Geometric Representations of Complex Numbers - Algebra 2

On the complex plane (real axis horizontal, imaginary axis vertical), the complex number $z = -3 + 4i$ is represented by the point $(a,b)$ where $z=a+bi$ . What is the modulus $|z|$ (the distance from the origin), and which point represents $z$ ?
Answer: Point $(-3,4)$ and $|z|=5$ This question tests your understanding of representing complex numbers geometrically on the complex plane and finding their modulus as the distance from the origin. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, -3 + 4i goes at (-3, 4), and the modulus |z| equals sqrt( $a^2$ + $b^2$ ), which is the distance from the origin using the Pythagorean theorem. For z = -3 + 4i, identify a = -3 and b = 4, so plot at (-3, 4) and |z| = sqrt(9 + 16) = sqrt(25) = 5; geometrically, it's 5 units from the origin in the second quadrant. Choice A correctly identifies the point (-3, 4) and modulus 5. A tempting distractor like Choice B swaps the coordinates to (4, -3), but remember, real part is horizontal (x-like) and imaginary is vertical (y-like)—don't reverse them! To plot any complex number a + bi, draw axes with real horizontal and imaginary vertical, move a units horizontally (positive right, negative left) and b units vertically (positive up, negative down), then mark the point; for modulus, always use sqrt( $a^2$ + $b^2$ ), which is always non-negative as it's a distance—keep practicing, you've got this!
On the complex plane, $z = 6 - 8i$ corresponds to the point $(6,-8)$ . Using $|a+bi|=\sqrt{a^2+b^2}$ , what is $|z|$ ?
Answer: $|z|=\sqrt{6^2+8^2}=10$ This question tests your understanding of calculating the modulus of a complex number using its geometric representation on the complex plane. The modulus |a + bi| is sqrt( $a^2$ + $b^2$ ), representing the distance from the origin to the point (a, b); for example, for 6 - 8i at (6, -8), it's the hypotenuse of a right triangle with legs 6 and 8. Here, a = 6, b = -8 (note the sign doesn't affect squares), so |z| = sqrt(36 + 64) = sqrt(100) = 10; geometrically, it's 10 units from the origin in the fourth quadrant. Choice B correctly applies the formula with addition inside the square root and computes sqrt(100) = 10. A tempting distractor like Choice A subtracts instead of adding the squares, leading to a negative under the root, but remember, modulus uses $a^2$ + $b^2$ since both are squared positives—subtraction would be incorrect for distance! To find any modulus, identify a and b, square each ( $a^2$ and $b^2$ ), add them, then take the square root—practice with positives and negatives, as signs vanish after squaring. Geometrically, visualize the point and draw the line to the origin; the length is the modulus—great job verifying this, keep building your skills!
A complex number $z$ has modulus 6 and argument $150°$ . When $z$ is multiplied by the complex number $\frac{1}{2} - \frac{\sqrt{3}}{2}i$ , what is the argument of the resulting product?
Answer: $90°$ When you see complex numbers with modulus and argument, think about multiplication in polar form. The key insight is that multiplying complex numbers adds their arguments and multiplies their moduli. First, you need the argument of $\frac{1}{2} - \frac{\sqrt{3}}{2}i$ . This complex number lies in the fourth quadrant since the real part is positive and the imaginary part is negative. Using $\tan^{-1}\left(\frac{-\sqrt{3}/2}{1/2}\right) = \tan^{-1}(-\sqrt{3}) = -60°$ , or equivalently $300°$ in standard position. When multiplying complex numbers in polar form, you add the arguments: $150° + 300° = 450°$ . Since arguments are typically expressed between $0°$ and $360°$ , you reduce: $450° - 360° = 90°$ . This confirms answer B is correct. Let's examine why the other answers are wrong. Answer A ( $210°$ ) might result from incorrectly subtracting the arguments: $150° + (-60°) = 90°$ , but then miscalculating to get $210°$ . Answer D ( $30°$ ) could come from subtracting $300°$ from $150°$ and taking the absolute value, showing confusion about the addition rule. Answer C ( $270°$ ) might result from adding $150°$ and $120°$ (using $120°$ instead of the correct $300°$ for the second argument). Remember: when multiplying complex numbers, always add their arguments. Convert to standard position ( $0°$ to $360°$ ) if needed, and be careful about which quadrant your complex numbers lie in when finding arguments.
