Splitting a number into two parts, and when two such numbers agree

What to assume they know

• An equation with no real solution, and the symbol invented to solve it — the form a + ib, and the relation i² = −1
• That a real number is positive, negative or zero, and that its square is never negative
• Solving two simultaneous linear equations in two unknowns
• Handling improper fractions and negative coefficients confidently

What they should be able to do

• Given a number in the form a + ib, state its real part and its imaginary part, and say that both are real numbers
• Explain why the imaginary part is a real number and does not carry the symbol i
• State the condition under which two complex numbers count as equal
• Justify componentwise equality by an argument, not by citing the definition
• Convert a single equation between complex expressions into two real equations by matching the parts
• Solve the resulting pair of real equations, including cases where one unknown must be substituted into the second equation
• Recognise, in a longer expression, which terms belong to the real slot and which to the imaginary slot before any simplification is attempted

Where it usually goes wrong

• "The imaginary part of 2 + i5 is 5i." It is 5. The symbol i is part of the notation for the form, not part of the value being named. Students who carry the i into Im z produce a complex-valued "part" and then cannot compare parts at all.
• "Equality of complex numbers is just one equation, like for reals." It is two. Losing the second is how a solvable two-unknown problem silently becomes an underdetermined one.
• "You can balance a shortfall in one part against a surplus in the other." The squaring argument in section 7 shows why not: it would require a real square to equal minus a real square.
• "Re and Im can be read straight off any expression." Only off the standard form. In 3 + i(2 + i) the visible 3 is not the real part.
• "Im z is somehow not a real number, because of the name." Both parts are real numbers. The name records the history of the notation, as the chapter's Historical Note on pp. 87–88 makes clear, not a property of the value.

Questions to check understanding

• State Re z and Im z for a given number, including cases where the imaginary part is negative, fractional or irrational
• Given an equation between two complex expressions with real unknowns, find the unknowns by matching parts
• Two-mark trap items that reward writing the imaginary part without the symbol i
• Simplify an expression to standard form first, then report its two parts
• Prove a short identity about the real part of a product, working from the multiplication rule of §4.3.3

Examples worth working on the board

Items marked verified are worked out here from the chapter's stated data.

• The naming example (§4.2, p. 76). For z = 2 + i5 the chapter states the real part as 2 and the imaginary part as 5. Note: the imaginary part is 5, not 5i — the printed value carries no symbol.
• The three sample numbers (§4.2, p. 76): 2 + i3; (−1) + i√3; 4 + i(−1/11). Verified by reading the form: real parts 2, −1, 4; imaginary parts 3, √3, −1/11. Use the third to make the point that an imaginary part can be negative and fractional, and the second to make the point that it can be irrational.
• The equality condition (§4.2, p. 76). Two numbers written a + ib and c + id count as equal exactly when a = c and, separately, b = d.
• Why it has to be componentwise — an added derivation, not printed in the chapter, which simply defines equality. Suppose a + ib and c + id are equal, with all four letters real. Then a − c and b − d satisfy a − c = i(d − b). Square both sides: the left is (a − c)², the right is i²(d − b)², which is −(d − b)². Verified: the left side is a real square, so it is zero or positive; the right side is minus a real square, so it is zero or negative; the only way both hold at once is that each is zero, giving a = c and b = d. Be exact about what this argument costs and what it earns. It subtracts one complex number from another and squares a complex expression, so it borrows the whole of §4.3, pp. 77–78 — and both of those operations are themselves defined slot by slot. Nor can it be a derivation of p. 76's definition, since in the book's order that definition is what licenses the hypothesis it starts from. What it does establish, granted an arithmetic in which i² = −1 and the ordinary laws hold, is that a number's two-part form is unique — so the componentwise reading is the only one the chapter's own algebra will tolerate.
• Example 1 (p. 77). Inputs exactly as printed: the equation 4x + i(3x − y) = 3 + i(−6), with x and y both real. Verified: matching the first slot gives 4x = 3, so x = 3/4; matching the second gives 3x − y = −6, so y = 3x + 6 = 9/4 + 24/4 = 33/4. The chapter prints these same two values, so the explanation can show the derivation and then point at the page.
• A deliberate near-miss for section 10. Ask what Re and Im are for the expression 3 + i(2 + i). Verified: i(2 + i) is 2i + i², which is −1 + 2i, so the number is 2 + 2i, giving Re = 2 and Im = 2 — not 3 and 2 + i. The lesson is that the parts can only be read once the expression is in the standard form. This example is added here, not the chapter's.
• Where the move is reused (Miscellaneous Exercise on Chapter 4, p. 86, Q8): two real unknowns x and y are wanted, subject to one condition — the product of (x − iy) with (3 + 5i) has to come out equal to whatever the conjugate of −6 − 24i turns out to be. Hand this over as an input only; it needs the conjugate from §4.4, and it is settled by exactly the matching step taught here.
• A statement whose proof is this move in reverse (Miscellaneous Exercise, p. 85, Q2): show that the real part of a product equals Re z₁ Re z₂ minus Im z₁ Im z₂. Verified against §4.3.3: the product rule places ac − bd in the first slot, and ac is Re z₁ Re z₂ while bd is Im z₁ Im z₂.

Figures to have open

• A two-slot template for a + ib with the Re and Im labels attached, reusable across the whole chapter. Standard schematic; the chapter has no figure here.
• A side-by-side panel for the equality condition: two templates stacked, with a tick on each slot pair independently. Standard schematic.
• A step panel for the squaring argument, with the left side shaded as never-negative and the right side shaded as never-positive, meeting at zero. Standard schematic; this argument is added here, not the book's.
• No textbook figure is required for this topic. Fig 4.1 to Fig 4.3 belong to §4.5 and are handled in Reading the plane: real axis, imaginary axis, and a mirror image for the conjugate.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 4, whose printed title is Complex Numbers and Quadratic Equations; §4.2 Complex Numbers, p. 76, for the parts and the equality condition.
• Example 1, p. 77, for the worked matching step.
• Summary, p. 87, for the compact restatement of the two parts.
• Miscellaneous Exercise on Chapter 4, p. 85 Q2 and p. 86 Q8, for later uses of the same move.