Knowledge organisers

Number and algebra: IB Maths AI SL 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

Standard form: A number written as a × 10ᵏ with 1 ≤ a < 10 and k an integer.
Compound interest: Interest is added to the balance, so each period's interest is earned on a bigger amount.
Percentage error: |approximate − exact| ÷ |exact| × 100%.
Amortisation: Paying off a loan in regular instalments that cover the interest and part of the amount borrowed.

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	$a\times10^k,$ $1\le a<10,\ k\in\mathbb Z$
Arithmetic sequenceIn the formula booklet	$u_n=u_1+(n-1)d$
Arithmetic seriesIn the formula booklet	$S_n=\tfrac n2\big(2u_1+(n-1)d\big)=\tfrac n2(u_1+u_n)$
Common difference	$d=u_{n+1}-u_n$
Geometric sequenceIn the formula booklet	$u_n=u_1r^{n-1}$
Geometric series, $r\ne1$In the formula booklet	$S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}$
Compound interest ($k$ times a year, $n$ years, $r$%)In the formula booklet	$FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}$
Depreciation at $r$% a year	$V=V_0\left(1-\tfrac{r}{100}\right)^n$

Exponents & logsIn the formula booklet	$a^x=b\iff x=\log_ab$
Log facts	$\log_{10}10^x=x,$ $\ln e^x=x,$ $e^{\ln x}=x$
Percentage errorIn the formula booklet	$\varepsilon=\left\|\frac{v_A-v_E}{v_E}\right\|\times100\%$
Bounds: a value rounded to the nearest $u$ lies in $[x-\tfrac u2,\ x+\tfrac u2)$

Worked example

$2000 is invested at 3% a year, compounded monthly. Find the value after 5 years, to the nearest cent.

FV = PV(1 + r/(100k))^(kn) with k = 12, n = 5
FV = 2000 × (1 + 0.03/12)⁶⁰ = 2000 × 1.0025⁶⁰

Answer: $2323.23

Common mistakes

Finance: compound-interest formula with regular payments, TVM signs and periods
Sequences: uₙ vs Sₙ vs n, and rounding n up
Giving bounds as [value − 0.5×unit, value + 0.5×unit] with the wrong unit of accuracy
Confusing percentage error with absolute error

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

You should be able to…

Convert numbers to and from the form a × 10ᵏ where 1 ≤ a < 10, and interpret very large and very small quantities in context.
Recognise an arithmetic sequence from its common difference and use uₙ = u₁ + (n − 1)d to find terms in contexts such as rows of seats.
Compare simple and compound interest, use the compound interest formula with different compounding periods and check answers with the calculator's finance solver.
Use the finance solver and a spreadsheet to model annuities and regular-payment investments, comparing the outcomes of different saving plans.
Solve polynomial equations such as 2x³ − 5x² + x + 3 = 0 with the calculator and decide which roots are valid in a given context.
Find bounds and percentage error for measured quantities and calculated results, and choose an appropriate degree of accuracy when reporting answers.

The printable sheet

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

Revise it next

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

Standard form	\(a\times10^k,\) \(1\le a<10,\ k\in\mathbb Z\)
Arithmetic sequenceIn the formula booklet	\(u_n=u_1+(n-1)d\)
Arithmetic seriesIn the formula booklet	\(S_n=\tfrac n2\big(2u_1+(n-1)d\big)=\tfrac n2(u_1+u_n)\)
Common difference	\(d=u_{n+1}-u_n\)
Geometric sequenceIn the formula booklet	\(u_n=u_1r^{n-1}\)
Geometric series, \(r\ne1\)In the formula booklet	\(S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}\)
Compound interest (\(k\) times a year, \(n\) years, \(r\)%)In the formula booklet	\(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
Depreciation at \(r\)% a year	\(V=V_0\left(1-\tfrac{r}{100}\right)^n\)

Exponents & logsIn the formula booklet	\(a^x=b\iff x=\log_ab\)
Log facts	\(\log_{10}10^x=x,\) \(\ln e^x=x,\) \(e^{\ln x}=x\)
Percentage errorIn the formula booklet	\(\varepsilon=\left\|\frac{v_A-v_E}{v_E}\right\|\times100\%\)
Bounds: a value rounded to the nearest \(u\) lies in \([x-\tfrac u2,\ x+\tfrac u2)\)