Number and Algebra (AI HL) — IB Math AI HL AI HL Study Guide

For: IB Math AI HL candidates sitting IB Math: Applications & Interpretation HL.

Covers: Approximation and percentage error, real-world sequence and series applications, exponential and logarithmic modeling, solving linear and non-linear systems with technology, and HL-only matrix applications for Markov chains and transformations.

You should already know: IGCSE / pre-DP math; comfort with applied problems and tech.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Math AI HL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Is Number and Algebra (AI HL)?

Number and Algebra is the foundational core topic of IB Math AI HL, combining numerical computation, pattern recognition, and algebraic modeling to solve real-world quantitative problems. It includes all SL core number and algebra content plus HL-exclusive matrix operations and applications, and appears in every paper of the AI HL exam, often integrated with statistics, calculus, and financial mathematics questions for higher-mark extended response items.

2. Approximation and percentage error

Approximation is the process of rounding a measured or calculated value to a practical level of precision, typically specified as decimal places (dp) or significant figures (sig fig) in exam questions. Errors between approximate and exact values are quantified using two core metrics:

• Absolute error: The absolute difference between the approximate ($V_A$) and exact ($V_E$) value: $|V_A-V_E|$
• Percentage error: The absolute error expressed as a percentage of the exact value, using the formula: $$\%\text{ error}=\frac{|V_A-V_E|}{|V_E|}\times 100$$

Worked Example

A builder measures the length of a beam as 3.7 m (2 sig fig). The exact manufactured length of the beam is 3.672 m. Calculate the percentage error in the builder’s measurement.

1. Calculate absolute error: $|3.7-3.672|=0.028$
2. Substitute into percentage error formula: $\frac{0.028}{3.672}\times 100\approx 0.76\%$ (2 dp)

Exam tip: Examiners regularly test that you use the exact value as the denominator, never the approximate value, even if the approximate value is given first in the question.

3. Sequences and series — applications

Sequences are ordered lists of numbers following a consistent rule, while series are the sum of terms in a sequence. The two most common sequence types tested in AI HL are arithmetic and geometric, both with frequent real-world applications:

• Arithmetic sequences: Have a constant absolute difference $d$ between terms, used for linear growth/decay (e.g. fixed annual salary raises, constant monthly loan repayments)
• $n$th term: $u_n=u_1+(n-1)d$
• Sum of first $n$ terms: $S_n=\frac{n}{2}(2u_1+(n-1)d)=\frac{n}{2}(u_1+u_n)$
• Geometric sequences: Have a constant proportional ratio $r$ between terms, used for exponential growth/decay (e.g. compound interest, population growth, radioactive decay)
• $n$th term: $u_n=u_1{r}^{n-1}$
• Sum of first $n$ terms: $S_n=\frac{u_1(r^n-1)}{r-1}$ for $r\ne 1$
• Infinite sum: $S_{\infty }=\frac{u_1}{1-r}$ for $|r|<1$

Worked Example

A coffee shop has 1200 customers in its first month of operation, and customer numbers grow by 5% per month. Calculate the total number of customers the shop will have in its first year of operation.

1. Identify as geometric series: $u_1=1200$, $r=1.05$, $n=12$
2. Substitute into sum formula: $S_{12}=\frac{1200(1.05^{12}-1)}{1.05-1}\approx \frac{1200(1.7959-1)}{0.05}\approx 19101$ total customers

4. Exponential and log models

Exponential and logarithmic models are used for processes where the rate of change is proportional to the current value of the quantity. The general form of an exponential model is: $$A(t)=A_0{b}^{kt}$$ Where $A_0$ is the initial value at $t=0$, $b$ is the base (use $b=e$ for continuous growth/decay, $b=2$ for doubling processes, $b=0.5$ for half-life decay), and $k$ is the growth/decay constant.

