IB Mathematics: Analysis and Approaches HL
Numbers and Algebra Formulae AA HL Only
A complete, exam-focused guide to the current higher-level-only Number and Algebra content: counting principles, the generalized binomial theorem, partial fractions, complex numbers, De Moivre’s theorem, proof and systems of linear equations.
Reviewed for the current IB Mathematics: Analysis and Approaches course (first assessment 2021). The IB has announced a revised course for first assessment in May 2029; the short update near the end explains how to avoid mixing the two versions.
The phrase “AA HL only” matters. An IB Mathematics: Analysis and Approaches higher level student studies all of the common SL content and then the additional higher-level material. This guide concentrates on that additional Number and Algebra layer, numbered AHL 1.10 to AHL 1.16 in the current guide. It is not a substitute for sequences, logarithms, financial applications and the other shared content that comes earlier in Topic 1.
The most useful way to revise this topic is not to collect isolated formulas. Learn what each expression means, identify the condition under which it is valid, and practise choosing it from an unfamiliar prompt. In an IB solution, a correct formula with no mathematical reasoning may not earn all available marks. Clear substitution, intermediate steps, exact notation and a conclusion in context are part of the answer.
AA HL Numbers and Algebra syllabus map
The syllabus labels below are useful because teachers, question banks and revision resources often refer to them. The labels are not separate exam compartments: an IB question can combine complex numbers with trigonometry, partial fractions with integration, or induction with sequences. The official guide recommends about 20 additional teaching hours for the AHL part of Topic 1, but the difficulty comes from connections, not simply the amount of content.
| Current syllabus item | Core content | What an exam may ask you to do |
|---|---|---|
| AHL 1.10 | Counting principles; permutations and combinations; binomial theorem for fractional and negative indices | Count arrangements or selections, find an expansion or coefficient, and state the valid interval |
| AHL 1.11 | Partial fractions | Decompose a rational expression and use it in later algebra or integration |
| AHL 1.12 | Complex numbers in Cartesian form; conjugate, modulus, argument and Argand diagram | Calculate, represent and interpret a complex number geometrically |
| AHL 1.13 | Polar/modulus–argument and Euler forms; operations and geometric effects | Convert forms, multiply or divide, and explain rotations and dilations |
| AHL 1.14 | Conjugate roots; De Moivre’s theorem; powers and roots of complex numbers | Find powers, solve equations such as zn=w, or locate roots on a circle |
| AHL 1.15 | Proof by induction and contradiction; disproof by counterexample | Construct a logically complete proof rather than only verify examples |
| AHL 1.16 | Systems of up to three linear equations in three unknowns | Use algebra, row reduction, matrices or technology and interpret the solution |
Before starting HL-only material, make sure your shared SL foundations are secure. Manipulating indices, solving equations, working with sequences, using logarithms and recognizing the ordinary binomial theorem recur throughout this guide. If these feel uncertain, review the AA prior-learning formulae before attempting mixed HL questions.
AHL 1.10: counting principles, permutations and combinations
Counting problems ask how many outcomes are possible without listing them all. The first decision is usually whether stages are combined by multiplication or alternatives are combined by addition. If an outfit consists of one of four shirts and one of three pairs of trousers, there are 4 × 3 = 12 outfits. If a student chooses either one of four art activities or one of three music activities, there are 4 + 3 = 7 choices. Real HL questions often add restrictions, so define the cases before calculating.
Factorials and ordered arrangements
For a positive integer n, n factorial is the product n(n−1)(n−2)…2·1, with 0! defined as 1. Arranging all n distinct objects in a line gives n! possible orders. If only r of the n objects are selected and order matters, use a permutation.
ⁿPᵣ = n!/(n−r)!
The words “arrange,” “rank,” “code” and “assign different positions” often signal that order matters. But do not decide from a keyword alone. Ask whether changing the order creates a genuinely different outcome. Choosing Maya, Leon and Ari for three unnamed committee places is one selection; assigning them president, secretary and treasurer creates six different role allocations.
Unordered selections
When r objects are chosen from n and order does not matter, divide the permutation count by r!, because every group has been counted once for each internal ordering.
Worked example: a committee with a restriction
A committee of four is selected from six Grade 11 students and five Grade 12 students. How many committees contain at least two Grade 12 students?
Split “at least two” into disjoint cases: exactly two, exactly three, or exactly four Grade 12 students.
= 10·15 + 10·6 + 5·1 = 215
The case split is part of the reasoning. Writing only 215 hides why no committee was missed or counted twice.
