- Interpret modulus as distance from the origin. - Ask whether a ray from the origin to the origin itself determines any unique direction.

1. The modulus is the distance from the origin to the point representing \(z\). For \(z=0\), that distance is \(0\). 2. An argument describes the direction of the ray from the origin to a nonzero complex number. The origin itself has no direction from the origin, so \(\arg(0)\) is undefined.

a) \(|z|=0\) b) \(\arg(0)\) is undefined because the zero complex number has no direction from the origin.

- Where is the point if its real part is zero? - Measure the direction counterclockwise from the positive real axis.

1. The point lies seven units below the origin on the imaginary axis. 2. Its distance from the origin is \(7\), and the angle measured counterclockwise from the positive real axis is \(270^\circ\).

\(|z|=7\), \(\arg(z)=270^\circ\)

- Use the two coordinates to determine the distance from the origin. - Decide which quadrant contains the point before choosing the angle.

1. The modulus is \(\sqrt{(-1)^2+1^2}=\sqrt{2}\). 2. The point is in Quadrant II and makes a \(45^\circ\) reference angle, so its argument is \(135^\circ\).

\(|z|=\sqrt{2}\), \(\arg(z)=135^\circ\)

- Find the length of the arrow from the origin to the point. - What angle points straight up from the positive real axis?

1. The modulus of \(2i\) is \(2\). 2. Its direction is straight up, at \(90^\circ\). Thus \(z=2(\cos90^\circ+i\sin90^\circ)\).

\(2(\cos90^\circ+i\sin90^\circ)\)

- The angle in the exponent tells you which direction the arrow points. - Use the cosine and sine of that angle as the two rectangular components.

1. By Euler's formula, \(3e^{i\pi}=3(\cos\pi+i\sin\pi)\). 2. Since \(\cos\pi=-1\) and \(\sin\pi=0\), the result is \(-3\).

\(z=-3\)

- Locate a negative purely imaginary number in the complex plane. - Rationalize a denominator containing \(i\). - Use the sign of the imaginary part to determine the argument on the vertical axis.

1. The point \(z=-2.5i\) lies on the negative imaginary axis, so \(|z|=2.5\) and \(\arg(z)=270^\circ\). 2. \(w=\frac{1}{-2.5i}=\frac{i}{-2.5i^2}=0.4i\). 3. The point \(w=0.4i\) lies on the positive imaginary axis, so \(|w|=0.4\) and \(\arg(w)=90^\circ\). 4. In general, if \(z=bi\) with \(b<0\), then \(\frac{1}{z}=-\frac{1}{b}i\), whose imaginary coefficient is positive. The argument changes from \(270^\circ\) to \(90^\circ\), a change of \(180^\circ\) modulo \(360^\circ\).

a) \(|z|=2.5\), \(\arg(z)=270^\circ\) b) \(w=0.4i\) c) \(|w|=0.4\), \(\arg(w)=90^\circ\); the argument changes by \(180^\circ\) modulo \(360^\circ\)

- Think of each complex number as an arrow starting at the origin. - Compare the arrow's length and direction with those of the unit vector \(1\). - To map one nonzero vector to another, compare their lengths and the angle between their directions.

1. For \(z_1=-3\), the modulus is \(3\) and the argument is \(180^\circ\), so the transformation is a dilation by \(3\) and a \(180^\circ\) rotation. 2. For \(z_2=4i\), the modulus is \(4\) and the argument is \(90^\circ\), so the transformation is a dilation by \(4\) and a \(90^\circ\) counterclockwise rotation. 3. For \(z_3=1+i\), the modulus is \(\sqrt{2}\) and the argument is \(45^\circ\), so the transformation is a dilation by \(\sqrt{2}\) and a \(45^\circ\) counterclockwise rotation. 4. For \(z_4=-\sqrt{3}+i\), the modulus is \(2\). Its argument is \(150^\circ\). Thus, the transformation is a dilation by \(2\) and a \(150^\circ\) counterclockwise rotation. 5. To map \(z_1\) to \(z_2\), the scale factor is \(\frac{4}{3}\). The counterclockwise rotation is \(90^\circ-180^\circ+360^\circ=270^\circ\).

