we have $(e^{i\theta }{)}^n=(e^{i\theta }{)}^{-m}=((e^{i\theta }{)}^{-1}{)}^m=(e^{-i\theta }{)}^m=e^{-im\theta }=e^{in\theta }$. Expanding the first and last exponentials gives the stated identity. $\square$

Proposition 2 (the $n$th roots of unity sum to zero). For $n\ge 2$, $\sum _{k=0}^{n-1}{e}^{2\pi ik/n}=0$.

Proof. Let $\omega =e^{2\pi i/n}$. The sum is the geometric series $1+\omega +{\omega }^2+\cdots +{\omega }^{n-1}$. Since $\omega \ne 1$ (because $n\ge 2$ gives $2\pi /n\in (0,\pi ]$), the finite geometric-sum formula gives

$$\sum _{k=0}^{n-1}{\omega }^k=\frac{{\omega }^n-1}{\omega -1}.$$

But ${\omega }^n=(e^{2\pi i/n}{)}^n=e^{2\pi i}=1$, so the numerator is $1-1=0$. The denominator is $\omega -1\ne 0$, so the quotient is $0$. $\square$

Proposition 3 (multiplication is multiplicative on the modulus and additive on the argument). If $z_1=r_1{e}^{i{\theta }_1}$ and $z_2=r_2{e}^{i{\theta }_2}$ with $r_1,r_2\ge 0$, then $z_1{z}_2=(r_1{r}_2)e^{i({\theta }_1+{\theta }_2)}$.

Proof. Using the exponential law — which here is the angle-addition formulas in disguise — $z_1{z}_2=r_1{r}_2{e}^{i{\theta }_1}{e}^{i{\theta }_2}=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}$. To verify $e^{i{\theta }_1}{e}^{i{\theta }_2}=e^{i({\theta }_1+{\theta }_2)}$ directly from the definition $e^{i\theta }=\cos \theta +i\sin \theta$:

$$(\cos {\theta }_1+i\sin {\theta }_1)(\cos {\theta }_2+i\sin {\theta }_2)=(\cos {\theta }_1\cos {\theta }_2-\sin {\theta }_1\sin {\theta }_2)+i(\sin {\theta }_1\cos {\theta }_2+\cos {\theta }_1\sin {\theta }_2),$$

which equals $\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2)$ by the addition formulas of 00.08.01. $\square$

Connections Master

• Unit-circle trigonometry 00.07.01. The polar form $z=r(\cos \theta +i\sin \theta )$ borrows its coordinates directly from the unit circle: the modulus-1 complex numbers $e^{i\theta }$ are exactly the points $(\cos \theta ,\sin \theta )$ on the unit circle of 00.07.01, and the argument $\theta$ is the same angle. The Pythagorean identity ${\cos }^2\theta +{\sin }^2\theta =1$ becomes the modulus statement $|e^{i\theta }|=1$.

• Trigonometric addition formulas 00.08.01. De Moivre's theorem is the angle-addition identities of 00.08.01 in disguise. The inductive proof of $(\cos \theta +i\sin \theta {)}^n$ runs on exactly the two identities $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$ and $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$, which is why multiplying two complex numbers adds their arguments.

• Complex numbers (introductory) 00.05.03. The lighter unit 00.05.03 introduces $i$, the complex plane, and rectangular arithmetic; the present unit adds polar form, the exponential form, De Moivre's theorem, and the roots of unity. Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$, derived from power series in 00.05.03, is the bridge that links the rectangular and polar descriptions.

• Real and complex analysis 02.01.01. This unit is the on-ramp the audit flagged for the analysis strand. The exponential law $e^{i(\alpha +\beta )}=e^{i\alpha }{e}^{i\beta }$ is the engine behind the Fourier decomposition of periodic functions, the power series for holomorphic functions, and contour integration. Every result about waves, heat conduction, and signal processing descends from the geometry of the complex plane built here.

• Quantum mechanics 12.01.01. Quantum states live in a complex Hilbert space: wavefunctions are complex-valued, amplitudes combine by complex addition, and time evolution multiplies each state by a phase $e^{-iEt/\hslash }$. The polar form introduced here is the literal algebra of quantum phases, and the roots of unity describe the discrete symmetries that organise particle spectra.

Historical & philosophical context Master

Complex numbers entered mathematics through the cubic, not the quadratic. Cardano's 1545 Ars Magna ^{[Cardano 1545]} gave the formula for the roots of $x^3=px+q$, and applying it to $x^3=15x+4$ produces the intermediate expression $\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$, in which a square root of a negative number appears even though the true root $x=4$ is perfectly real. This was the first sign that the "impossible" objects do real work. Cardano himself was uneasy; he called such quantities sophistic and used them only formally.

Bombelli's 1572 L'Algebra ^{[Bombelli 1572]} took the decisive step of treating $\sqrt{-1}$ as an algebraic object obeying the ordinary rules, working out its arithmetic, and showing that the two cube roots above combine to give the real answer $4$. Bombelli's rules for adding, subtracting, multiplying, and dividing these objects are exactly the rules in the Formal definition section. The philosophy was pragmatic: the objects computed correct answers, so they deserved arithmetic, regardless of whether they "existed."

Euler's 1748 Introductio in analysin infinitorum ^{[Euler 1748]} unified the field by proving $e^{i\theta }=\cos \theta +i\sin \theta$ from the power series for the exponential, sine, and cosine, and by writing down $e^{i\pi }+1=0$. This gave complex numbers an analytic footing independent of cubics and linked them permanently to rotation and waves.

The geometric representation came late. Wessel in 1797 (published 1799) ^{[Wessel 1799]} and Argand in 1806 ^{[Argand 1806]} independently described the complex plane, showing that $a+bi$ is a point and that multiplication adds angles. The picture was not widely accepted until Gauss, in his 1831 Theoria residuorum biquadraticorum ^{[Gauss 1831]}, publicly identified complex numbers with ordered pairs of reals and used the plane systematically. Once the geometry was in place, the earlier algebraic and analytic discoveries fused into a single subject, and the road opened to Riemann's complex analysis and to the twentieth-century use of complex Hilbert spaces in quantum mechanics.

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