For “at least one” conditions, a complement is often faster. Count all possible selections and subtract those with none of the required type. For arrangements with repeated objects, first imagine that every object is distinct, then divide by the factorial of each repetition count. That method may appear in a contextual counting problem even when the formula is not the central point.
Counting habits that prevent errors
- State whether order matters and why.
- Keep cases mutually exclusive before adding them.
- Use the complement for “at least one” when direct casework is long.
- Check that r is not greater than n and that your answer is an integer.
- Estimate the scale: a restricted count cannot exceed the unrestricted total.
AHL 1.10: generalized binomial theorem
The shared course introduces finite binomial expansions for non-negative integer powers. At HL, the exponent can be fractional or negative. The resulting expansion is generally infinite, and it is valid only where the transformed variable has absolute value less than 1. The convergence condition is not an optional extra: it is part of the statement.
Here n may be rational, including a negative integer or a fraction. The coefficients are produced by the descending product n(n−1)(n−2)… rather than by treating n as the top of an ordinary combinations button. If the expression is (a+bx)ⁿ, factor out a first:
Worked example: fractional index and interval
Find the first four terms of (1−2x)1/2 and state the interval of validity.
Use n=1/2 and replace x in the general expansion by −2x.
= 1 − x − ½x² − ½x³ + …
Because |−2x|<1, the interval is |x|<1/2, or −1/2<x<1/2.
Worked example: coefficient after multiplication
Find the coefficient of x² in (1+3x)(1−x)−2.
First expand (1−x)−2=1+2x+3x²+4x³+… . The x² term in the product comes from 1·3x² and 3x·2x. Therefore the coefficient is 3+6=9. You do not need to multiply every displayed term.
A calculator can check coefficients numerically, but it does not replace a valid series argument. On a no-technology paper, show the coefficient construction. On a technology paper, retain exact fractions unless a decimal is requested. If an approximation is required, explain why the selected x-value lies inside the interval and mention the order to which the series was truncated.
AHL 1.11: partial fractions
Partial fractions reverse the process of combining rational expressions. A complicated fraction is written as a sum of simpler fractions. In the current AA HL scope, the denominator involves no more than two distinct linear factors, and the numerator has lower degree than the denominator. If it does not, perform polynomial division first.
The symbol ≡ emphasizes an identity: the equality holds for every permitted x, not just for a single solution. Multiply both sides by the original denominator, then compare coefficients or substitute convenient values.
Worked example: cover-up values with verification
Express (5x−1)/[(x−2)(x+1)] in partial fractions.
Set x=2: 9=3A, so A=3. Set x=−1: −6=−3B, so B=2.
Check by recombining: 3(x+1)+2(x−2)=5x−1. This one-line check catches most sign mistakes.
Partial fractions frequently reappear in calculus. After decomposition, each simple rational term may integrate to a logarithm. That is why it is worth learning the structure, not only a shortcut. See the AA HL-only calculus formulae for the integration connection.
AHL 1.12: complex numbers in Cartesian form
A complex number extends the real number system by defining i²=−1. In Cartesian form, z=a+bi, where a is the real part and b is the imaginary part. Write Re(z)=a and Im(z)=b. Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal.
The Argand diagram represents z=a+bi as the point (a,b), or as a vector from the origin. The horizontal axis is the real axis and the vertical axis is the imaginary axis. This picture connects algebra to geometry: addition translates vectors, multiplication changes size and direction, and roots appear as equally spaced points on a circle.
Conjugate, modulus and argument
|z|=√(a²+b²), and arg(z)=θ.
The conjugate reflects a point across the real axis. The product z z̄=a²+b²=|z|² is real, so conjugates are used to simplify division. The modulus is the distance from the origin. An argument is an angle from the positive real axis; the principal argument is written Arg(z) and is normally selected from the convention stated in your course materials.
Worked example: division in Cartesian form
Simplify (3+4i)/(1−2i).
Multiply numerator and denominator by the conjugate 1+2i:
A quick check is to multiply (−1+2i)(1−2i); the result is 3+4i.
Quadratic and polynomial roots
If a polynomial has real coefficients and a+bi is a non-real root, then a−bi is also a root. This conjugate-root result is powerful when constructing a polynomial or finding an unknown root. For example, a real-coefficient cubic with roots 2+i and 2−i must have those roots as a pair. Their quadratic factor is:
Do not apply the conjugate-pair rule automatically if coefficients are complex. The condition “real coefficients” is essential and is often a mark-bearing observation.
AHL 1.13: polar, modulus–argument and Euler forms
Cartesian form is convenient for addition and subtraction. Polar or Euler form is usually better for multiplication, division, powers and roots. If z=a+bi has modulus r and argument θ, then:
a=r cos θ, b=r sin θ, r=√(a²+b²).