1. \(z_1\): dilation \(3\), rotation \(180^\circ\); \(z_2\): dilation \(4\), rotation \(90^\circ\); \(z_3\): dilation \(\sqrt{2}\), rotation \(45^\circ\); \(z_4\): dilation \(2\), rotation \(150^\circ\). 2. Dilation by \(\frac{4}{3}\) and rotation by \(270^\circ\) counterclockwise.

- Track the effects of a dilation and a rotation separately. - Apply the transformations in the stated order. - Determine the final quadrant. - Convert from polar to rectangular form using sine and cosine.

1. The original number has modulus \(4\) and argument \(0^\circ\). 2. The dilation changes the modulus to \(4\cdot0.5=2\). The rotations add, giving \(30^\circ+120^\circ=150^\circ\). 3. Thus, \(z_2=2(\cos150^\circ+i\sin150^\circ)=-\sqrt{3}+i\).

a) \(r=2\) and \(\theta=150^\circ\) b) \(z_2=-\sqrt{3}+i\)

- For multiplication, multiply moduli and add arguments. - For division, divide moduli and subtract arguments. - Add \(360^\circ\) to a negative argument to place it in the required interval. - Use sine and cosine to convert to rectangular form.

1. For the product, multiply moduli and add arguments: \(z_3=6(\cos150^\circ+i\sin150^\circ)=-3\sqrt{3}+3i\). 2. For the quotient, divide moduli and subtract arguments: the modulus is \(\frac{2}{3}\), and the argument is \(30^\circ-120^\circ=-90^\circ\), or \(270^\circ\) in the required interval. 3. Therefore, \(z_4=\frac{2}{3}(\cos270^\circ+i\sin270^\circ)=-\frac{2}{3}i\).

\(z_3=6(\cos150^\circ+i\sin150^\circ)=-3\sqrt{3}+3i\) \(z_4=\frac{2}{3}(\cos270^\circ+i\sin270^\circ)=-\frac{2}{3}i\)

- The modulus of the multiplier gives the dilation factor. - Locate the multiplier \(2i\) on the complex plane to determine its rotation angle. - Apply the fixed multiplier to the given complex number only after identifying the transformation it represents.

1. The modulus of \(a=2i\) is \(2\), so the scale factor is \(2\). An argument is \(\frac{\pi}{2}\), so the rotation is \(\frac{\pi}{2}\) counterclockwise. 2. \(w=2i(3-4i)=6i-8i^2=8+6i\).

1. Scale factor \(2\); rotation \(\frac{\pi}{2}\) counterclockwise 2. \(w=8+6i\)

- Read the real and imaginary coordinates from the graph first. - The modulus is the distance from the origin to the point. - Use the quadrant and the slope of the ray from the origin to determine the argument.

1. The graph shows \(z=3+3i\). Its modulus is \(r=\sqrt{3^2+3^2}=3\sqrt{2}\). 2. The point lies on the line \(y=x\) in Quadrant I, so \(\theta=\frac{\pi}{4}\). 3. Therefore, \(z=3\sqrt{2}e^{i\pi/4}\).

\(z=3\sqrt{2}e^{i\pi/4}\)

- Determine the quadrant before selecting an angle from the reference angle. - Distinguish one representative in a specified interval from the entire family of coterminal arguments. - A full revolution changes an angle without changing the represented direction.

1. \(|z|=\sqrt{(-2)^2+(2\sqrt{3})^2}=4\). 2. The point is in Quadrant II. Its reference angle is \(\pi/3\), so the argument in \([0,2\pi)\) is \(2\pi/3\). 3. Every coterminal angle gives the same direction, so all arguments are \(\theta=\frac{2\pi}{3}+2\pi k\), where \(k\in\mathbb{Z}\).

a) \(|z|=4\) and \(\theta=\frac{2\pi}{3}\) b) \(\theta=\frac{2\pi}{3}+2\pi k\), where \(k\in\mathbb{Z}\)

- Use the standard conversion factor between degrees and radians. - Coterminal arguments differ by a whole revolution. - Keep the modulus unchanged when only the angle unit or representative changes.

1. Convert degrees to radians: \(210^\circ\cdot\frac{\pi}{180^\circ}=\frac{7\pi}{6}\). Thus the requested exponential form is \(6e^{i7\pi/6}\). 2. Subtract one full revolution: \(\frac{7\pi}{6}-2\pi=-\frac{5\pi}{6}\). This is a negative coterminal argument for the same direction.

a) \(6e^{i7\pi/6}\) b) One valid negative argument is \(-\frac{5\pi}{6}\).