Logarithms are used to solve for unknown exponents in these models, using the core power rule and change of base formula:

1. Power rule: ${\log }_b{a}^c=c{\log }_{b}a$
2. Change of base: ${\log }_{b}a=\frac{\ln a}{\ln b}$ (for use with GDCs that only have natural log functions)

Worked Example

A radioactive isotope has a half-life of 16 years. How long will it take for a 40 g sample to decay to 12 g?

1. Set up model: $m(t)=40\times 0.5^{t/16}$
2. Set equal to 12: $0.5^{t/16}=0.3$
3. Take natural log of both sides: $\frac{t}{16}\ln 0.5=\ln 0.3$
4. Solve for $t$: $t=16\times \frac{\ln 0.3}{\ln 0.5}\approx 27.8$ years

5. Solving systems with technology

AI HL allows graphic display calculator (GDC) use for all papers except Paper 1, and examiners explicitly expect you to use technology to solve systems of equations efficiently, rather than solving manually. You will be tested on two system types:

1. Linear systems (up to 3 variables): Input the coefficient matrix and constant vector into your GDC’s system solver function to get exact or approximate solutions
2. Non-linear systems: Use your GDC’s graphing function to find intersection points of two curves, or use the numeric solver for systems with polynomial, exponential, or trigonometric terms

Worked Example

A school sells three ticket types for a play: student ($s$), adult ($a$), and senior ($p$). 2 student, 3 adult, and 1 senior ticket cost $72;1student,2adult,and4seniorticketscost$78; 3 student, 1 adult, and 2 senior tickets cost $66. Find the price of each ticket type.

1. Write the system of equations: $$\{\begin{array}{l}2s+3a+p=72\\ s+2a+4p=78\\ 3s+a+2p=66\end{array}$$
2. Input to GDC system solver: $s=10$, $a=18$, $p=8$
3. Verify: $2(10)+3(18)+8=72$, which matches the first equation.

6. Matrices (HL only) — applications

Matrices are rectangular arrays of numbers used to store and manipulate structured data, and are exclusive to AI HL. The three most common matrix applications tested are:

1. Transformation matrices: Used to apply rotations, reflections, scaling, and translations to 2D and 3D coordinates for design and modeling problems
2. Markov chains: Transition matrices model state changes over time for processes like customer loyalty, weather patterns, and transport mode use
3. Input-output models: Used in economics to calculate supply and demand across multiple industry sectors

For Markov chains, the state after $n$ steps is calculated as $S_n=S_0\times T^n$, where $S_0$ is the initial state vector, and $T$ is the transition matrix (rows = current state, columns = next state).

Worked Example

A city has two grocery store chains, GreenMart and FreshCo. Each month, 75% of GreenMart customers return, 25% switch to FreshCo; 65% of FreshCo customers return, 35% switch to GreenMart. Initial market share is 60% GreenMart, 40% FreshCo. What is the market share after 2 months?

1. Transition matrix $T=\left(\begin{array}{cc}0.75& 0.35\\ 0.25& 0.65\end{array}\right)$, initial state $S_0=[0.6,0.4]$
2. Calculate $S_1=S_0\times T=[0.6\ast 0.75+0.4\ast 0.35,0.6\ast 0.25+0.4\ast 0.65]=[0.59,0.41]$
3. Calculate $S_2=S_1\times T=[0.59\ast 0.75+0.41\ast 0.35,0.59\ast 0.25+0.41\ast 0.65]=[0.586,0.414]$
4. After 2 months, GreenMart has 58.6% share, FreshCo has 41.4% share.