Which set of points on the complex plane represents all complex numbers $z$ such that $|z|=5$ ? (Recall $|z|$ is the distance from the origin.)
Answer: A circle of radius $5$ centered at the origin This question tests your understanding of the geometric set where |z| = 5, the modulus as distance from the origin. |z| = sqrt( $a^2$ + $b^2$ ) = 5 means all points (a, b) exactly 5 units from (0, 0), forming a circle of radius 5 centered at the origin. Geometrically, it's every complex number whose distance to zero is 5, like points on the circumference. Choice A correctly describes this as a circle of radius 5 at the origin. A tempting distractor like Choice B shifts the center to (5, 0), but |z| = 5 is centered at origin, not offset—|z - 5| would be centered at 5 + 0i. To describe sets like |z| = r, recognize it's a circle radius r at origin; for |z - c| = r, it's centered at c. This is key for geometric interpretations—keep practicing, you've got a strong foundation!
Which set of complex numbers satisfies $|z|=4$ ? (Recall: $|a+bi|$ is the distance from $(a,b)$ to the origin.)
Answer: All points on the circle centered at the origin with radius $4$ . This question tests your understanding of sets of complex numbers defined by modulus conditions and their geometric representations. The equation |z| = 4 means all complex numbers z whose distance from the origin equals 4—this describes a circle centered at the origin with radius 4! For any complex number z = a + bi satisfying |z| = 4, we have square root of (a squared + b squared) = 4, which means a squared + b squared = 16: this is the equation of a circle with center (0, 0) and radius 4 in the coordinate plane. Choice B correctly identifies this as all points on the circle centered at the origin with radius 4, recognizing that constant modulus means constant distance from origin. Choice A incorrectly describes a vertical line a = 4 (where real part is always 4), but points like (4, 3) have modulus square root of (16 + 9) = 5, not 4—only the point (4, 0) on this line has modulus 4! The key insight about modulus equations: (1) |z| = r describes a circle centered at origin with radius r, (2) |z - w| = r describes a circle centered at w with radius r, (3) |z| < r describes the interior of the circle (disk), (4) |z| > r describes the exterior. These geometric interpretations make complex inequalities visual: for example, 2 < |z| < 5 is an annulus (ring) between two concentric circles!
In the complex plane, the points representing $z = 5 + 12i$ and its complex conjugate $\overline{z}$ are plotted. What is the geometric relationship between these two points and the real axis?
Answer: They are reflections of each other across the real axis, equidistant from it When you encounter complex conjugates in the complex plane, you're dealing with one of the most fundamental geometric relationships in complex number theory. The complex conjugate of $z = 5 + 12i$ is $\overline{z} = 5 - 12i$ , which means you keep the real part the same but change the sign of the imaginary part. In the complex plane, $z = 5 + 12i$ corresponds to the point $(5, 12)$ , while $\overline{z} = 5 - 12i$ corresponds to $(5, -12)$ . Notice that both points have the same x-coordinate (real part) but opposite y-coordinates (imaginary parts). This creates a perfect reflection across the real axis (x-axis). Both points are exactly 12 units away from the real axis, making them equidistant from it. Option A is incorrect because the points aren't symmetric about the origin—they have the same real part, so they're not equidistant from the imaginary axis. Option B confuses the axes; these points would be reflections across the imaginary axis only if their real parts were opposites. Option D describes a rotation, but rotating $(5, 12)$ by $180°$ about $(5, 0)$ would indeed give $(5, -12)$ —however, this is unnecessarily complicated since the primary relationship is the simpler reflection. Remember this pattern: complex conjugates always create mirror images across the real axis. When you see any complex number and its conjugate plotted, immediately think "reflection across the x-axis" and you'll quickly identify the correct geometric relationship.
Let $z=4-3i$ . The complex conjugate $\overline{z}$ is obtained by reflecting the point for $z$ across the real axis. Which statement correctly describes $\overline{z}$ and its point on the complex plane?
Answer: $\overline{z}=4+3i$ , plotted at $(4,3)$ This question tests your understanding of complex conjugates and their geometric reflection on the complex plane. The conjugate of a + bi is a - bi, which reflects the point (a, b) over the real axis to (a, -b); for z = 4 - 3i at (4, -3), the conjugate is 4 + 3i at (4, 3). Geometrically, this flips the sign of the imaginary part while keeping the real part, mirroring across the horizontal axis. Choice A correctly gives \bar{z} = 4 + 3i at (4, 3). A tempting distractor like Choice B negates both parts to -4 - 3i at (-4, -3), but conjugation only changes the imaginary sign, not the real—it's not a full inversion! To find the conjugate, change the sign of the i term only; plot by keeping x-coordinate and flipping y-coordinate over the x-axis. Both z and \bar{z} have the same modulus since reflection preserves distance—wonderful, you're grasping this reflection idea!