- First decide the quadrant from the angle in the exponent. - Convert the angle to cosine and sine components to distinguish the plotted points.

1. The angle \(\frac{7\pi}{4}\) points into Quadrant IV. 2. The rectangular coordinates are \(2\sqrt{2}(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})=(2,-2)\). 3. This is point \(A\).

Point \(A\); \(z=2-2i\)

- How does adding a complete turn change an angle's direction? - Once the angle lies in the required interval, use sine and cosine to find the two components.

1. Adding one full turn to \(-330^\circ\) gives the equivalent angle \(30^\circ\). 2. Thus \(z=2(\cos30^\circ+i\sin30^\circ)\). 3. Since \(\cos30^\circ=\frac{\sqrt{3}}{2}\) and \(\sin30^\circ=\frac12\), we get \(z=\sqrt{3}+i\).

\(z=2(\cos30^\circ+i\sin30^\circ)=\sqrt{3}+i\)

- Multiply the real and imaginary terms by \(i\) separately. - Compare the original and new points' distances from the origin. - How has the position of the new arrow changed around the origin?

1. Multiply: \(i(3+2i)=3i+2i^2=-2+3i\). 2. The original point is \((3,2)\), and the new point is \((-2,3)\). Both have distance \(\sqrt{3^2+2^2}=\sqrt{13}\) from the origin. 3. The coordinate change \((3,2)\mapsto(-2,3)\) is a \(90^\circ\) counterclockwise turn.

\(iz=-2+3i\); both distances are \(\sqrt{13}\); the arrow turns \(90^\circ\) counterclockwise.

- What happens to the distances from the origin when two complex numbers are multiplied? - Think of each factor's angle as a rotation, then combine the rotations.

1. The modulus of the product is \(2\cdot3=6\). 2. The arguments add: \(\frac{\pi}{3}+\frac{\pi}{6}=\frac{\pi}{2}\). 3. Hence the product is \(6e^{i\pi/2}\).

\(6e^{i\pi/2}\)

- How does the sign of a polar radius affect the direction of the plotted point? - Change to a positive distance before interpreting it as the modulus of a complex number. - Use cosine and sine for the rectangular coordinates.

1. A negative radius points in the opposite direction, so \((-2,30^\circ)\) is the same point as \((2,210^\circ)\). 2. The rectangular coordinates are \((2\cos210^\circ,2\sin210^\circ)=(-\sqrt{3},-1)\). 3. Thus the complex number is \(-\sqrt{3}-i=2(\cos210^\circ+i\sin210^\circ)\).

Rectangular: \(-\sqrt{3}-i\); trigonometric polar: \(2(\cos210^\circ+i\sin210^\circ)\)

- Which coordinate changes sign in a reflection across the real axis? - Does a reflection change distance from the origin? - Locate the new ray before writing its angle in the required interval.

1. Conjugation changes the sign of the imaginary part, giving \(\overline{z}=-1-\sqrt{3}i\). 2. The modulus of each number is \(\sqrt{1+3}=2\). The original argument is \(\frac{2\pi}{3}\), while reflection across the real axis gives the conjugate an argument of \(\frac{4\pi}{3}\). 3. Hence \(\overline{z}=2e^{i4\pi/3}\). The modulus stays the same and the argument is negated modulo \(2\pi\).

\(\overline{z}=-1-\sqrt{3}i=2e^{i4\pi/3}\); modulus unchanged at \(2\), argument reflected from \(\frac{2\pi}{3}\) to \(\frac{4\pi}{3}\).

- Think about the coordinate condition for the imaginary axis. - Interpret the magnitude of a difference as distance. - Because both points lie on the same axis, reduce the distance equation to an absolute-value equation.