7. Common Pitfalls (and how to avoid them)

• Wrong move: Using the approximate value as the denominator in percentage error calculations. Why: Students mix up $V_A$ and $V_E$ when reading questions quickly. Correct move: Always divide by the more precise/exact value, even if the approximate value is given first.
• Wrong move: Using arithmetic sequence formulas for percentage growth/decay. Why: Students confuse constant absolute change with constant proportional change. Correct move: Look for keywords like "percentage increase" or "doubles every X years" to use geometric sequences, and "fixed increase of $X" for arithmetic sequences.
• Wrong move: Mixing up row and column order for transition matrices. Why: Students forget that transition matrix rows represent the original state, columns represent the new state. Correct move: Test your matrix with a 1-step calculation first to confirm order before calculating long-term values.
• Wrong move: Rounding intermediate values before the final answer. Why: Students try to save time by rounding early, leading to large cumulative errors. Correct move: Store all intermediate values in your GDC, only round the final answer to 3 significant figures (the IB default unless specified otherwise).
• Wrong move: Solving non-linear systems manually instead of using technology. Why: Habit from pre-DP math where algebraic solving was required. Correct move: AI HL explicitly allows GDC use for non-linear systems, so use graphing or solver functions to save time and avoid algebraic errors.

8. Practice Questions (IB Math AI HL Style)

Question 1

A student calculates the area of a circle with radius 4.2 cm (2 sig fig) as $55.4{\text{ cm}}^2$. The exact area of the circle using the precise radius of 4.18 cm is $54.99{\text{ cm}}^2$. (a) Calculate the absolute error in the student’s calculation. (b) Calculate the percentage error, giving your answer to 2 decimal places.

Solution

(a) Absolute error = $|55.4-54.99|=0.41{\text{ cm}}^2$ (b) $\%\text{ error}=\frac{0.41}{54.99}\times 100\approx 0.75\%$

Question 2

A teacher starts a retirement fund with a $5000deposit,andadds$2000 to the fund at the end of every year. The fund earns 4% annual interest, compounded annually. Calculate the total value of the fund after 20 years, to the nearest dollar.

Solution

This is a geometric series of annual deposits, where each deposit earns interest for a different number of years:

• Initial deposit: $5000\times 1.04^{20}$
• Year 1 deposit: $2000\times 1.04^{19}$
• ...
• Year 19 deposit: $2000\times 1.04^1$
• Total value = $5000(1.04^{20})+2000\times \frac{1.04(1.04^{19}-1)}{1.04-1}\approx 10955.62+60039.22=\$70995$

Question 3

The transition matrix below shows student attendance patterns for university lectures, where rows represent attendance status one week (attend, skip), columns represent status the next week: $$T=\left(\begin{array}{cc}0.8& 0.3\\ 0.2& 0.7\end{array}\right)$$ Initially, 90% of students attend lectures, 10% skip. (a) Find the percentage of students skipping lectures after 3 weeks, to 1 decimal place. (b) Find the long-term steady state percentage of students who attend lectures, to the nearest whole number.

Solution

(a) Initial state $S_0=[0.9,0.1]$ $S_1=[0.9\ast 0.8+0.1\ast 0.3,0.9\ast 0.2+0.1\ast 0.7]=[0.75,0.25]$ $S_2=[0.75\ast 0.8+0.25\ast 0.3,0.75\ast 0.2+0.25\ast 0.7]=[0.675,0.325]$ $S_3=[0.675\ast 0.8+0.325\ast 0.3,0.675\ast 0.2+0.325\ast 0.7]=[0.6375,0.3625]$ 36.3% of students skip after 3 weeks. (b) Steady state $S=[s,1-s]=S\times T$ $s=0.8s+0.3(1-s)\to s=0.5s+0.3\to 0.5s=0.3\to s=0.6$ Long-term attendance rate is 60%.

9. Quick Reference Cheatsheet

10. What's Next

Number and Algebra is the backbone of all remaining AI HL topics, so mastering these concepts will simplify your study of later units significantly. Exponential and log models are used extensively in calculus for related rates and optimization problems, while matrix operations are applied to graph theory, 3D modeling, and statistical regression in later HL-exclusive units. You will also see sequence and series questions combined with financial mathematics topics like amortization and annuities in Papers 2 and 3, where extended response questions often integrate 2-3 topics for high marks.

If you have any gaps in your understanding, or want to practice more exam-style questions tailored to your weak areas, you can ask Ollie, our AI tutor, for personalized help at any time. You can also find more topic-specific study guides and past paper practice on the homepage to build your confidence for the IB Math AI HL exams.