Number Sets & Approximation — SL 1.1 Number sets, approximation, and rounding in context

Numbers are so familiar that it's easy to forget they were invented — or, more accurately, discovered and refined over thousands of years. The earliest humans used natural numbers (1, 2, 3, …) for counting livestock and tracking days. As civilisations developed trade, debt, and measurement, entirely new categories of numbers had to be imagined. The story of number sets is really the story of mathematicians asking, "What if we need a number that doesn't exist yet?"

Alongside the expansion of number types came the practical challenge of approximation. Ancient Babylonian scribes carved tables of square roots into clay tablets around 1800 BCE — and they had to decide how many decimal places were "good enough." That same question surfaces every time you read a population figure in the news or round a chemistry measurement in the lab.

So the central questions of SL 1.1 are actually ancient ones: What kind of number am I dealing with? and How precisely should I express it? Understanding number sets tells you what's possible; mastering approximation tells you what's practical.

Before diving into calculations, you need a clear mental map of how mathematicians classify numbers. Each number set is a collection of numbers sharing a common property, and smaller sets nest inside larger ones like Russian dolls. On top of that classification sits the toolkit of approximation: rounding, decimal places (d.p.), and significant figures (s.f.).

Notice that the real number set ℝ is the outermost ellipse. It contains every number you'll encounter on the IB Applications and Interpretation course at Standard Level. Inside it, the rationals ℚ include any number that can be expressed as the ratio of two integers, such as ¾ or −2 (which is −2/1). The integers ℤ are the whole numbers plus their negatives. Finally, the natural numbers ℕ are the non-negative counting numbers (the IB includes 0 in ℕ). Irrational numbers like π and √2 sit in ℝ but cannot be written as fractions, so they live outside the ℚ ellipse.

The IB syllabus specifies three main ways to express an approximate value: rounding to a given number of decimal places, rounding to a given number of significant figures, and rounding to a given power of 10 (e.g., the nearest hundred). It also requires you to calculate the resulting error.

Counting Significant Figures

1. Rule 1: All non-zero digits are significant. Example: 345 has 3 s.f.
2. Rule 2: Zeros between non-zero digits are significant. Example: 3 007 has 4 s.f.
3. Rule 3: Leading zeros are NOT significant. Example: 0.0042 has 2 s.f.
4. Rule 4: Trailing zeros after a decimal point ARE significant. Example: 2.50 has 3 s.f.
5. Rule 5: Trailing zeros in a whole number without a decimal point are ambiguous — use scientific notation to clarify. Example: write 2.50 × 10³ for 3 s.f.

A common IB exam question gives you a list of numbers and asks you to place each one in its smallest correct number set. The table below provides a structured reference.

A scientist measures the mass of a chemical sample as 0.05738 g. She records the value to 3 significant figures. Find the rounded value and then calculate the percentage error introduced by rounding.

Choosing between decimal places and significant figures isn't arbitrary — each method suits different scenarios. The table below highlights when to use each approach and what pitfalls to watch for.

Number sets and approximation form the foundation for nearly everything else in the IB AI course. Understanding where a value lives on the number set hierarchy affects how you model problems, and handling rounding correctly prevents compounding errors through multi-step calculations.

At Higher Level, students encounter complex numbers (ℂ), which extend the real numbers by introducing i = √(−1). Even at SL, the habit of classifying numbers and controlling precision will follow you through every topic — from probability distributions to calculus-adjacent modelling in the Internal Assessment.