To find θ, use the signs of a and b to identify the quadrant. A calculator’s inverse tangent alone may return an angle with the wrong quadrant. An atan2 function can help on a technology paper, but your written answer should still respect the requested range and units. Unless a question states otherwise, radians are the natural language of Euler form and calculus.
Worked example: Cartesian to polar
Write z=−1+√3 i in modulus–argument and Euler forms.
The modulus is √(1+3)=2. The point lies in quadrant II and has reference angle π/3, so θ=2π/3.
Multiplication and division as transformations
(r₁cis θ₁)/(r₂cis θ₂)=(r₁/r₂)cis(θ₁−θ₂), r₂≠0.
Multiplying by r cis θ scales every modulus by r and rotates every argument by θ. Division scales by 1/r and rotates by −θ. This geometric language can turn a long coordinate calculation into a short explanation. It also prepares you for transformations in the AA HL-only functions guide.
AHL 1.14: De Moivre’s theorem, powers and roots
De Moivre’s theorem raises a complex number in polar form to an integer power by raising its modulus and multiplying its argument. The current guide also considers its extension to rational exponents and its use in finding powers and roots.
Worked example: a high power
Evaluate (1+i)8.
Since 1+i=√2 cis(π/4),
Polar form avoids repeated Cartesian multiplication and makes the real result immediately plausible.
Finding all nth roots
Arguments are periodic: θ and θ+2πk describe the same direction. To solve zⁿ=R cis φ, include that full family before dividing the argument by n.
There are n distinct nth roots when R>0. They have the same modulus R1/n and are separated by an angle 2π/n, so they form the vertices of a regular n-gon centred at the origin.
Worked example: cube roots
Solve z³=8i. Write 8i=8 cis(π/2). Then the root modulus is 2 and the arguments are:
Therefore z=2cis(π/6), 2cis(5π/6), or 2cis(3π/2).
Plotting the three answers gives equally spaced points on the circle |z|=2. If your answers are not equally spaced, recheck the 2πk term.
De Moivre can also establish trigonometric identities by expanding (cos θ+i sin θ)ⁿ algebraically and equating real or imaginary parts. In a proof question, write which parts you are comparing and preserve exact coefficients. This connects directly with the AA functions formulae and compound-angle work.
AHL 1.15: mathematical proof
Proof is not a list of examples. A proof gives a chain of reasoning that establishes a statement for every object in the stated domain. The current course includes proof by mathematical induction, proof by contradiction and the use of a counterexample to show that a universal statement is not always true.
Proof by mathematical induction
Induction is appropriate for statements indexed by integers, usually n≥n₀. Think of a ladder: establish the first rung, prove that any occupied rung leads to the next, and then conclude that every rung from the starting point is reached.
- Statement: define P(n) clearly and identify the domain.
- Base case: verify P(n₀).
- Inductive hypothesis: assume P(k) is true for an arbitrary permitted k.
- Inductive step: use that assumption to prove P(k+1).
- Conclusion: invoke induction and restate exactly what has been proved.
Worked example: sum formula by induction
Prove that 1+3+5+…+(2n−1)=n² for every positive integer n.
Base case: for n=1, the left side is 1 and the right side is 1²=1.
Hypothesis: assume 1+3+…+(2k−1)=k² for some positive integer k.
Step: the next odd term is 2(k+1)−1=2k+1. Therefore
Thus P(k) implies P(k+1). Since the base case is true, the identity holds for all positive integers n by mathematical induction.
The sentence “assume true for n=k” is not enough. Your algebra in the inductive step must actually use the hypothesis. Another common error is proving P(k) again instead of P(k+1). Mark the new term explicitly before simplifying.
Proof by contradiction
To prove a statement by contradiction, assume its logical negation. Develop consequences until they conflict with a known fact, a definition or the assumption itself. Then reject the negation and conclude the original statement is true. The contradiction must be named; “this is impossible” without explaining why is incomplete.
Outline: why √3 is irrational
Assume √3=a/b for coprime positive integers a and b. Squaring gives a²=3b², so 3 divides a² and hence divides a. Write a=3c. Substitution gives 9c²=3b², so b²=3c² and 3 divides b. Then a and b share a factor of 3, contradicting the assumption that a/b was in lowest terms. Therefore √3 is irrational.
Disproof by counterexample
A universal claim such as “for every real x…” can be disproved with one valid counterexample. State the proposed value, show that it satisfies the conditions, and calculate why the conclusion fails. It is not sufficient to write a number with no explanation. A counterexample disproves a universal statement; several supporting examples never prove one.
