The five number sets
A set is just a collection of numbers that share a property. In MYP 4 you work with five sets, each with its own symbol. Start with the plainest idea, counting, and build outward.
- Natural numbers \(\mathbb{N}\)
- The counting numbers \(1, 2, 3, 4, \dots\) (some courses also include 0). No fractions, no negatives.
- Integers \(\mathbb{Z}\)
- The naturals plus zero plus their negatives: \(\dots, -2, -1, 0, 1, 2, \dots\). Still whole, but now they can be negative.
- Rational numbers \(\mathbb{Q}\)
- Any number that can be written as a fraction \(\dfrac{a}{b}\) with integers \(a\) and \(b\), and \(b \neq 0\). Their decimals either stop or repeat.
- Irrational numbers
- Numbers that cannot be written as such a fraction. Their decimals run forever with no repeating pattern, for example \(\sqrt{2}\) and \(\pi\).
- Real numbers \(\mathbb{R}\)
- Everything on the number line: all rationals and all irrationals together.
How the sets nest
The sets are not separate boxes; the smaller ones sit inside the larger ones like Russian dolls. Every natural number is also an integer, every integer is also rational, and every rational is real. In symbols:
The irrationals are the one exception to the chain: they are real, but they sit outside \(\mathbb{Q}\). Together the rationals and irrationals fill the whole real line with no gaps, which is why \(\mathbb{R}\) is the outermost doll.
Belonging is cumulative
Because the sets nest, a number belongs to every set from its smallest home outward. The number 5 is natural, so it is automatically also an integer, rational and real. List all of them, not just the smallest.
Deciding where a number lives
To classify a number, ask three quick questions in order: Is it a whole counting number? Is it whole but possibly negative? Can it be written as a fraction? The first "yes" that fits gives its smallest set.
Classify each number as fully as you can: \(6\), \(-4\), \(\tfrac{3}{5}\), \(\sqrt{9}\), \(\sqrt{7}\).
A root sign does not mean irrational
Check whether the root simplifies first. \(\sqrt{9}\), \(\sqrt{16}\) and \(\sqrt{0.25}\) all give tidy rational values. Only roots of non-perfect-squares such as \(\sqrt{2}\) or \(\sqrt{7}\) are irrational.
Recurring decimals as fractions
A key fact: every recurring decimal is rational, because you can always rewrite it as a fraction. The trick is to multiply by a power of 10 so the repeating tails line up, then subtract them away.
Write the recurring decimal \(0.\overline{45} = 0.454545\dots\) as a fraction in lowest terms.
You can check it: dividing \(5 \div 11\) does give \(0.4545\dots\), so the fraction is correct. The rule of thumb it reveals: a block of \(n\) repeating digits sits over \(n\) nines, so \(0.\overline{45} = \tfrac{45}{99}\) before you cancel.
Where this is assessed
Classifying numbers and justifying why a value is rational or irrational is Criterion A knowledge, while writing the sets with correct symbols and notation is Criterion C communicating. The recurring-decimal method rewards clear, labelled working.
Check yourself
1. List every set that \(-7\) belongs to. +
It is whole but negative, so it is not natural. It is an integer, rational and real number.
2. Is \(\sqrt{2}\) rational or irrational? Explain. +
2 is not a perfect square, so \(\sqrt{2} = 1.41421\dots\) never terminates or repeats. It cannot be written as a fraction of integers, so it is irrational.
3. Write \(0.\overline{4} = 0.444\dots\) as a fraction in lowest terms. +
Let \(x = 0.444\dots\). One digit repeats, so \(10x = 4.444\dots\). Subtract: \(10x - x = 4\), giving \(9x = 4\), so \(x = \mathbf{\tfrac{4}{9}}\). Checking, \(4 \div 9 = 0.444\dots\).
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