Adding and subtracting componentwise, and undoing an addition

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
• Splitting a number into two parts, and when two such numbers agree — real part, imaginary part, and componentwise equality
• Fluent addition and subtraction of signed fractions with unlike denominators
• The meaning of an identity element and an inverse element for an operation, from earlier work on number systems

What they should be able to do

• Add two numbers written in the form a + ib and present the result in that same form
• State why adding two members of this system lands you back inside it, arguing from the closure of real addition rather than by example
• Name the additive identity of the system and show that it leaves any number unchanged
• Write down the additive inverse of a given complex number and verify that the two sum to the identity
• Define subtraction in terms of addition, and use the definition rather than a separate rule
• Show by a printed pair of examples that reversing the order of a subtraction changes the result
• Explain the force of the phrase "for all" in each stated law, and why a single example does not establish one

Where it usually goes wrong

• "Simplify 2 + 3i to 5i, or to 5." The two parts are not addable. Every operation in this chapter keeps them apart, and the equality condition of §4.2 depends on their staying apart.
• "Subtraction of complex numbers is commutative, since addition is." The chapter prints the counterexample itself: the same two numbers subtracted both ways give answers differing in sign in both slots.
• "Closure is trivial and can be skipped." It is the claim that the system is self-contained under the operation. Without it, an addition could in principle produce something outside the system, and every later argument that chains operations together would need re-checking.
• "0 + i0 is a different object from the number zero." It is the identity of this system, and the chapter writes it as 0 for exactly that reason. The real number zero sits inside the system as the number whose second part vanishes.
• "A law is proved by checking an example." Each stated law is quantified over every complex number. An example can refute a law; it cannot establish one. The reason these laws hold is that real addition already has them.
• "The negative of a + ib is a − ib." That is the conjugate, introduced later in §4.4 on p. 81 for a different purpose. The additive inverse changes the sign of both slots, not one.

Questions to check understanding

• Express a sum or difference of two given complex numbers in the form a + ib, with fractional or negative parts
• Write down the additive inverse of a given complex number and verify it
• Name the additive identity and state the property that characterises it
• One-mark items on whether a stated property holds for all complex numbers
• Give a counterexample showing that subtraction does not commute
• Multi-term items combining three or more numbers with mixed signs and brackets, where the associative law is what licenses regrouping

Examples worth working on the board

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

• The addition rule (§4.3.1, p. 77). For a + ib and c + id, the sum has a + c in the first slot and b + d in the second. The rule and the closure claim are stated together on the page, in that order.
• The printed sum (§4.3.1, p. 77): (2 + i3) + (−6 + i5) gives −4 + i8. Verified: 2 + (−6) = −4 and 3 + 5 = 8. Use this as the rule running once in each slot, not as a result to be memorised.
• The additive identity and inverse (§4.3.1, p. 77). The identity is written 0 + i0 and abbreviated 0. The inverse of a + ib is the number with −a in the first slot and −b in the second. Verified: adding them gives 0 + i0 in both slots at once, which is the identity.
• The subtraction definition (§4.3.2, p. 77). The difference of two numbers is defined as the first plus the negative of the second. There is no separate subtraction rule on the page; there is only this reduction.
• The order-matters pair (§4.3.2, p. 77), which is the section's best asset because both halves are printed: (6 + 3i) − (2 − i) gives 4 + 4i, and (2 − i) − (6 + 3i) gives −4 − 4i. Verified: first slots 6 − 2 = 4 and 2 − 6 = −4; second slots 3 − (−1) = 4 and −1 − 3 = −4. Same two inputs, and each slot of the answer changes sign when the order is swapped.
• Exercise 4.1 items that are pure addition and subtraction (p. 83). Q5: (1 − i) − (−1 + i6). Q6: (1/5 + i 2/5) − (4 + i 5/2). Q7: [(1/3 + i 7/3) + (4 + i 1/3)] − (−4/3 + i). Verified: Q5 gives 2 − 7i, since 1 − (−1) = 2 and −1 − 6 = −7. Q6 gives −19/5 − (21/10)i, since 1/5 − 4 = −19/5 and 2/5 − 5/2 = −21/10. Q7 gives 17/3 + (5/3)i, since 1/3 + 4 + 4/3 = 17/3 and 7/3 + 1/3 − 1 = 5/3. These are worked out here on the printed items; the chapter prints no answers on these pages.
• A closure demonstration to show. Take any two of the exercise items and note that each answer, however ugly the fractions, is again one real number plus i times another real number. Verified on Q6, whose parts −19/5 and −21/10 are both perfectly ordinary rationals. The point is that the shape survives, not that the numbers are tidy.
• A non-example worth showing. Ask what "2 + 3i" collapses to as a single number. Verified: nothing — 2 and 3 sit in different slots and there is no rule in the chapter that combines them, which is exactly why the form has two slots. This non-example is added here.

Figures to have open

• A two-track adder: two horizontal real number lines stacked, one for each slot, with the two inputs marked and the sum landing on each. Standard schematic; this is the explanation's device for making "twice over" visible. The chapter has no figure in §4.3.
• A slot-aligned template pair for the order-matters demonstration, with sign changes highlighted in both slots. Standard schematic.
• A short table of the five additive properties with a one-line reason column pointing back to the corresponding property of real addition. Standard schematic; the reason column is added here, not the book's.
• No textbook figure is required for this topic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 4, whose printed title is Complex Numbers and Quadratic Equations; §4.3 Algebra of Complex Numbers with §4.3.1 and §4.3.2, p. 77.
• Exercise 4.1, items 5, 6 and 7, p. 83.
• Summary, p. 87, which restates the addition rule alongside the multiplication rule.
• Forward pointer inside the same chapter: the conjugate, easily confused with the additive inverse, is §4.4 on p. 81.