AHL 1.16: systems of linear equations
The current course includes systems with a maximum of three linear equations in three unknowns. You should be able to solve them algebraically and technologically, for example with elimination, row reduction or matrices. A solution is an ordered triple that satisfies every equation simultaneously.
Three planes in three-dimensional space may meet at one point, have no common intersection, or share infinitely many points. Algebraically, row reduction reveals these cases:
| Reduced result | Meaning | Conclusion |
|---|---|---|
| A pivot for every variable | One value each for x, y and z | Unique solution |
| A row such as 0=5 | Contradiction | No solution; system is inconsistent |
| A zero row and at least one free variable | One equation was dependent | Infinitely many solutions, normally parameterized |
Worked example: elimination
Solve x+y+z=6, 2x−y+z=3 and x+2y−z=2.
Subtract the first equation from the second: x−2y=−3. Subtract the first equation from the third: y−2z=−4. Hence x=2y−3 and y=2z−4. Substitute into the first equation:
Then y=2 and x=1, so (x,y,z)=(1,2,3). Verify in all three original equations, not only the last one used.
On a technology paper, a matrix command can give the result quickly. You still need to define the matrix or equations, report the solution correctly and interpret it if the variables have context. Technology is also useful for checking an algebraic solution, but an unexplained screen value may not demonstrate the method requested.
What is in the IB formula booklet—and what must you understand?
The official guide states that all formulae required for the course appear in the mathematics formula booklet and that a clean copy is available during examinations. That does not mean every question becomes a substitution exercise. The booklet does not decide which formula applies, explain restrictions, choose a branch of an argument, organize a proof or interpret a matrix result.
| Area | Booklet support | Your responsibility |
|---|---|---|
| Permutations and combinations | Standard factorial, permutation and combination relationships | Decide whether order matters, build correct cases and handle restrictions |
| Generalized binomial expansion | Coefficient pattern and convergence form | Normalize to 1+u, substitute carefully, find the needed term and state |u|<1 |
| Partial fractions | Notation may support the topic, but decomposition is a method | Choose the correct form, find constants and verify the identity |
| Complex numbers | Key Cartesian/polar/Euler and De Moivre relationships | Choose the correct quadrant, argument range, roots and representation |
| Proof | No formula can supply a logical proof | State assumptions, justify implications and write a valid conclusion |
| Linear systems | General matrix knowledge supports the work | Model the equations, perform or explain the method, and classify the solution |
Use the same formula booklet edition your school supplies. During practice, locate a formula before using it, then close the booklet and explain its variables and conditions aloud. This builds navigation speed without encouraging blind substitution. Do not annotate the examination copy; the official requirement is a clean booklet.
How this topic appears in AA HL exams
Number and Algebra can appear alone or inside a multi-topic problem. A generalized expansion may be used to estimate a value; partial fractions may unlock an integral; complex roots may connect to geometry or trigonometry; induction may prove a result first discovered from a sequence. Paper 3 in particular rewards sustained reasoning across connected steps.
For the current assessment model, AA HL includes external papers with and without technology and an internal mathematical exploration. Always follow the instructions printed on your own paper. A graphic display calculator is a mathematical tool, not a replacement for exact algebra. If a question says “show that,” “prove,” “hence” or “find the exact value,” plan to display a logical route.
A practical method for every structured question
- Decode: underline the command term and write the target beside it.
- Classify: identify the structure—ordered count, infinite expansion, rational decomposition, complex transformation, proof or linear system.
- Choose: select a formula or theorem and write its conditions.
- Execute: keep exact values and show enough intermediate work to make the method visible.
- Check: test restrictions, recombine fractions, substitute a solution, or inspect geometry.
- Conclude: answer the command term with correct notation, units and context.
Common mistakes and fast corrections
Permutation or combination?
Do not choose from a keyword. Ask whether swapping selected objects changes the outcome. If roles or positions are distinct, order matters.
Missing convergence interval
For a generalized binomial series, identify u after factoring and state |u|<1. The condition belongs with the expansion.
Wrong complex quadrant
Sketch the point (a,b) before finalizing the argument. A correct reference angle in the wrong quadrant is still wrong.
Missing complex roots
Use φ+2πk before dividing by n, with k=0 to n−1. Plotting should show equal angular spacing.
Incomplete induction
Include base case, hypothesis, step and conclusion. The step must visibly use the assumed P(k).
Calculator-only systems answer
Define the equations or matrix, show the requested method, classify the solution and verify it in context.
Build accuracy before speed. A strong revision session contains a short concept review, several questions of one type, one mixed unfamiliar problem and an error-log update. Record the cause of each error—concept, algebra, calculator, notation or time—not just the final correction.
