Proof that arguments add under multiplication

Builds on Argument of a complex number Modulus of a complex number Modulus is multiplicative The complex numbers form a field Angle addition formulas Pythagorean trigonometric identity Sine and cosine Equality of complex numbers Floor function Sine is 2π-periodic Principal value of the argument

Proof

We multiply the two trigonometric forms and use the angle addition formulas.

Let $z \neq 0$ and $w \neq 0$ . Let $r = |z|$ and $s = |w|$ be their moduli, so $r > 0$ and $s > 0$ .
Let $\theta$ be an argument of $z$ and $\varphi$ an argument of $w$ . By definition, $z = r (\cos \theta + i \sin \theta)$ and $w = s (\cos \varphi + i \sin \varphi)$ .
Multiplication in $\C$ is commutative and associative, since $\C$ is a field. So $z w = r s \, (\cos \theta + i \sin \theta)(\cos \varphi + i \sin \varphi)$ .
Expanding with $i^2 = -1$ , the last product is $(\cos \theta \cos \varphi - \sin \theta \sin \varphi) + (\sin \theta \cos \varphi + \cos \theta \sin \varphi) i$ .
By the angle addition formulas, this equals $\cos(\theta + \varphi) + i \sin(\theta + \varphi)$ .
Since the modulus is multiplicative, $|z w| = r s > 0$ . In particular $z w \neq 0$ .
By steps 3 to 6, $z w = |z w| \bigl(\cos(\theta + \varphi) + i \sin(\theta + \varphi)\bigr)$ . So $\theta + \varphi$ is an argument of $z w$ .
It remains to show that each argument $\psi$ of $z w$ differs from $\chi = \theta + \varphi$ by a multiple of $2 \pi$ .
Both $\psi$ and $\chi$ are arguments of $z w$ . Divide both forms by the real number $|z w| > 0$ and compare parts. By equality of complex numbers, $\cos \psi = \cos \chi$ and $\sin \psi = \sin \chi$ .
The power series of sine and cosine give $\cos(-x) = \cos x$ and $\sin(-x) = -\sin x$ , since $(-x)^{2n} = x^{2n}$ and $(-x)^{2n+1} = -x^{2n+1}$ .
Let $\delta = \psi - \chi$ . By the angle addition formulas and step 10, $\cos \delta = \cos \psi \cos \chi + \sin \psi \sin \chi$ . By step 9 and the Pythagorean identity, this is $\cos^2 \chi + \sin^2 \chi = 1$ .
In the same way, $\sin \delta = \sin \psi \cos \chi - \cos \psi \sin \chi = \sin \chi \cos \chi - \cos \chi \sin \chi = 0$ .
Let $m = \floor{(\pi - \delta) / (2 \pi)}$ , using the floor function. Then $2 \pi m \le \pi - \delta < 2 \pi m + 2 \pi$ . So $\delta' = \delta + 2 \pi m$ satisfies $-\pi < \delta' \le \pi$ .
Since $\sin$ and $\cos$ are $2\pi$ -periodic, applied $\abs{m}$ times, $\cos \delta' = 1$ and $\sin \delta' = 0$ .
So $1 = 1 \cdot (\cos \delta' + i \sin \delta')$ , and $|1| = 1$ . Thus $\delta'$ is an argument of $1$ in $(-\pi, \pi]$ .
Also $0$ is an argument of $1$ in $(-\pi, \pi]$ , since $\cos 0 = 1$ and $\sin 0 = 0$ by the power series.
The principal value of the argument is unique, so $\delta' = 0$ . Hence $\psi = \theta + \varphi - 2 \pi m$ .
So $\theta + \varphi$ is an argument of $z w$ , and every argument of $z w$ is congruent to $\theta + \varphi$ modulo $2 \pi$ . This is $\arg(z w) \equiv \arg z + \arg w \pmod{2 \pi}$ .

Idea

Writing $z$ and $w$ in trigonometric form, their product has modulus $r s$ . The angle addition formulas turn the product of the two unit factors into the unit factor of the angle $\theta + \varphi$ . Two arguments of the same number always differ by a multiple of $2\pi$ , which gives the congruence.