Which set of points on the complex plane represents all complex numbers $z$ such that $|z|=4$ ? (Here $|z|$ is the distance from the origin.)
Answer: A circle of radius 4 centered at the origin. This question tests your understanding of how equations involving modulus describe geometric shapes on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! The modulus |z| represents the distance from the origin to the point z, so the equation |z| = 4 describes all points that are exactly 4 units from the origin—this is the definition of a circle! The set |z| = 4 represents: (1) All complex numbers z = a + bi where square root of (a squared + b squared) = 4. (2) Squaring both sides: a squared + b squared = 16. (3) This is the equation of a circle centered at origin with radius 4! Points include 4 + 0i, 0 + 4i, -4 + 0i, 0 - 4i, and infinitely many others like 2√2 + 2√2i. Choice A correctly identifies this as a circle of radius 4 centered at the origin. Choice B describes only one point (4, 0) on the circle, not the entire circle—this would be the set {z : z = 4}. Choice C places the center at (4, 0) instead of origin—that would be |z - 4| = 4. Choice D describes points with real part 4, which would be {z : Re(z) = 4}, a vertical line. Modulus equations create circles: (1) |z| = r is a circle centered at origin with radius r. (2) |z - c| = r is a circle centered at complex number c with radius r. (3) The equation describes all points at fixed distance from center. Think of it as all possible endpoints when you spin a rope of length r around the center!
Let $z = 2 - 5i$ . The conjugate of $z$ is $\overline{z} = 2 + 5i$ . How are the points representing $z$ and $\overline{z}$ related geometrically on the complex plane?
Answer: $\overline{z}$ is a reflection of $z$ across the real axis. This question tests your understanding of the geometric relationship between a complex number and its conjugate on the complex plane. The conjugate of a + bi is a - bi, which reflects the point (a, b) across the real axis (horizontal axis), flipping the sign of the imaginary part while keeping the real part the same! For example, the conjugate of 3 + 2i is 3 - 2i, reflecting (3,2) to (3,-2). For z = 2 - 5i at (2, -5), its conjugate 2 + 5i is at (2, 5), which is indeed a reflection across the real axis, as the real parts match and imaginary parts are opposites. Choice C correctly describes this as a reflection across the real axis. Choice A might tempt you if you confuse it with reflection across the imaginary axis, which would flip the real part's sign instead, like turning 2 - 5i into -2 - 5i, but that's not the conjugate. To find and interpret conjugates: (1) For a + bi, change to a - bi, (2) Plot both: they'll be symmetric over the real axis, and note they have the same modulus since distances from origin are equal. You're doing wonderfully; this insight will help with many complex number properties—keep it up!
The complex number $w = -2 + 2\sqrt{3}i$ has modulus $4$ and argument $120°$ . What is the result of $w^3$ expressed in rectangular form?
Answer: $64$ When you encounter a complex number raised to a power, you have two main approaches: multiply it out in rectangular form, or use polar form with De Moivre's theorem. Since the problem gives you the modulus and argument, polar form is more efficient. Using De Moivre's theorem, when a complex number has modulus $r$ and argument $\theta$ , then $z^n = r^n \cdot (\cos(n\theta) + i\sin(n\theta))$ . Here, $w^3 = 4^3 \cdot (\cos(3 \cdot 120°) + i\sin(3 \cdot 120°)) = 64 \cdot (\cos(360°) + i\sin(360°))$ . Since $360°$ is a full rotation, $\cos(360°) = 1$ and $\sin(360°) = 0$ . Therefore, $w^3 = 64 \cdot (1 + 0i) = 64$ , which is answer D. Looking at the wrong answers: A) $32 - 32\sqrt{3}i$ uses the wrong power for the modulus ( $4^2 = 16$ scaled incorrectly) and miscalculates the trigonometric values. B) $-64$ gets the correct modulus cubed but uses $\cos(180°) = -1$ instead of $\cos(360°) = 1$ , suggesting the student computed $3 \times 60°$ instead of $3 \times 120°$ . C) $-32 + 32\sqrt{3}i$ combines both errors from A and uses an incorrect modulus calculation. Study tip: When using De Moivre's theorem, always check if your final angle is a multiple of $360°$ (or $2\pi$ radians). These "full rotations" simplify to real numbers, eliminating the imaginary component entirely.