1. A point on the imaginary axis has \(\operatorname{Re}(z)=0\). Its argument is \(90^\circ\) when the imaginary part is positive and \(270^\circ\) when the imaginary part is negative. 2. Write \(z=bi\). Then \(|bi-1.5i|=|b-1.5|=4.5\). 3. Thus, \(b-1.5=4.5\) or \(b-1.5=-4.5\), giving \(b=6\) or \(b=-3\). Therefore, \(z=6i\) or \(z=-3i\). 4. For \(6i\), the modulus is \(6\) and the argument is \(90^\circ\). For \(-3i\), the modulus is \(3\) and the argument is \(270^\circ\).

a) \(\operatorname{Re}(z)=0\); \(\arg(z)\in\{90^\circ, 270^\circ\}\) b) \(z\in\{6i, -3i\}\) c) \(|6i|=6\), \(\arg(6i)=90^\circ\); \(|-3i|=3\), \(\arg(-3i)=270^\circ\)

- Use polar form to translate a geometric description into a complex number. - Find the modulus and argument of \(1-i\). - Under multiplication, moduli multiply and arguments add. - A negative argument represents a clockwise rotation.

1. \(w=4(\cos120^\circ+i\sin120^\circ)=-2+2\sqrt{3}i\). 2. For \(z=1-i\), \(|z|=\sqrt{2}\) and \(\arg(z)=-45^\circ\). Thus, multiplication by \(z\) dilates by \(\sqrt{2}\) and rotates \(45^\circ\) clockwise. 3. \(|v|=|w||z|=4\sqrt{2}\), and \(\theta=120^\circ-45^\circ=75^\circ\).

1. \(w=-2+2\sqrt{3}i\) 2. Dilation by \(\sqrt{2}\) and rotation \(45^\circ\) clockwise 3. \(|v|=4\sqrt{2}\) and \(\theta=75^\circ\)

- Find each modulus and choose its argument in the stated interval. - Under multiplication in polar form, moduli multiply and arguments add. - Isolate \(w\) by division. - The modulus and argument of \(w\) determine the geometric transformation.

1. \(|z_1|=2\) and \(\arg(z_1)=\frac{\pi}{3}\), so \(z_1=2e^{i\pi/3}\). 2. \(|z_2|=4\) and \(\arg(z_2)=\frac{3\pi}{2}\), so \(z_2=4e^{i3\pi/2}\). 3. \(w=\frac{z_2}{z_1}=\frac{-4i}{1+i\sqrt{3}}=-\sqrt{3}-i\). 4. The scale factor is \(|w|=\frac{|z_2|}{|z_1|}=2\). The rotation angle is \(270^\circ-60^\circ=210^\circ\), equivalent to \(150^\circ\) clockwise.

a) \(z_1=2e^{i\pi/3}\) and \(z_2=4e^{i3\pi/2}\) b) \(w=-\sqrt{3}-i\) c) Dilation by \(2\) and rotation \(210^\circ\) counterclockwise

- Use the signs of the coordinates in Quadrant III. - Apply the Pythagorean theorem to find the missing component. - Use a reference angle and adjust it to the correct quadrant.

1. From \(5^2=(-3)^2+b^2\), \(b^2=16\). Because \(z\) is in Quadrant III, \(b=-4\). 2. The reference angle is \(\arctan\left(\frac{4}{3}\right)\approx53.13^\circ\). Therefore, \(\arg(z)\approx180^\circ+53.13^\circ=233.13^\circ\). 3. Thus, \(z\approx5(\cos233.13^\circ+i\sin233.13^\circ)\).

1. \(b=-4\) 2. \(\arg(z)\approx233.13^\circ\) 3. \(z\approx5(\cos233.13^\circ+i\sin233.13^\circ)\)

- Under multiplication, moduli multiply. - Under multiplication, arguments add. - Use right-triangle relationships to find the initial arguments. - Interpret the multiplier as a transformation.

1. \(|z_1|=\sqrt{2}\) and \(\arg(z_1)=45^\circ\). Also, \(|z_2|=2\) and \(\arg(z_2)=30^\circ\). 2. In rectangular form, \((1+i)(\sqrt{3}+i)=(\sqrt{3}-1)+(1+\sqrt{3})i\). 3. In polar form, the modulus is \(\sqrt{2}\cdot2=2\sqrt{2}\), and the argument is \(45^\circ+30^\circ=75^\circ\). Thus, \(z=2\sqrt{2}(\cos75^\circ+i\sin75^\circ)\). 4. Multiplication by \(z_2\) dilates by a factor of \(2\) and rotates \(30^\circ\) counterclockwise.

