Polar and exponential forms of complex numbers

Cartesian form $x+yi$ displays horizontal and vertical components. Polar form $r(\cos\theta+i\sin\theta)$ displays size and direction.

For a non-zero complex number, $r>0$. Arguments that differ by a whole turn represent the same direction, so $\theta+2k\pi$ gives the same complex number for every integer $k$.

Polar form is especially efficient for multiplication and division because moduli scale while arguments rotate. Cartesian form remains efficient for addition and subtraction.

Always state whether an angle is a principal argument, a particular argument, or the general family of arguments.

Cis notation abbreviates the trigonometric factor: $\operatorname{cis}\theta=\cos\theta+i\sin\theta$. Therefore $z=r\operatorname{cis}\theta$.

The notation is compact but carries exactly the same information as full polar form. It does not mean the product of three variables c, i, and s.

Because sine and cosine are both $2\pi$-periodic, $\operatorname{cis}(\theta+2k\pi)=\operatorname{cis}\theta$.

On the unit circle, $\operatorname{cis}\theta$ has modulus 1 and argument $\theta$. Multiplication by r scales this unit vector to modulus r.

Euler's relation connects exponential and trigonometric notation: $e^{i\theta}=\cos\theta+i\sin\theta$. Hence $\operatorname{cis}\theta=e^{i\theta}$.

The exponential form of a non-zero complex number is $z=re^{i\theta}$, where $r=|z|>0$ and θ is an argument.

The familiar exponential index laws explain polar multiplication and division: multiplying adds arguments, while dividing subtracts them.

Exponential form is a representation, not a numerical approximation. Keep angles exact in terms of $\pi$ when exact values are available.

A non-zero complex number has infinitely many arguments because adding $2\pi$ completes one full turn. The multivalued notation $\arg z$ can describe this entire family.

The principal argument $\operatorname{Arg}z$ is the unique representative in a specified interval. In these notes the convention is $-\pi<\operatorname{Arg}z\le\pi$.

Under this convention, a negative real number has principal argument $\pi$, not $-\pi$. A negative imaginary number has principal argument $-\frac{\pi}{2}$.

Some calculators or texts use $0\le\theta<2\pi$. If so, convert the result to the convention requested by the question.

Start with $z=x+yi$. First calculate $r=\sqrt{x^2+y^2}$, then determine a quadrant-correct argument θ.

A reliable manual method is: sketch the point, find a positive reference angle from $\tan^{-1}(\frac{|y}{x|})$, then adjust it for the quadrant.

Once r and θ are known, the three equivalent outputs are $r(\cos\theta+i\sin\theta)$, $r\operatorname{cis}\theta$, and $re^{i\theta}$.

For points on an axis, read the direction directly rather than dividing by zero. For example, $-3i$ has modulus 3 and principal argument $-\frac{\pi}{2}$.

To convert from polar form, evaluate $x=r\cos\theta$ and $y=r\sin\theta$, then write $z=x+yi$.

Use exact unit-circle values whenever the angle is a standard angle. Do not replace $\frac{\sqrt3}{2}$ by a rounded decimal unless approximation is requested.

For a non-standard angle, use a calculator in radian mode and keep extra digits during the calculation. Round only the final real and imaginary parts as instructed.

Negative sine or cosine values automatically place the Cartesian result in the correct quadrant.

If $z_1=r_1\operatorname{cis}\theta_1$ and $z_2=r_2\operatorname{cis}\theta_2$, then their product has modulus $r_1r_2$ and argument $\theta_1+\theta_2$.

Geometrically, multiplication by $r_2\operatorname{cis}\theta_2$ scales every distance from the origin by $r_2$ and rotates every direction by $\theta_2$.

After adding arguments, an answer may be normalized to the requested principal interval by adding or subtracting $2\pi$.

The same rule follows immediately from exponential form: $r_1e^{i\theta_1}r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}$.

Polar form makes integer powers systematic. If $z=r\operatorname{cis}\theta$, then each multiplication multiplies the modulus by $r$ and adds another copy of the argument $\theta$. Therefore $z^n=r^n\operatorname{cis}(n\theta)$ for every integer $n$, with $z\ne0$ when $n<0$.

The same rule in exponential form is $(re^{i\theta})^n=r^n e^{in\theta}$. This follows directly from the ordinary index laws and is a form of De Moivre's theorem.

After multiplying the argument by $n$, normalize it to the requested interval only at the end. For an exact Cartesian answer, use exact sine and cosine values after the power has been simplified.

Geometrically, raising to the nth power raises the distance from the origin to the nth power and multiplies the direction angle by $n$. This gives a fast reasonableness check for both magnitude and quadrant.

