Free Math Worksheets

- 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 \(360^\circ\)

- View each number as a vector from the origin. - The vector length gives the dilation factor. - Measure the rotation from the positive real axis. - Use the Pythagorean theorem for the modulus and determine the correct quadrant for the angle.

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 \(2\). The point is in Quadrant II with reference angle \(30^\circ\), so its argument is \(150^\circ\). Thus, the transformation is a dilation by \(2\) and a \(150^\circ\) counterclockwise rotation.

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. \(|z_1|=3\), \(\arg(z_1)=180^\circ\); \(|z_2|=4\), \(\arg(z_2)=90^\circ\); \(|z_3|=\sqrt{2}\), \(\arg(z_3)=45^\circ\); \(|z_4|=2\), \(\arg(z_4)=150^\circ\)

- 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 \(\varphi=150^\circ\) 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, \(z_4=\frac{2}{3}(\cos270^\circ+i\sin270^\circ)=-\frac{2}{3}i\).

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

- Interpret the magnitude of a difference as distance. - Identify the region described by a fixed maximum distance. - Translate the condition on the imaginary part into a condition on \(y\). - Identify both regions and determine their intersection.

1. The condition \(|z-(1+2i)|\le2\) describes the disk centered at \((1, 2)\) with radius \(2\). 2. The condition \(\operatorname{Im}(z)\ge2\) describes the half-plane on or above the line \(y=2\). 3. The line \(y=2\) passes through the center of the disk, so it divides the disk into two equal halves. The intersection is the upper semicircular region. 4. Its area is \(\frac{1}{2}\pi(2)^2=2\pi\).

The region is the upper half of the disk centered at \((1, 2)\) with radius \(2\), and its area is \(2\pi\) square units.

- 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 \(\arg(v)=120^\circ-45^\circ=75^\circ\).

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

- Find each modulus and argument from its rectangular form. - Under multiplication in polar form, moduli multiply and arguments add. - Isolate \(w\) by division. - The modulus and argument of \(w\) determine the 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 \(\arg(z_2)-\arg(z_1)=270^\circ-60^\circ=210^\circ\), 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

- Solve \(z_1w=z_2\) by division. - Use a conjugate to make the denominator real. - The modulus of \(w\) is the scale factor. - The argument of \(w\) is the rotation angle.

1. \(w=\frac{z_2}{z_1}=\frac{2+4i}{3+i}\cdot\frac{3-i}{3-i}=\frac{10+10i}{10}=1+i\). 2. \(|w|=\sqrt{1^2+1^2}=\sqrt{2}\), and \(\arg(w)=45^\circ\). 3. Multiplication by \(w\) dilates every vector from the origin by \(\sqrt{2}\) and rotates it \(45^\circ\) counterclockwise.

a) \(w=1+i\) b) \(k=\sqrt{2}\), \(\alpha=45^\circ\) c) Dilation by \(\sqrt{2}\) and rotation \(45^\circ\) 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. \(z\approx5(\cos233.13^\circ+i\sin233.13^\circ)\)

- The modulus of the multiplier gives the scale factor. - Determine the direction of a positive imaginary number in the complex plane. - Use a conjugate to simplify a complex denominator. - Isolate \(z\) as in an ordinary linear equation.

1. The modulus of \(a=2i\) is \(2\), so the scale factor is \(2\). Its argument is \(\frac{\pi}{2}\), so the rotation is \(90^\circ\) counterclockwise. 2. \(w=2i(3-4i)=6i-8i^2=8+6i\). 3. \(z=\frac{4}{1+i}\). Multiplying by the conjugate gives \(z=\frac{4(1-i)}{(1+i)(1-i)}=\frac{4-4i}{2}=2-2i\).

1. Scale factor \(2\); rotation angle \(90^\circ\) counterclockwise 2. \(w=8+6i\) 3. \(z=2-2i\)

- 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 \(90^\circ\).