1. \(|z_1|=\sqrt{2}\), \(\arg(z_1)=45^\circ\); \(|z_2|=2\), \(\arg(z_2)=30^\circ\) 2. \(z=(\sqrt{3}-1)+(1+\sqrt{3})i=2\sqrt{2}(\cos75^\circ+i\sin75^\circ)\) 3. Dilation by \(2\) and rotation \(30^\circ\) counterclockwise

- Convert a given modulus and angle to rectangular form. - Use the exact values of sine and cosine at \(120^\circ\). - Locate \(i\) in the complex plane and determine its modulus and argument.

1. The multiplier has modulus \(2\) and argument \(120^\circ\), so \(w=2(\cos120^\circ+i\sin120^\circ)=-1+\sqrt{3}i\). 2. \(z'=(-1+\sqrt{3}i)(3+2i)=(-3-2\sqrt{3})+(3\sqrt{3}-2)i\). 3. The number \(i\) has modulus \(1\) and argument \(90^\circ\). Multiplying by \(i\) multiplies the original modulus by \(1\) and adds \(90^\circ\) to its argument.

1. \(w=-1+\sqrt{3}i\) 2. \(z'=(-3-2\sqrt{3})+(3\sqrt{3}-2)i\) 3. Since \(|i|=1\) and \(\arg(i)=90^\circ\), multiplication by \(i\) preserves length and rotates \(90^\circ\).

- Read the modulus and argument of \(z\) from its position relative to the origin and axes. - In polar form, multiplication combines radial scale factors and angular rotations directly. - Normalize the resulting argument only if it falls outside the required angle interval.

1. The graph shows \(z=-2+2i\). Its modulus is \(2\sqrt{2}\), and its Quadrant II argument is \(\frac{3\pi}{4}\). Thus \(z=2\sqrt{2}e^{i3\pi/4}\). 2. Multiply the moduli: \(\sqrt{3}\cdot2\sqrt{2}=2\sqrt{6}\). 3. Add the arguments: \(-\frac{\pi}{6}+\frac{3\pi}{4}=\frac{7\pi}{12}\). Therefore, \(wz=2\sqrt{6}e^{i7\pi/12}\).

a) \(z=2\sqrt{2}e^{i3\pi/4}\) b) \(wz=2\sqrt{6}e^{i7\pi/12}\)

- Think geometrically about reflection across the real axis before changing the angle. - For a reciprocal, consider separately what must happen to the modulus and to the argument so that multiplication returns \(1\). - Convert a negative angle to the required interval only at the end.

1. Conjugation reflects a complex number across the real axis, so it keeps the modulus and negates the argument. Thus \(\overline{z}=5e^{-i2\pi/5}=5e^{i8\pi/5}\). 2. A reciprocal replaces the modulus by its reciprocal and negates the argument. Thus \(\frac{1}{z}=\frac{1}{5}e^{-i2\pi/5}=\frac{1}{5}e^{i8\pi/5}\). 3. In both cases the argument changes from \(2\pi/5\) to its reflected representative \(8\pi/5\). Conjugation leaves modulus \(5\) unchanged, while reciprocation changes it to \(1/5\).

a) \(\overline{z}=5e^{i8\pi/5}\) b) \(\frac{1}{z}=\frac{1}{5}e^{i8\pi/5}\) Conjugation keeps the modulus and negates the argument; reciprocation takes the reciprocal modulus and negates the argument.

- What are the distance and direction of the point for \(z\)? - Think about which scale and turn would undo multiplication by \(z\). - Use the original angle's sine and cosine to return to rectangular components.

1. The modulus is \(|z|=\sqrt{5}\), and the argument is \(\alpha=\arctan(2)\), in Quadrant I. Thus \(z=\sqrt{5}e^{i\alpha}\). 2. The reciprocal has modulus \(\frac{1}{\sqrt{5}}\) and argument \(-\alpha\), or \(2\pi-\alpha\) in \([0,2\pi)\). Its point is in Quadrant IV. 3. Since \(\cos\alpha=\frac{1}{\sqrt{5}}\) and \(\sin\alpha=\frac{2}{\sqrt{5}}\), we get \(w=\frac{1}{\sqrt{5}}(\cos\alpha-i\sin\alpha)=\frac{1}{5}-\frac{2}{5}i\).

\(w=\frac{1}{5}-\frac{2}{5}i\); Quadrant IV; \(|z|=\sqrt{5}\), \(|w|=\frac{1}{\sqrt{5}}\). The argument changes from \(\alpha=\arctan(2)\) to \(2\pi-\alpha\).