Reading the plane: real axis, imaginary axis, and a mirror image for the conjugate

What to assume they know

• Splitting a number into two parts, and when two such numbers agree — real part and imaginary part, and componentwise equality
• Size and reflection: two quantities that turn algebra into geometry — the modulus as a sum of two squares under a root, and the conjugate as a single sign change
• Plotting a point from its coordinates on a pair of perpendicular axes, and the distance of a point from the origin
• That the coordinate plane's two axes are set at right angles and meet at the origin

What they should be able to do

• Convert a complex number into a coordinate pair and back
• Plot given complex numbers as points and read given points back as numbers
• Name the plane in which this is done, using both names the chapter prints
• Explain why the modulus of a number equals the distance of the corresponding point from the origin, rather than merely asserting it
• Identify which complex numbers correspond to points on each of the two axes, and name those axes
• Locate the point matching the conjugate of a given number, and state the relationship between the two points
• Say precisely where this section of the book stops

Where it usually goes wrong

• "The Argand plane is a special new plane, different from the coordinate plane used in geometry." It is the same plane. What is new is that each point now has a complex number attached to it.
• "Points on the vertical axis are not really numbers." They are complex numbers whose first part is zero, and the coordinate measured along that axis is an ordinary real number. The axis is named for the part it carries, not for the kind of quantity marked on it.
• "The conjugate reflects across the vertical axis." It reflects across the horizontal one. The sign that changes is the one attached to i, which is the vertical coordinate, so the point moves vertically and the horizontal coordinate is untouched.
• "Every complex number sits off both axes." The chapter's own six points include one on each axis. Those are the cases where one part vanishes.
• "A number and its conjugate are always different points." Not when the second part is zero. The real numbers are exactly the points the reflection leaves where they are.
• "The modulus formula is a coincidence that happens to look like the distance formula." They are the same formula. The correspondence is what makes them so.
• "The section will give me the polar form, because the heading says so." It will not. See the last section and the notes below.

Questions to check understanding

• Plot given complex numbers on a labelled pair of axes and read given points back as numbers
• State which axis a given number lies on, and the condition on its parts
• Given a number, mark the point of its conjugate and describe the relationship in one line
• Find the distance from the origin of the point matching a given number
• Identify the complex numbers that are unchanged by conjugation, using the picture
• One-mark items on the two names the chapter gives the plane

Examples worth working on the board

Items marked verified are worked out here from the chapter's stated data; details marked read off the printed page were taken from the page image, because the lettering sits inside the artwork.

• Fig 4.1 (§4.5, p. 83). Read off the printed page: a pair of perpendicular axes with the horizontal one labelled at both ends and the vertical one labelled at both ends, meeting at a point marked O. Both axes carry evenly spaced unit tick marks on either side of the origin, and every point is plotted against that scale — checked on the printed page, where D lies two ticks along the horizontal axis, C one tick up the vertical, and E five ticks left and two down. The scale is not decoration: this figure is the only one of the three that has one. Six plotted points, each carrying its letter and its coordinate pair inside the artwork: A at (2, 4), B at (−2, 3), C at (0, 1), D at (2, 0), E at (−5, −2) and F at (1, −2). The running text lists the matching complex numbers in the same order: 2 + 4i, −2 + 3i, 0 + 1i, 2 + 0i, −5 − 2i and 1 − 2i.
• The correspondence (§4.5, p. 83). Any two reals, taken in order, fix one point of the plane, and any point fixes one such pair; the number carrying those two parts is then attached to that point.
• The two names (§4.5, p. 83). The chapter gives the plane two names in one sentence, and both are examinable.
• Fig 4.2 (§4.5, p. 84). Read off the printed page: a bare pair of axes — no tick marks anywhere, so this is not Fig 4.1's frame — the origin labelled O and additionally marked with the pair (0, 0), a single point P in the upper right region labelled with its two coordinates, and a segment drawn from the origin to P carrying the square root of the sum of the two squared coordinates written along it. No angle is drawn at the origin and no arc appears anywhere in the figure.
• Why the two formulas coincide. Verified: the distance of a point from the origin is the square root of the sum of the squares of its two coordinates, and the modulus of §4.4 is the square root of the sum of the squares of the two parts. Under the correspondence the coordinates are the parts, so the two expressions are the same expression with different names on the letters. That is the reason, and it is stronger than observing that the answers match.
• The two axes (§4.5, p. 84). The horizontal axis holds exactly the numbers whose second part is zero; the vertical axis holds exactly those whose first part is zero. The chapter names them on this page. Verified against Fig 4.1: D, at (2, 0), matches 2 + 0i and sits on the horizontal axis; C, at (0, 1), matches 0 + 1i and sits on the vertical one.
• Fig 4.3 (§4.5, p. 84). Read off the printed page: a bare pair of axes again, unticked like Fig 4.2's, with the origin marked O, a point P above the horizontal axis labelled with its coordinates, a point Q below it labelled with the same first coordinate and the negated second, and three drawn segments joining the origin to P, P to Q, and Q back to the origin, so the figure closes into a triangle. The segment from P to Q is vertical and is cut in half by the horizontal axis, which is what makes the reflection visible.
• The reflection statement (§4.5, p. 84). The point of the conjugate is the mirror image, in the horizontal axis, of the point of the number.
• Distances for the six points — arithmetic added here, not the chapter's. Verified: the moduli of the six numbers are the square roots of 20, 13, 1, 4, 29 and 5 respectively, so A lies 2√5 from the origin, C lies exactly 1 away and D exactly 2 away. Putting these on Fig 4.1 turns a plotting exercise into a distance exercise at no extra cost.
• Conjugates of the six points — again an added extension. Verified: the conjugate of A's number is 2 − 4i, sitting at (2, −4), directly below A and the same distance from the horizontal axis; the conjugate of D's number is D itself, because its second part is already zero. That second case is worth showing: the points fixed by the reflection are exactly the points of the real axis.

Figures to have open

• Fig 4.1 redrawn as a clean schematic: two perpendicular axes, unit tick marks on both of them either side of the origin, and the six points plotted to that scale with their letters and coordinate pairs. This is the chapter's own figure (p. 83) and the section's data lives inside it; redraw rather than reproduce, and keep all six labels, since the lettering carries the pairing between the points and the listed numbers. Keep the ticks too — the overlay figure below writes distances on radial segments, and a distance means nothing on an unscaled frame.
• Fig 4.2 redrawn: axes, origin marked with its coordinate pair, one point in the upper right, and the joining segment labelled with the modulus expression (p. 84). Redraw as a schematic. Do not add an angle at the origin — the printed figure has none, and adding one would illustrate material this book does not carry.
• Fig 4.3 redrawn: a point above the horizontal axis, its reflection below, the three joining segments, and a tick showing the axis cutting the vertical segment in half (p. 84). Redraw as a schematic.
• An overlay of the six points of Fig 4.1 with their distances from the origin written on the radial segments. Standard schematic; the distances are arithmetic added here, not the chapter's.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 4, whose printed title is Complex Numbers and Quadratic Equations; §4.5, pp. 83–84, with Fig 4.1 on p. 83 and Fig 4.2 and Fig 4.3 on p. 84.
• Backward pointers inside the same chapter: the modulus and the conjugate are defined in §4.4 on p. 81; the two parts of a number are named in §4.2 on p. 76.
• Historical Note, pp. 87–88, for Hamilton's treatment of the form a + ib as a pair taken in order, which is the same identification this section draws.