Knowledge organisers

Number and algebra: IB Maths AI 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

Matrix: A rectangular array of numbers; an m × n matrix has m rows and n columns.
Inverse matrix: A⁻¹ satisfies AA⁻¹ = I; a square matrix has an inverse only if its determinant is not zero.
Eigenvalue: A number λ with Av = λv for some non-zero vector v; find it from det(A − λI) = 0.
Complex number: z = a + bi with i² = −1; it can also be written in polar form re^(iθ).

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.

Standard form	\(a\times10^k,\) \(1\le a<10\)
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	\(u_n=u_1r^{n-1},\) \(S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}\)
Sum to infinity, \(\|r\|<1\)In the formula booklet	\(S_\infty=\frac{u_1}{1-r}\)
Compound interestIn the formula booklet	\(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)

Percentage errorIn the formula booklet	\(\varepsilon=\left\|\frac{v_A-v_E}{v_E}\right\|\times100\%\)
Exponents & logsIn the formula booklet	\(a^x=b\iff x=\log_ab\)
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	\(\log_ba=\frac{\log_ca}{\log_cb}\)
Complex numbersIn the formula booklet	\(z=a+bi=r(\cos\theta+i\sin\theta)=re^{i\theta}=r\,\cis\theta\)
Modulus, argument	\(\|z\|=\sqrt{a^2+b^2},\) \(\arg z=\theta,\) \(\tan\theta=\tfrac ba\ \text{(check quadrant)}\)
Products	\(\|z_1z_2\|=\|z_1\|\|z_2\|,\) \(\arg(z_1z_2)=\arg z_1+\arg z_2\)
Sinusoids: \(A\cos(\omega t)+B\cos(\omega t+\phi)\) = real part of \((A+Be^{i\phi})e^{i\omega t}\), so amplitudes and phases add as complex numbers
2×2 determinant; inverseIn the formula booklet	\(\det A=ad-bc,\) \(A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)
3×3 determinant	\(\det A=a\begin{vmatrix}e&f\\h&i\end{vmatrix}\) \({}-b\begin{vmatrix}d&f\\g&i\end{vmatrix}\) \({}+c\begin{vmatrix}d&e\\g&h\end{vmatrix}\)

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

Worked example

Find the eigenvalues of A = (2 1; 1 2).

det(A − λI) = (2 − λ)² − 1 = 0
2 − λ = ±1

Answer: λ = 1 or λ = 3

Common mistakes

Finance on the GDC: signs, months vs years, P/Y and C/Y, and the rounding asked for
Markov chains: columns, the initial state, the right power of T
Writing a number in standard form with a coefficient outside 1 ≤ a < 10
Giving an error interval as an inequality the wrong way round

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

You should be able to…

Convert to and from standard form a × 10ᵏ and calculate with it in scientific and financial contexts, reading calculator notation correctly.
Recognise a geometric sequence from its common ratio and use uₙ = u₁rⁿ⁻¹ to find terms and when a value is first exceeded.
Use the finance app to calculate loan repayments, amortization schedules and annuity values, and compare repayment options.
Plot complex numbers on an Argand diagram and find their modulus and argument, with the argument in radians.
Find determinants and inverses of 2 × 2 matrices by hand and larger ones with technology, and solve systems of equations in the form AX = B.
Find the steady state of a regular Markov chain by solving equations or using eigenvectors, and interpret long-term behaviour in context.

The printable sheet

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

Revise it next

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