For $z_2\ne0$, division divides the moduli and subtracts the arguments. This is usually shorter than rationalizing a complex denominator in Cartesian form.

The reciprocal of $r\operatorname{cis}\theta$ is $\frac1r\operatorname{cis}(-\theta)$: the scale is inverted and the rotation is reversed.

The conjugate of $r\operatorname{cis}\theta$ is $r\operatorname{cis}(-\theta)$. Conjugation preserves modulus and reflects the direction in the real axis.

Normalize the final argument only after subtracting. Preserve exact angles when possible.

Multiplying z by a positive real number k stretches its position vector by factor k without changing its direction. Multiplying by a unit complex number $\operatorname{cis}\phi$ rotates it by $\phi$ without changing its length.

Multiplication by $k\operatorname{cis}\phi$ combines both transformations: a dilation of scale factor k centered at the origin and a rotation of angle $\phi$.

Division by $k\operatorname{cis}\phi$ scales by $\frac{1}{k}$ and rotates by $-\phi$.

This interpretation is a strong reasonableness check. A product modulus or direction inconsistent with the predicted transformation signals an arithmetic error.

Polar complex numbers are useful when a quantity has both magnitude and phase. In alternating-current models, a voltage, current, or impedance can be represented as a phasor $re^{i\theta}$. The modulus gives amplitude and the argument gives phase angle.

Multiplication and division then describe physical scaling and phase shift. For example, the phasor model $I=\frac{V}{Z}$ divides voltage magnitude by impedance magnitude and subtracts the impedance phase from the voltage phase.

Repeated rotations and dilations are modeled by powers. If each step multiplies a state by $ke^{i\phi}$, then after $n$ steps the state is $z_n=z_0k^n e^{in\phi}$. This appears in iterative graphics, rotating mechanisms, signal processing and planar motion.

Phasors simplify operations only when the model's assumptions apply. State the meaning and units of the modulus and argument, and convert to Cartesian form before adding phasors with different directions.

Positive real numbers have principal argument 0, negative real numbers have principal argument $\pi$, positive pure imaginary numbers have principal argument $\frac{\pi}{2}$, and negative pure imaginary numbers have principal argument $-\frac{\pi}{2}$.

The zero complex number may be written as $0(\cos\theta+i\sin\theta)$ for any θ, but this does not give zero a well-defined argument. For uniqueness, polar form is normally discussed for $z\ne0$.

A negative quantity should not be used as the modulus. If an expression begins with a negative radial coefficient, make the modulus positive and add $\pi$ to the argument.

Angles exactly at a principal-interval boundary must follow the stated convention. With $-\pi<\operatorname{Arg}z\le\pi$, use $\pi$ instead of $-\pi$.

Mixed questions often present values in different forms. Convert them to the form that makes the requested operation simplest, then convert the final answer only if required.

For multiplication or division, identify each modulus and argument before operating. For addition or subtraction, convert each number to Cartesian form first.

Use exact standard-angle values throughout. If one angle is non-standard, retain full calculator precision until the final answer.

A complete AI HL solution communicates the chosen form, quadrant reasoning, operation rule, and final argument convention.

Set the GDC to complex mode, radian angle mode, and the output form required by the question. Common output choices are rectangular $a+bi$ and polar $re^{i\theta}$.

Use rectangular-to-polar and polar-to-rectangular commands where available. On other models, calculate $r=\sqrt{x^2+y^2}$ and use a two-argument angle function such as atan2$y,x$.

Check whether the calculator reports an argument in $( -\pi,\pi ]) or ([0,2\pi)$, then convert to the required convention.

For exact standard angles, provide exact mathematical values even if the display gives decimals. Use the GDC to check, not replace, exact reasoning.

After multiplication or division, verify the modulus and the expected rotation independently. This catches incorrect angle signs quickly.

Most errors come from an incorrect quadrant, an inconsistent principal interval, degree mode, a negative modulus, or applying polar rules to addition and subtraction.

A clear conversion solution should show the modulus, a reference-angle or atan2 calculation, the quadrant, and the selected argument.

A clear operation solution should identify the two moduli and arguments, state the multiplication or division rule, normalize the final argument if requested, and retain exact values.

Always distinguish $\arg z$, which may represent every coterminal argument, from $\operatorname{Arg}z$, which denotes the chosen principal argument.

Attempt each problem before reading the steps. The sequence moves from basic conversion to mixed representation, geometric interpretation and multi-step operation problems.

Unless stated otherwise, give exact values and principal arguments in ($-\pi,\pi]$.