Fourteen-day AA HL Numbers and Algebra revision plan
| Days | Focus | Evidence of mastery |
|---|---|---|
| 1–2 | Counting principles, permutations, combinations and restricted cases | You can justify order, create disjoint cases and use complements |
| 3–4 | Fractional and negative binomial expansions | You can find a requested coefficient and derive the valid interval |
| 5 | Partial fractions | You can decompose, recombine and connect to a simple integral |
| 6–7 | Cartesian complex arithmetic, conjugates, modulus and argument | You can calculate and plot without quadrant errors |
| 8–9 | Polar/Euler form, multiplication, division and transformations | You can switch forms deliberately and explain the geometry |
| 10 | De Moivre, powers and all nth roots | You can generate, plot and verify every root |
| 11–12 | Induction, contradiction and counterexample | Your proof has explicit assumptions, justified steps and conclusion |
| 13 | Three-variable linear systems | You can solve algebraically and technologically and identify special cases |
| 14 | Timed mixed set and error-log repair | You can select methods without topic labels and correct recurring errors |
After each two-day block, attempt one previously unseen problem without notes. Mark it against an official-style markscheme, then rewrite only the steps that lost marks. For broader preparation, pair this page with the AA SL/HL calculus formulae and the IB Mathematics AA vs AI comparison guide.
Current syllabus versus the course first assessed in 2029
The current AA course was first assessed in May 2021. The IB has announced an updated course launching in February 2027, first taught from August 2027 and first assessed in May 2029. The published update says the course retains its overall character but removes selected content. For higher level, the announced removals include financial applications, proof by counterexample, bivariate data and Euler’s method. It also adjusts the number of marks on external papers and the duration of Paper 3.
This page intentionally teaches the current AHL 1.10–1.16 sequence. If your final examination is May 2029 or later, confirm your school’s new subject guide and formula booklet before treating any numbered syllabus list as final. The underlying mathematics remains useful, but the assessment boundary may differ. Never combine a 2021-course topic list with a 2029-course assessment table.
Frequently asked questions
What are the AA HL-only Number and Algebra topics?
In the current guide they are AHL 1.10–1.16: counting and the generalized binomial theorem; partial fractions; complex numbers in Cartesian, polar and Euler forms; conjugate roots, De Moivre, powers and roots; proof by induction and contradiction plus counterexamples; and systems of up to three linear equations in three unknowns.
Do AA HL students also need the SL Number and Algebra content?
Yes. HL contains the common SL content plus additional higher-level material. “HL only” means this page focuses on the additional layer, not that the shared content can be skipped.
Are formulas provided in the IB Mathematics formula booklet?
The official guide states that all required formulae are in the mathematics formula booklet and that students receive a clean copy in examinations. You must still recognize the correct formula, understand its conditions and show the reasoning required by the question.
Must I memorize De Moivre’s theorem?
Know what it means and be able to use it fluently even if the relationship appears in the booklet. The difficult part is usually choosing the correct argument, including every root and presenting exact form—not recalling a line of symbols.
Why must a binomial expansion include an interval?
For a fractional or negative exponent the expansion is generally infinite and converges only when the substituted quantity u satisfies |u|<1. If the original expression is not 1+x, factor it first and translate that condition back to x.
Can a GDC solve complex-number and matrix questions?
Technology can calculate or check many results on permitted papers, but it does not supply a mathematical explanation. Retain exact values, define your input, respect branch and angle conventions, and show any method the command term requires.
What is the fastest check for complex nth roots?
All n roots should have the same modulus and equal angular spacing of 2π/n. Substituting each root into zⁿ=w is the definitive algebraic check.
What is the difference between induction and examples?
Examples verify only selected cases. Induction proves a base case and a general implication P(k)⇒P(k+1), so every integer case from the starting value follows.
Is proof by counterexample still assessed?
It belongs to the current course. The IB update says it is removed from the revised course first assessed in May 2029. Check your examination session before deciding what to revise.
How should I practise this topic?
Work by structure first, then mix topics. Explain method choice, keep an error log and complete timed unfamiliar questions. Use official specimen or past-paper material through your school and check your responses against the relevant markscheme.
Official references and next guides
This guide was checked against the official IB Mathematics: Analysis and Approaches guide, the IB’s Diploma Programme mathematics overview, and the official 2029 curriculum update. RevisionTown is an independent educational resource and is not affiliated with or endorsed by the International Baccalaureate Organization.
Continue with Functions Formulae AA HL Only, Calculus Formulae AA HL Only, or compare pathways in the AA vs AI guide.