On the complex plane, the distance from the origin to the point representing $z=a+bi$ equals $|z|=\sqrt{a^2+b^2}$ . What is the distance from the origin to the point representing $z=-5-12i$ ?
Answer: $13$ This question tests your understanding of representing complex numbers geometrically on the complex plane and finding their modulus (absolute value) as the distance from the origin. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem—the real and imaginary parts form legs of a right triangle, modulus is hypotenuse! To find the distance from origin to z = -5 - 12i: (1) Identify real part a = -5 and imaginary part b = -12. (2) Apply distance formula: |z| = √((-5)² + (-12)²) = √(25 + 144) = √169 = 13. (3) The point (-5, -12) is 13 units from the origin—this forms a 5-12-13 Pythagorean triple! Choice A correctly gives 13 as the distance. Choice B gives √119, which would come from an arithmetic error, choice C gives 17 which might come from incorrectly adding 5 + 12, and choice D shows √(25 - 144) which incorrectly subtracts instead of adds the squared terms—remember, the formula uses a² + b², not a² - b²! When finding modulus, always add the squares of both parts: (1) Square the real part, (2) Square the imaginary part, (3) Add these squares together, (4) Take the square root of the sum. The signs of a and b don't matter after squaring since (-5)² = 25 and (-12)² = 144 are both positive. This distance formula works because complex numbers form a right triangle with the axes, making the Pythagorean theorem our perfect tool!
A complex number $z=a+bi$ is plotted as the point $(a,b)$ on the complex plane. If $z=-2-5i$ , which point is its location?
Answer: (-2, -5) This question tests your understanding of plotting complex numbers on the complex plane as ordered pairs. A complex number a + bi is at (a, b), with a on the real (horizontal) axis and b on the imaginary (vertical) axis; for example, -2 - 5i is at (-2, -5), in the third quadrant. For z = -2 - 5i, a = -2 (left 2 units) and b = -5 (down 5 units), so the point is (-2, -5). Choice B correctly places it at (-2, -5). A tempting distractor like Choice A uses positives (2, 5), but don't forget the negative signs— -2 means left, -5 means down! To plot any z = a + bi, start at origin, move a horizontally (negative left) and b vertically (negative down); practice with examples like 1 + 2i at (1, 2) or -3i at (0, -3). Geometrically, this representation treats complex numbers as vectors from the origin—keep going, you're doing great!
A force is modeled as the complex number $F=3+4i$ , where $3$ is the horizontal component and $4$ is the vertical component. What is the magnitude of the force (the modulus $|F|$ )?
Answer: $\sqrt{3^2+4^2}=5$ This question tests your understanding of applying complex numbers to model physical quantities and calculating magnitude using the modulus. When a force F = 3 + 4i represents horizontal and vertical components, the magnitude |F| gives the total force strength using the Pythagorean theorem—just like finding the length of the hypotenuse! To find the magnitude of F = 3 + 4i: (1) Identify components: horizontal (real) = 3, vertical (imaginary) = 4. (2) Apply modulus formula: |F| = square root of (3 squared + 4 squared) = square root of (9 + 16) = square root of 25 = 5. (3) Physical interpretation: a force with 3 units horizontal and 4 units vertical has total magnitude 5 units. Choice A correctly calculates the magnitude as square root of (3 squared + 4 squared) = 5, recognizing this as a 3-4-5 right triangle—a classic Pythagorean triple! Choice C incorrectly adds 3 + 4 = 7, which would be like saying a 3-foot horizontal rope and 4-foot vertical rope stretched at right angles have total length 7 feet—but the diagonal distance is 5 feet, not 7! Complex numbers in physics: (1) Forces, velocities, and other vectors can be represented as complex numbers, (2) Real part = horizontal component, imaginary part = vertical component, (3) Modulus = magnitude (total strength/speed), (4) Argument = direction angle from positive horizontal. This representation makes vector addition simple (just add complex numbers) and connects algebra to physical applications!
On the complex plane, the distance between $z=-1+2i$ and $w=3-1i$ is $|z-w|$ . What is $|z-w|$ ?
