AA Topic 1: number and algebra

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Shared with Applications

Both guides state 1.1 to 1.5 as content common to the two courses, so these are the same pages Applications students use rather than a second copy that could drift.

1.1 · Standard form SHARED a × 10^k, the two conditions, and the arithmetic. 10⁶ is a thousand times 10³, three short steps along. 1.2 · Arithmetic sequences SHARED The nth term, both sum formulae, and working backwards. The off-by-one formula lands one term late for ever. 1.3 · Geometric sequences SHARED The nth term and the sum, with the ratio found by dividing. Same first two terms, 51 times apart by n = 10. 1.4 · Compound interest SHARED Growth, depreciation, and real value after inflation. It grew to 1628.89 and buys 1212.03. 1.5 · Exponents and logarithms SHARED The index laws, and a logarithm as the answer to "what power?". One point on y = 2^x and its reflection.

Analysis and Approaches, Standard Level

1.6 · Deductive proof Numerical and algebraic proof, and laying out left-hand side to right-hand side. A formula gives a prime thirty-nine times running, then fails. 1.7 · Rational exponents SAME CONTENT Fractional and negative powers. The laws of logarithms, which AA also meets here, are on their own page. 16 to the power a half is 4, and the curve never crosses the axis. 1.8 · The sum to infinity SAME CONTENT When a geometric series converges, and the condition |r| < 1. A ball bounces for ever and travels exactly 18 metres. 1.9 · The binomial theorem Pascal's triangle, ⁿC_r, and finding one term without expanding. Every number is the two above it, so a forgotten row costs twenty seconds.

Higher Level

1.10 · Counting principles HL Permutations, combinations, and the binomial theorem extended to fractional and negative indices. Ten choices fan out into sixty arrangements. The gap is exactly 3!. 1.11 · Partial fractions HL Splitting a fraction over two distinct linear factors. One awkward curve is two simple ones added, each owning an asymptote. 1.12 · Complex numbers SAME CONTENT Cartesian form, conjugate, modulus and argument, and the four operations. Multiply by i four times and walk round a square. 1.13 · Polar and Euler form SAME CONTENT cis and exponential form, and converting between all three. Multiply the lengths, add the angles. 1.14 · Complex roots and de Moivre HL Conjugate pairs, powers, and the nth roots of a complex number. Find one root and the rest are rotations of it. 1.15 · Proof by induction HL The four-part structure, for sums and for divisibility. Remove the base case and nothing falls, however good the step is. 1.16 · Systems of linear equations HL Row reduction, and all three possible outcomes. 0 = 0 is not a mistake. It means the solutions form a whole line.

Where the two courses part company in Topic 1

	Analysis and Approaches	Applications and Interpretation
proof and induction	yes, 1.6 and HL 1.15	not on the course
binomial theorem	SL, 1.9	not on the course
partial fractions	HL, 1.11	not on the course
de Moivre and nth roots	HL, 1.14	not on the course
amortization and annuities	not on the course	SL, 1.7
matrices and eigenvalues	not on the course	HL, 1.14 and 1.15
approximation and bounds	not on the course	SL, 1.6

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