Reading path

To understand this page from the ground up, read these 277 pages in order. The foundations come first. The path takes the proofs Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that cos pi = -1 and sin pi = 0, Proof of the equality of complex numbers, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the angle addition formulas by differentiation, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the continuity of the inverse of a strictly monotone continuous function, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof that one-sided limits of a monotone function exist, Proof that sine and cosine are 2π-periodic with smallest period 2π, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof of the existence and uniqueness of the principal value of the argument, Proof that a composition of continuous functions is continuous, Proof of the Pythagorean trigonometric identity by differentiation, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a function with zero derivative on an open interval is constant, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof of the constant multiple rule.

Binomial coefficient definition
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Even and odd functions definition
Factorial definition
Group concept
Imaginary unit object
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Periodic function definition
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Strictly monotone function definition
Proof of Pascal's rule from the recursive definition proof
Real and imaginary parts definition
Set definition
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Pascal's rule statement
Complex conjugate definition
Ordered pair definition
Commutative ring definition
Unit of a ring definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Addition of natural numbers definition
Biconditional definition
Truth table concept
Modus ponens concept
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
Field definition
Continuous function definition
Differentiable function definition
One-sided derivative definition
Subset definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Mathematical proof concept
Order on the natural numbers definition
Propositional logic concept
The rationals form an ordered field statement
Uniqueness quantifier definition
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Partial sum definition
Power series definition
Equality of complex numbers statement
Factorial formula for binomial coefficients statement
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Order on the reals definition
Commutativity of addition statement
Tautology definition
Proof of strong induction from ordinary induction proof
Sum of a finite geometric series statement
Uniform convergence definition
Radius of convergence definition
Convergent series definition
Sine and cosine definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Proof that the complex numbers form a field proof
Function definition
Logical equivalence definition
Uniform convergence on a set definition
Strong induction statement
Sum of a series definition
Series of functions definition
Absolute convergence definition
Binomial theorem statement
Preorder definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
The complex numbers form a field statement
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Monotone function definition
Epsilon-delta definition of a limit statement
Double negation law statement
Well-ordering principle statement
Total order definition
Composition of continuous functions is continuous statement
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that one-sided limits of a monotone function exist proof
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that convergent sequences are Cauchy proof
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Argument of a complex number definition
Proof that a complex number times its conjugate is the square of its modulus proof
Differentiable functions are continuous statement
Proof of the continuity of the inverse of a strictly monotone continuous function proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Floor function definition
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Squeeze theorem statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
One-sided limits of a monotone function exist statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Proof that the modulus is multiplicative proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Continuity of the inverse of a strictly monotone continuous function statement
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Modulus is multiplicative statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof of the ratio test proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product rule statement
Linearity of differentiation statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the constant multiple rule proof
Proof of the power rule by induction proof
Boundedness theorem statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Proof of Rolle's theorem proof
Absolutely convergent series converge statement
Rolle's theorem statement
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof that a function with positive derivative is strictly increasing proof
A function with zero derivative is constant statement
Uniform convergence of derivatives allows term-by-term differentiation statement
A function with positive derivative is strictly increasing statement
Proof of term-by-term differentiation proof
Term-by-term differentiation statement
Proof that power series are differentiable inside the radius of convergence proof
Power series are differentiable inside the radius of convergence statement
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
Derivative of cosine statement
Derivative of sine statement
Proof of the angle addition formulas by differentiation proof
Proof of the Pythagorean trigonometric identity by differentiation proof
pi object
Pythagorean trigonometric identity statement
Angle addition formulas statement
Arccosine definition
Proof that cos pi = -1 and sin pi = 0 proof
cos pi = -1 statement
Proof that sine and cosine are 2π-periodic with smallest period 2π proof
Sine is 2π-periodic statement
Proof of the existence and uniqueness of the principal value of the argument proof
Principal value of the argument statement
Proof that arguments add under multiplication proof

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Arguments add under multiplication

Tag	`S3PV26SY4Z`
URI	`m2://S3PV26SY4Z`
Kind	proof
Writing	W0 stub
MSC	30A99
Proves	Arguments add under multiplication
Uses	Argument of a complex number, Modulus of a complex number, Modulus is multiplicative, The complex numbers form a field, Angle addition formulas, Pythagorean trigonometric identity, Sine and cosine, Equality of complex numbers, Floor function, Sine is 2π-periodic, Principal value of the argument
Changed	4 November 2026

Proof that arguments add under multiplication

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