Answer: $\sqrt{( -1-3)^2+(2-(-1))^2}=\sqrt{25}=5$ This question tests your understanding of the distance between two points on the complex plane using |z - w|. The formula is sqrt( $(a - c)^2$ + $(b - d)^2$ ) for z = a + bi and w = c + di; here, z = -1 + 2i at (-1, 2), w = 3 - i at (3, -1), so |z - w| = sqrt( $(-1 - 3)^2$ + $(2 - (-1))^2$ ) = sqrt(16 + 9) = sqrt(25) = 5. Geometrically, it's the line segment length between (-1, 2) and (3, -1). Choice A correctly subtracts and computes with (2 + 1) = 3 for the imaginary difference. A tempting distractor like Choice D adds moduli |z| + |w|, but distance is |z - w|, not the sum—sum would be longer unless collinear. Always compute z - w first, then modulus; or use distance formula on points, double-checking subtractions. You're building great intuition for distances—excellent!
A complex number $z$ satisfies $|z| = 3$ and $\text{arg}(z) = 60°$ . When $z$ is divided by $1 + i\sqrt{3}$ , what is the modulus of the resulting quotient?
Answer: $\frac{3}{2}$ obtained by dividing the modulus of $z$ by the modulus of the divisor When you encounter complex number division problems involving modulus and argument, remember that the modulus of a quotient equals the quotient of the moduli. This fundamental property makes these calculations much more straightforward than they might initially appear. First, let's find the modulus of the divisor $1 + i\sqrt{3}$ . Using the formula $|a + bi| = \sqrt{a^2 + b^2}$ , we get $|1 + i\sqrt{3}| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$ . Since $|z| = 3$ and we're dividing by a complex number with modulus 2, the modulus of the quotient is simply $\frac{|z|}{|1 + i\sqrt{3}|} = \frac{3}{2}$ . This confirms answer choice B. Let's examine why the other options are incorrect. Choice A gives $\frac{3\sqrt{3}}{2}$ , which appears to incorrectly incorporate the $\sqrt{3}$ component from the divisor into the final modulus calculation. Choice C yields $\frac{\sqrt{3}}{2}$ , suggesting a confusion where perhaps the coefficient 3 from $|z|$ was replaced with $\sqrt{3}$ . Choice D results in $\frac{3}{4}$ , which might arise from incorrectly calculating the modulus of the divisor as 4 instead of 2. The key insight here is that you don't need to perform the full complex division or work with trigonometric forms. When finding the modulus of a quotient, simply divide the moduli of the numerator and denominator. This property saves significant time on exams.
Find the modulus of the complex number $z = 6 - 8i$ . (Recall: $|a+bi|=\sqrt{a^2+b^2}$ , the distance from the origin to $(a,b)$ on the complex plane.)
Answer: $|z|=\sqrt{6^2+8^2}=10$ This question tests your understanding of calculating the modulus (absolute value) of a complex number using the distance formula on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem—the real and imaginary parts form legs of right triangle, modulus is hypotenuse! To find modulus of z = 6 - 8i: (1) Identify real part a = 6 and imaginary part b = -8 (note the negative!). (2) Apply modulus formula: |z| = square root of (6 squared + (-8) squared) = square root of (36 + 64) = square root of 100 = 10. (3) Geometric interpretation: the point (6, -8) is 10 units from origin. This forms a 6-8-10 right triangle, a multiple of the famous 3-4-5 triangle! Choice B correctly calculates |z| = square root of (6 squared + 8 squared) = 10. Choice A incorrectly subtracts the squares instead of adding: square root of (6 squared - 8 squared) would give square root of negative, which isn't real! Modulus uses addition under the square root. Choice C adds the absolute values directly: 6 + 8 = 14, forgetting the squares and square root—distance requires Pythagorean theorem! Choice D makes an arithmetic error: square root of (36 + 64) = square root of 100 = 10, not square root of 28. Modulus calculation checklist: (1) Square both parts (signs don't matter after squaring). (2) Add the squares. (3) Take square root. The result is always non-negative since it represents distance!
Vector addition of complex numbers corresponds to adding their coordinates on the complex plane. Using $a+bi\leftrightarrow (a,b)$ , what is the sum $(2+i)+(1+3i)$ and which point represents it?
