Knowledge organisers

Number and algebra: IB Maths AA HL knowledge organiser

Everything to know about number and algebra on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Complex number: z = a + bi, where i² = −1; a is the real part and b the imaginary part.
Modulus and argument: |z| is the distance of z from the origin on an Argand diagram; arg z is the angle it makes with the positive real axis.
Proof by induction: Show a statement is true for the first case, then that true for n = k means true for n = k + 1.
Permutation: An arrangement where order matters: ⁿPᵣ = n!/(n − r)!.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

ArithmeticIn the formula booklet	\(u_n=u_1+(n-1)d,\) \(S_n=\tfrac n2\big(2u_1+(n-1)d\big)=\tfrac n2(u_1+u_n)\)
GeometricIn the formula booklet	\(u_n=u_1r^{n-1},\) \(S_n=\frac{u_1(1-r^n)}{1-r},\) \(S_\infty=\frac{u_1}{1-r}\ (\|r\|<1)\)
Compound interestIn the formula booklet	\(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
Exponents & logsIn the formula booklet	\(a^x=b\iff x=\log_ab,\) \(a^x=e^{x\ln a},\) \(\log_aa^x=x=a^{\log_ax}\)
Log lawsIn the formula booklet	\(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^m=m\log_ax\)
Change of baseIn the formula booklet	\(\log_ba=\frac{\log_ca}{\log_cb}\)
Binomial, \(n\in\mathbb N\)In the formula booklet	\((a+b)^n=\sum_{r=0}^n\tbinom nra^{n-r}b^r,\) \(\tbinom nr=\frac{n!}{r!(n-r)!}\)
PermutationsIn the formula booklet	\({}^nP_r=\frac{n!}{(n-r)!}\)
Binomial, \(n\in\mathbb Q\), \(\|x\|<1\)In the formula booklet	\((1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\cdots\)
\((a+bx)^n=a^n\left(1+\tfrac bax\right)^n\), valid for \(\|x\|<\left\|\tfrac ab\right\|\)
Partial fractions	\(\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\)
Complex numbersIn the formula booklet	\(z=a+bi=r(\cos\theta+i\sin\theta)=re^{i\theta}=r\,\mathrm{cis}\,\theta\)
Modulus, argument, conjugate	\(r=\|z\|=\sqrt{a^2+b^2},\) \(\tan\theta=\tfrac ba,\) \(z^=a-bi,\) \(zz^=\|z\|^2\)
Products	\(\|z_1z_2\|=\|z_1\|\|z_2\|,\) \(\arg(z_1z_2)=\arg z_1+\arg z_2,\) \(\arg\tfrac{z_1}{z_2}=\arg z_1-\arg z_2\)
De MoivreIn the formula booklet	\(\big[r(\cos\theta+i\sin\theta)\big]^n=r^n(\cos n\theta+i\sin n\theta)=r^ne^{in\theta}\)
\(n\)th roots of \(re^{i\theta}\)	\(r^{1/n}e^{i(\theta+2\pi k)/n},\) \(k=0,1,\dots,n-1\)

More formulas are on the full IB Maths AA HL formula sheet.

Worked example

Write z = 1 + i√3 in the form re^(iθ) and hence find z³.

r = √(1² + (√3)²) = 2 and θ = arctan(√3/1) = π/3
z = 2e^(iπ/3), so z³ = 2³e^(iπ) = 8 × (−1)

Answer: z = 2e^(iπ/3) and z³ = −8

Common mistakes

Proof by induction: weak structure and "let n = k"
Complex numbers and polynomials: conjugate roots, all the roots, general solutions
Confusing common difference d with common ratio r
Using the arithmetic sum formula for a geometric series

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Recognise an arithmetic sequence from its common difference and use uₙ = u₁ + (n − 1)d to find a term, the first term or the number of terms.
Simplify with the laws of exponents, including negative and fractional powers, and solve exponential equations by matching bases or taking logarithms.
Expand expressions such as (1 + x)⁻¹ and (4 − x)^(1/2) as infinite series, stating the values of x for which each expansion is valid.
Plot complex numbers on an Argand diagram, find the modulus and argument, and convert between Cartesian and r(cos θ + i sin θ) forms.
Count arrangements with restrictions, such as items together, apart or in fixed positions, using ⁿPᵣ and ⁿCᵣ.
Prove results such as the irrationality of √2 by contradiction, and disprove false statements by finding a single counterexample.

The printable sheet

Number and algebra knowledge organiser for IB Maths AA HL: one A4 page of key definitions, formulas, a worked example and common mistakes — Number and algebra knowledge organiser (IB Maths AA HL), A4. Download the PDF.

Revise it next

Other IB Maths AA HL topics: Functions · Geometry and trigonometry · Statistics and probability · Calculus · All IB Maths AA HL organisers