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IB Math AI HL · Topic 1

Number and algebra

Everything in the SL topic — sequences, finance, logarithms, approximation, solving with technology — plus a second layer that changes what 'number' means: complex numbers, matrices and eigenvalues. The algebra is more abstract, the GDC is still allowed, and exact forms matter more than they did at SL.

What the syllabus covers

Fifteen assessed sections: SL 1.1 to 1.8, shared with AI SL, and AHL 1.9 to 1.15, which are HL only.

SL 1.1Standard formOperations with numbers of the form a × 10ᵏ, where 1 ≤ a < 10 and k is an integer.
SL 1.2Arithmetic sequences and seriesnth term and partial sums from the formulae, sigma notation, and real contexts where growth is close to — but not exactly — arithmetic.
SL 1.3Geometric sequences and seriesThe same apparatus with a constant ratio: nth term, partial sums, sigma notation, and their applications.
SL 1.4Financial applicationsCompound interest and annual depreciation, read as geometric sequences in disguise.
SL 1.5Exponents and logarithmsLaws of exponents with integer exponents; logarithms in base 10 and e, evaluated numerically with technology.
SL 1.6ApproximationDecimal places and significant figures, upper and lower bounds of rounded values, percentage errors, estimation.
SL 1.7Amortization and annuitiesLoan repayments and savings plans computed with the GDC's time-value-of-money solver.
SL 1.8Solving equations with technologySystems of up to three linear equations, and polynomial equations, solved on the GDC.
AHL 1.9Laws of logarithmsThe product, quotient and power laws, and writing an expression as a single logarithm.
AHL 1.10Rational exponentsThe laws of exponents extended to fractional powers, and the link to roots.
AHL 1.11Sum of an infinite geometric sequenceThe sum to infinity of a convergent series, and the condition for it to exist.
AHL 1.12Complex numbers in Cartesian formThe number i, real and imaginary parts, conjugates, the four operations, and the complex plane.
AHL 1.13Polar and exponential formsModulus and argument, r(cos θ + i sin θ) and re^{iθ}, converting between forms, and adding sinusoidal waves as complex numbers.
AHL 1.14Matrices and systems of equationsSums, products and scalar multiples, the determinant and inverse, and solving AX = B.
AHL 1.15Eigenvalues, eigenvectors and powers of matricesThe characteristic equation of a 2×2 matrix, eigenvectors, diagonalisation, and Aⁿ.

How it is examined

HL keeps the technology of AI SL, and adds work that has to be done in exact form and written in the right notation.

  • A GDC is allowed on every HL paper. But when a form is stated — 'in the form r ln p', 'as a single logarithm' — the answer is exact, and the calculator only checks it.

  • Complex numbers move between three forms: Cartesian a + bi, modulus–argument r(cos θ + i sin θ), and exponential re^{iθ}. The argument is in radians, and the principal argument lies in (−π, π].

  • Matrices are examined as algebra, not only arithmetic: products in the right order, the determinant and inverse of a 2×2 matrix, and solving AX = B by multiplying by A⁻¹ on the left.

  • Eigenvalue questions follow a fixed route: the characteristic equation det(A − λI) = 0, the eigenvectors for each λ, then A = PDP⁻¹ and Aⁿ = PDⁿP⁻¹.

  • A sum to infinity (AHL 1.11) exists only when |r| < 1. Questions that ask why it exists, or for the range of r that allows it, are testing that condition.

  • The rounding rule is the same as at SL: exact or three significant figures, and a rounded answer is written to the figures requested.

A marked question, mark by mark

A change of form where the method and the number are right, and the unit of the angle is not.

Practise the new algebra on fresh exercises

From SL 1.1 to AHL 1.15, each generated exercise comes back mark by mark, with a reason for each — and complex numbers, matrices and vectors can be answered in the forms a mark scheme accepts.

Start your trial

New to the codes? M1, A1, R1 explained →

Exponential form · AHL 1.13

Seed 9C41

Let z = 1 + √3 i.

  1. (a)Find the modulus and the argument of z.
  2. (b)Hence write z in the form re^{iθ}.

Your work

|z| = √(1² + (√3)²) = 2

arg z = tan⁻¹(√3) = 60°

z = 2e^{i·60°}

Marks awarded

  • M1Mark awarded. Modulus and argument found from the real and imaginary parts.
  • A1Mark awarded. |z| = 2.
  • A1Mark not awarded. The argument is given in degrees. The exponential form needs radians: arg z = π/3, so z = 2e^{iπ/3}.

Where marks are actually lost

The ideas are new at HL, but the marks go the way they always did: on form, order and unit.

Writing the argument in degrees in polar or exponential form.

CostThe exponential form is defined with radians. e^{i·60°} is not the number you mean, and the answer mark for the argument goes.

Taking the argument straight from tan⁻¹ without checking the quadrant.

Costtan⁻¹ returns an angle between −π/2 and π/2. For a number with a negative real part the argument is out by π, and the accuracy mark is lost.

Multiplying matrices in the wrong order, or assuming AB = BA.

CostMatrix multiplication is not commutative. The wrong order gives a different matrix, and every later line follows from it.

Writing the columns of P in a different order from the eigenvalues in D.

CostA = PDP⁻¹ only holds when each column of P matches its eigenvalue on the diagonal of D. Swapping them gives a different matrix.

Other Math AI HL topics

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