Algebra · step 8 of 13

Sets of numbers ℕ, ℤ, ℚ, ℝ

The standard number systems nest: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ. Each extension solves an equation the previous one could not — ℤ allows subtraction, ℚ division, ℝ limits, ℂ square roots of negatives. Knowing which set you are working in decides whether an equation has solutions, and interval notation is how subsets of ℝ get written concisely.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identify the smallest containing setA number is rational iff it can be written p/q with integers p and q ≠ 0. Terminating and recurring decimals are rational.
Use interval notation preciselySquare brackets include endpoints, round brackets exclude. Infinity always takes a round bracket.
Prove irrationality by contradictionAssume p/q in lowest terms, derive that both p and q are even, and contradict the "lowest terms" assumption.
Watch the ℕ conventionSome courses include 0 in ℕ and some do not. State which you are using when it matters.

Worked example

Prove √2 is irrational.

Suppose √2 = p/q in lowest terms, so p and q share no common factor.
Then p² = 2q², so p² is even, hence p is even; write p = 2m.
Substituting: 4m² = 2q², so q² = 2m² and q is even too.
But then p and q share the factor 2, contradicting lowest terms.

Answer. √2 cannot be written as a ratio of integers, so it is irrational.

Where marks get dropped

These are the specific errors that cost credit on sets of numbers ℕ, ℤ, ℚ, ℝ questions — QED's rubric penalises each of them separately.

Assuming a decimal that does not obviously repeat is irrational. 1/7 = 0.142857… repeats with period 6 and is perfectly rational.
Writing [3, ∞] with a square bracket at infinity. Infinity is not a number and is never included.
Claiming ℚ is not dense because it has gaps. ℚ is dense in ℝ — between any two rationals lies another — yet still countable and full of holes.

Practise this until it is automatic

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Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

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Sets of numbers ℕ, ℤ, ℚ, ℝ — frequently asked questions

Is every repeating decimal rational?

Yes. Multiply by a power of 10 to shift the repeat, subtract, and solve — the result is always a ratio of integers.

What does ℝ \ ℚ mean?

The irrationals: reals that are not rational, such as √2, π and e. They are uncountable, so in a precise sense almost every real is irrational.

Why extend to ℂ?

So that every polynomial has a root — the fundamental theorem of algebra. ℝ cannot solve x² + 1 = 0, and ℂ is the smallest field that fixes this.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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