Which school algebra identities survive the enlargement, and why

What to assume they know

• Multiplying, then inverting: how division becomes possible — multiplication, and the distributive and commutative laws as stated for complex numbers in §4.3.3
• Adding and subtracting componentwise, and undoing an addition — addition and the way the two slots behave
• The expansion and factorisation identities for squares, cubes and a difference of two squares, from earlier algebra
• What it means for a statement to be quantified over every member of a system

What they should be able to do

• State the identity the chapter proves in full, for arbitrary complex numbers
• Follow the printed proof and name, for each line, the law that permits it
• Explain why the same argument establishes the corresponding identity with a minus sign
• State the four further identities the chapter asserts without proof
• Give the reason those four also hold, without reproving each one
• Apply a cube identity to a numerical case in which one of the two terms carries the symbol i
• Use the difference-of-two-squares identity to turn a complex denominator into a real one
• Say what kind of real-number statement does not transfer to this system, and why the transfer argument does not reach it

Where it usually goes wrong

• "These are new identities that happen to look like the old ones." They are the same identities. Only the justification changed, from "true for reals" to "derivable from laws the chapter has granted for complex numbers".
• "The printed proof is a formality; the identity is obvious." It is the only derivation in §4.3 set out under a proof heading, and it is what licenses the other four. Skipping it leaves a student unable to say why any of them hold.
• "Everything true of real numbers is true of complex numbers." Not so. The transfer works for statements derivable from the laws of §4.3 alone. A statement about one number being greater than another is not among them, because this chapter never sets up such a comparison between two complex numbers in the first place.
• "The cross terms cancel because of i." They do not cancel; the middle term is twice the product, exactly as for real numbers. Commutativity is used to combine the two middle terms, not to remove them.
• "For the cube I should multiply the bracket out three times." The identity is faster and less error-prone, and Example 3 shows it applied with the second term carrying i.
• "A fourth power needs a fourth-power identity." Square, then square again. Exercise 4.1 item 8 is built for this.

Questions to check understanding

• Expand a square or a cube of a two-term complex expression and give the result in the form a + ib
• Evaluate a fourth power by squaring twice
• Prove the identity for the square of a difference, naming the law used at each step
• Use a difference of two squares to turn a complex denominator into a real one
• Short-answer items asking which law justifies a nominated line of a printed derivation
• Multi-step items where an identity has to be recognised inside a longer expression before it can be used

Examples worth working on the board

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

• The proved identity (§4.3.7, p. 80). The square of a sum of two complex numbers equals the sum of their squares together with twice their product, asserted for every pair of complex numbers. It is the only general statement in §4.3 that the chapter sets out under a proof heading.
• The proof's skeleton (§4.3.7, p. 80), which is the content of this topic. Write the square as the product of the bracket with itself; expand once by distributing the bracket over the two terms; expand each of those by distributing again, which produces four terms; use commutativity of multiplication to see that the two middle terms are the same thing; collect them into twice the product. Verified as a valid derivation: three of the four steps are labelled on the page with the law they use, and every law they use was stated for complex numbers earlier in §4.3.
• The four asserted identities (§4.3.7, p. 80): the square of a difference; the cube of a sum; the cube of a difference; and the factorisation of a difference of two squares into a product of a sum and a difference.
• The chapter's closing claim (§4.3.7, p. 80): many further identities true for real numbers can be shown to hold for complex numbers as well. The explanation's job is to supply the missing because — because their proofs use only the laws §4.3 has already granted.
• The stress test. Verified by inspection of the proof: it never divides, never takes a square root, never compares two quantities as larger and smaller, and never uses that a square is non-negative. It uses only rearrangement. That is why nothing about it is specific to the real line.
• Example 3 (p. 81). Input: the cube of (5 − 3i). Verified using the cube-of-a-difference identity with the two terms 5 and 3i: the first term cubed is 125; three times the square of the first times the second is 225i, entering with a minus; three times the first times the square of the second is 15 × 9i², which is −135; the cube of the second is 27i³, which is −27i, entering with a minus and so contributing +27i. Collecting, the first slot is 125 − 135 = −10 and the second is −225 + 27 = −198. The chapter reaches the same pair of values.
• Exercise 4.1 items driven by these identities (p. 83). Q8: the fourth power of (1 − i). Q9: the cube of (1/3 + 3i). Q10: the cube of (−2 − i/3). Verified: Q8 gives −4 + i0, because the square of (1 − i) is −2i and squaring that gives 4i², which is −4 — a fourth power is fastest as a square of a square. Q9 gives −242/27 − 26i, since the four cube terms are 1/27, i, −9 and −27i. Q10 gives −22/3 − (107/27)i, since the number is the negative of (2 + i/3), whose cube has first slot 8 − 2/3 and second slot 4 − 1/27.
• The difference of squares clearing a denominator (Example 5, p. 82). Input: the second route printed there multiplies above and below by 2 + 3i and reads the denominator as a difference of two squares. Verified: 2² − (3i)² is 4 − 9i², that is 4 + 9 = 13. The identity is doing the work of turning a complex denominator into a real one, which is why this topic sits just before the modulus and conjugate.

All values marked verified are worked out here on the printed items; the chapter prints no answers on these pages.

Figures to have open

• A proof ladder: the identity at the top, four rungs below it, each rung carrying the law that justifies it in a side column. Standard schematic, and the core image of this topic; the chapter prints the proof as running text with the laws in brackets.
• A transfer diagram: a box of granted laws, an arrow to a box of consequences, with the four asserted identities inside the second box and an order relation drawn outside both. Standard schematic; the diagram is an added argument.
• A term-by-term expansion panel for a cube of a two-term bracket, reusable for Example 3 and for Exercise 4.1 items 9 and 10. Standard schematic.
• 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; §4.3.7, p. 80, for the proved identity, the proof, the four further identities and the closing claim.
• Example 3, p. 81, for a cube worked with the identity.
• Example 5, p. 82, for a difference of squares used on a denominator.
• Exercise 4.1, items 8, 9 and 10, p. 83.
• Backward pointers inside the same chapter: the distributive and commutative laws the proof cites are stated in §4.3.3 on p. 78.