Answer: $3+4i$ , represented by $(3,4)$ This question tests your understanding of adding complex numbers geometrically as vector addition on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! For example, 3 + 2i goes at (3, 2), and -1 - 4i goes at (-1, -4). The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem—the real and imaginary parts form legs of a right triangle, modulus is the hypotenuse! For (2 + i) + (1 + 3i) = (2+1) + (1+3)i = 3 + 4i at (3,4); geometrically, add vectors tail-to-head from origin. Choice A correctly sums to 3 + 4i and plots at (3,4). Choice D has 3 + 4i but plots (4,3)—don't swap axes, real is horizontal! To add z1 = a + bi and z2 = c + di, sum reals a+c, imaginaries b+d, plot at (a+c, b+d); it's like vector addition—superb, you're adding with confidence!
On the complex plane, addition corresponds to vector addition. If $z_1=2+i$ and $z_2=1+3i$ , what point represents $z_1+z_2$ ?
Answer: $(3,4)$ This question tests your understanding of adding complex numbers geometrically as vector addition on the complex plane. Adding z1 + z2 means adding their real parts and imaginary parts separately, or tail-to-head vector addition; for z1 = 2 + i at (2, 1) and z2 = 1 + 3i at (1, 3), sum is (2+1) + (1+3)i = 3 + 4i at (3, 4). Geometrically, place z2's tail at z1's head, and the resultant goes from origin to z2's head. Choice C correctly computes (3, 4). A tempting distractor like Choice A uses (3, 2), perhaps swapping imaginaries, but add corresponding parts: reals 2+1=3, imaginaries 1+3=4—don't mix them! To add any two complex numbers, add reals and imaginaries separately, then plot the sum; geometrically, it's parallelogram addition too. Mastering this helps with vector concepts—great effort, you're progressing well!
The modulus of $z=a+bi$ is $|z|=\sqrt{a^2+b^2}$ . Which complex number has modulus $5$ ?
Answer: $3+4i$ This question tests your understanding of finding complex numbers with a given modulus on the complex plane. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! The modulus (absolute value) of a + bi equals square root of (a squared + b squared), which is the distance from the origin to point (a, b) using the Pythagorean theorem. To check which complex number has modulus 5, we calculate |a + bi| = √(a² + b²) for each choice: (A) |1 + 4i| = √(1² + 4²) = √(1 + 16) = √17 ≠ 5. (B) |3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5 ✓. (C) |5 + 5i| = √(5² + 5²) = √(25 + 25) = √50 ≠ 5. (D) |-5 + 2i| = √((-5)² + 2²) = √(25 + 4) = √29 ≠ 5. Choice B correctly identifies 3 + 4i as having modulus 5—this is the famous 3-4-5 Pythagorean triple! Choice A might tempt students who confuse 1² + 4² = 17 with 1 + 4 = 5. Choice C has both parts equal to 5, but |5 + 5i| = 5√2, not 5. Choice D has real part -5, but the modulus depends on the sum of squares, not individual parts. Finding complex numbers with a given modulus means finding points on a circle: all complex numbers with |z| = 5 lie on a circle of radius 5 centered at the origin. The 3-4-5 triple gives us one such point, and there are infinitely many others like 5 + 0i, 0 + 5i, and 4 - 3i!
Let $z=-2+7i$ and $w=1-1i$ . What is the distance between the points representing $z$ and $w$ on the complex plane?
Answer: $\sqrt{73}$ This question tests your understanding of finding distances between complex numbers on the complex plane using the modulus of their difference. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part: the complex number a + bi is plotted at point (a, b), just like ordered pairs! To find the distance between two complex numbers z and w, we calculate |z - w|, which gives the straight-line distance between their corresponding points. For z = -2 + 7i at point (-2, 7) and w = 1 - 1i at point (1, -1): (1) First find z - w = (-2 + 7i) - (1 - 1i) = -2 + 7i - 1 + 1i = -3 + 8i. (2) Then calculate |z - w| = |-3 + 8i| = √((-3)² + 8²) = √(9 + 64) = √73. (3) This represents the distance between points (-2, 7) and (1, -1) on the complex plane. Choice A correctly gives √73 as the distance. Choice B gives √58, which might come from a calculation error. Choice C gives √10, which is too small for points this far apart. Choice D shows the calculation √(3² + 6²) = √45, but this uses incorrect values—the difference is -3 + 8i, not 3 + 6i! The distance formula for complex numbers mirrors the familiar distance formula from coordinate geometry: d = √((x₂-x₁)² + (y₂-y₁)²). This connection shows how complex numbers unify algebra and geometry, letting us solve geometric problems using algebraic tools and visualize algebraic concepts geometrically!

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