Argument identities are congruences modulo $2\pi$. When $\arg$ denotes the principal argume

Question

Argument identities are congruences modulo $2\pi$. When $\arg$ denotes the principal argument, normalise the resulting angle to $( -\pi,\pi ]$.

Operation	Modulus	Argument modulo 2π
Conjugate	$\|z^*\|=\|z\|$	$\arg(z^*)\equiv-\arg z$
Negation	$\|-z\|=\|z\|$	$\arg(-z)\equiv\arg z+\pi$, $z\ne0$
Product	$\|z_1z_2\|=\|z_1\|\|z_2\|$	$\arg(z_1z_2)\equiv\arg z_1+\arg z_2$, $z_1z_2\ne0$
Quotient	$\left\|\frac{z_1}{z_2}\right\|=\frac{\|z_1\|}{\|z_2\|}$	$\arg\!\left(\frac{z_1}{z_2}\right)\equiv\arg z_1-\arg z_2$, $z_1z_2\ne0$
Integer power	$\|z^n\|=\|z\|^n$	$\arg(z^n)\equiv n\arg z$, where the expression is defined

General arguments and the nature of a number

The following are families of general arguments, not principal arguments. In every row, $k\in\mathbb Z$.

Nature	General argument θ
Purely imaginary, nonzero	$\theta=\frac\pi2+k\pi$
Real, nonzero	$\theta=k\pi$
Positive real	$\theta=2k\pi$
Negative real	$\theta=\pi+2k\pi$
Positive imaginary	$\theta=\frac\pi2+2k\pi$
Negative imaginary	$\theta=-\frac\pi2+2k\pi$

Check that the complex number is nonzero before taking an argument.
Apply the algebraic argument law modulo 2π.
If a principal argument is requested, add or subtract multiples of 2π until the result lies in (−π,π].

Enrichment — outside the H2 syllabus. This source material supports understanding of modulus and argument; it is not presented as required examination content.

Trigonometric form

\[z=x+yi=r(\cos\theta+i\sin\theta),\qquad x=r\cos\theta,\quad y=r\sin\theta.\]

Euler’s formula and exponential form

\[e^{i\theta}=\cos\theta+i\sin\theta,\qquad z=re^{i\theta}.\]

Form	Expression
Cartesian	$z=x+yi$
Trigonometric	$z=r(\cos\theta+i\sin\theta)$
Exponential	$z=re^{i\theta}$

Here $r=|z|\ge0$. For $z\ne0$, a principal representation takes $\theta=\arg z\in(-\pi,\pi]$; unrestricted arguments differ by integer multiples of $2\pi$.

Accepted Answer

$\text{Conjugate: }|z^*|=|z|,\ \arg(z^*)\equiv-\arg z\pmod{2\pi}$ $\text{Product: }|z_1z_2|=|z_1||z_2|,\ \arg(z_1z_2)\equiv\arg z_1+\arg z_2\pmod{2\pi}$ $\text{Quotient: }\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|},\ \arg\left(\dfrac{z_1}{z_2}\right)\equiv\arg z_1-\arg z_2\pmod{2\pi}$ $\text{Power: }|z^n|=|z|^n,\ \arg(z^n)\equiv n\arg z\pmod{2\pi}$

Operation	Modulus	Argument modulo 2π
Conjugate	\(\|z^*\|=\|z\|\)	\(\arg(z^*)\equiv-\arg z\)
Negation	\(\|-z\|=\|z\|\)	\(\arg(-z)\equiv\arg z+\pi\), \(z\ne0\)
Product	\(\|z_1z_2\|=\|z_1\|\|z_2\|\)	\(\arg(z_1z_2)\equiv\arg z_1+\arg z_2\), \(z_1z_2\ne0\)
Quotient	\(\left\|\frac{z_1}{z_2}\right\|=\frac{\|z_1\|}{\|z_2\|}\)	\(\arg\!\left(\frac{z_1}{z_2}\right)\equiv\arg z_1-\arg z_2\), \(z_1z_2\ne0\)
Integer power	\(\|z^n\|=\|z\|^n\)	\(\arg(z^n)\equiv n\arg z\), where the expression is defined

Modulus, Argument and Complex-Number Operations

Modulus, Argument and Complex-Number Operations

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Nature	General argument θ
Purely imaginary, nonzero	\(\theta=\frac\pi2+k\pi\)
Real, nonzero	\(\theta=k\pi\)
Positive real	\(\theta=2k\pi\)
Negative real	\(\theta=\pi+2k\pi\)
Positive imaginary	\(\theta=\frac\pi2+2k\pi\)
Negative imaginary	\(\theta=-\frac\pi2+2k\pi\)

Form	Expression
Cartesian	\(z=x+yi\)
Trigonometric	\(z=r(\cos\theta+i\sin\theta)\)
Exponential	\(z=re^{i\theta}\)