# Read complex multiplication in Book Isle: expand four products before replacing i squared Language: en Canonical: [Read complex multiplication in Book Isle: expand four products before replacing i squared]() 中文：[中文 HTML]() English: [English HTML]() Published: 2026-11-15T00:00:00+08:00 Modified: 2026-11-15T09:48:08+08:00 Author: Bison Check a complex product, real and imaginary parts and conjugation without treating i as a cancellable unit label. English translation of the [Chinese original](), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained. In complex numbers i is not a unit label like metres or seconds. Multiplying two i terms changes the real part. Read original cards in Book Isle, expand before combining and check with conjugation and multiplication by i. Calculate on paper or in external notes; displayed mathematics is not automatic computation, and a complex-plane sketch supplies no real circuit measurement. ## Fix two numbers and the imaginary-unit rule Use `z=2+3i`, `w=1-2i` and `i^2=-1` in the complex numbers. Record source, example, signs and brackets. Separate real parts and coefficients of i: two/three and one/minus two. i is a defined imaginary unit, not an arbitrary variable or removable physical-unit label. Open readable text and preserve the original conditions rather than overwrite them while solving. ## Write all four products Expand into two, minus four i, plus three i and minus six i squared. Check each factor meets both terms instead of omitting cross-products. The last coefficient minus six and i squared come separately from three i times minus two i; it is not simply minus six. A paper four-cell multiplication aid is optional, without an app-generated table. Cells are products, not four independent answer points in the complex plane. ## Replace i squared and track both signs Minus six i squared becomes minus six times minus one, plus six; with two it gives eight. Minus four i plus three i gives minus i, so the product is eight minus i. Show the replacement rather than lose the first minus sign. In standard form the real part is eight and imaginary coefficient minus one; distinguish that coefficient from the whole term minus i. Do not drop i and write eight minus one. ## Keep addition separate from multiplication Adding the same z/w gives three plus i, distinct from eight minus i. Label the operation so two inputs and a result do not lose their relationship. Reversing factor order should preserve this complex product and can check signs, but does not change w’s minus to plus. One example checks arithmetic; general commutativity comes from the rules, not that single trial. ## Distinguish conjugation from negating everything The conjugate of z is two minus three i, not minus two minus three i. Multiplying z by its conjugate cancels cross-terms and gives four plus nine, thirteen, not the original zw because w is different. The conjugate of eight minus i is eight plus i, also obtained by multiplying conjugates of both original inputs. Label identities and purposes so thirteen does not replace the original answer simply because another calculation contains two and three. Also check squared moduli: z gives thirteen, w five and eight minus i sixty-five, equal to thirteen times five. Its modulus is the square root of sixty-five, not sixty-five itself. ## Check multiplication by i as a separate control Separately i times z equals two i plus three i squared, minus three plus two i. With real part horizontal and i coefficient vertical, (two,three) becomes (minus three,two), a counterclockwise quarter-turn in standard coordinates. Screen-down y requires a separate display conversion, not altered algebra. This is not zw and requires no interactive complex-plane rotation. Paper points can be checked against the text. ## Check missing cross-terms and ambiguous glyphs A result minus four suggests only real-real and coefficient-coefficient products were used. Minus four minus i suggests a double-sign error replacing i squared. Enlarge an ambiguous i/one glyph rather than guess. Revisit conjugate versus whole negation using both components, not colours alone. Share the original inputs and expansion rather than a full private textbook. External notation suffices without a special keyboard or invented complex calculator/device test. ## Reopen and verify three calculation chains Reopen, verify minus two i and recover eight minus i from four products. Then check conjugate product thirteen and i times z, minus three plus two i. Retain inputs, operation and substitution for each, explaining why thirteen is not zw. Recalculate changed signs rather than overwrite new text with old screenshots. Bookmarks and opening words locate the source; another reader should trace signs and components without automatic mathematical interpretation. ## Related guides and sources [Book Isle binomial expansion: map coefficients, exponents and substitution checks](); [Follow worked examples in Book Isle: track symbols, units and substitutions](). Checked on November 15, 2026: [Apple public information]() lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the [official App Store]() for published capabilities; verify controls in the installed version. Method reference: [OpenStax: complex numbers and multiplication](). Materials and numbers are original exercises. Record external reports, paper reasoning and actual reading observations separately, without inventing missing conditions. Sources do not establish new automatic analysis by the app. No step-by-step physical-device, private-server or all-output-environment testing is claimed.