Rational and irrational numbers

The real numbers fill every point of the number line “without gaps”: each point corresponds to one number and each number to one point. Historically the number system grew in stages: natural numbers for counting, integers for subtraction, rationals for division, and irrationals for taking roots and measuring lengths. Exam questions often ask you to sort numbers into sets, convert a recurring decimal into a fraction, compare numbers with roots and decide whether a number is rational.

§1. The chain of number sets

• Natural numbers N = {1, 2, 3, …} — for counting.
• Integers Z — the natural numbers, their opposites and zero.
• Rational numbers Q — numbers of the form p/q (p ∈ Z, q ∈ N). Every integer is rational too: 5 = 5/1.
• Irrational numbers — numbers that cannot be written as p/q: √2 = 1.4142…, π = 3.1415…, e = 2.718….
• Real numbers R — the rationals and irrationals together.

√n (n natural) is rational only when n is a perfect square: √16 = 4, but √5 is irrational. Likewise ∛n is rational only when n is a perfect cube.

§2. Decimal expansions: terminating, recurring, non-recurring

Why is a rational number recurring? In long division of p by q the remainders can only be 0, 1, …, q − 1. Within q steps a remainder repeats, and from then on the digits repeat: 1/7 = 0.(142857).
A fraction in lowest terms gives a terminating decimal only if its denominator has no prime factors other than 2 and 5: 3/8 = 0.375, but 1/3 = 0.(3).
Converting a recurring decimal to a fraction:

Rule: the numerator is “the digits after the point minus the part before the repeating block”; the denominator has as many nines as there are digits in the block, followed by as many zeros as digits before it.

Worked example

Convert 0.(27) and 0.2(3) to fractions.

1. 0.(27) = 27/99 = 3/11.
2. 0.2(3) = (23 − 2)/90 = 21/90.
3. Simplify: 21/90 = 7/30.

§3. Why √2 is irrational

Suppose √2 = p/q in lowest terms. Then p² = 2q², so p² is even and hence p is even: p = 2k. Then 4k² = 2q², q² = 2k², so q is even too. Both even — the fraction can be simplified, contradicting the assumption. Therefore √2 is not rational.
This is proof by contradiction: assume the opposite and reach a contradiction. In the same way √3, √5 and the root of any natural number that is not a perfect square are shown to be irrational.

§4. What type of number results

• Rational ± rational and rational × rational are rational; dividing by a non-zero rational also gives a rational.
• Rational + irrational is always irrational: 1 + √2. (If 1 + √2 = r were rational, then √2 = r − 1 would be rational.)
• Non-zero rational × irrational is irrational: 3√2.
• Irrational ± irrational or × may be rational: √2 + (2 − √2) = 2, √2·√8 = 4.

Worked example

Is (√3 − 1)² + 2√3 rational?

1. Expand the square: (√3 − 1)² = 3 − 2√3 + 1 = 4 − 2√3.
2. 4 − 2√3 + 2√3 = 4.
3. Answer: yes, the expression equals 4 — rational.

§5. Comparing and estimating

To compare expressions with roots, move factors under the root or square both (for positive numbers the order is preserved):

Worked example (from the official sample)

Arrange a = 6, b = 4√2, c = 2√10 in increasing order.

1. Squares: a² = 36, b² = 16·2 = 32, c² = 4·10 = 40.
2. 32 < 36 < 40.
3. So b < a < c.

Estimating: between which integers is √50? 49 < 50 < 64 → 7 < √50 < 8; 50 is close to 49, so √50 ≈ 7.07.

§6. The number line, intervals and rounding

−2 < x ≤ 5 → (−2; 5]. Infinity is not a number and cannot be “reached”, so it always takes a round bracket.
Standard form: a·10ⁿ, 1 ≤ a < 10: 0.000 32 = 3.2·10⁻⁴, 45 000 = 4.5·10⁴.
Rounding: if the first digit dropped is ≥ 5, increase the previous digit by one: 3.1415 ≈ 3.14, 2.718 ≈ 2.72. In practical calculations irrational numbers are always replaced by approximations.

§7. Operations on intervals and density

Solving systems and alternatives of inequalities requires the intersection of intervals (∩, numbers in both) and the union (∪, numbers in at least one). The safest way is to draw them on the number line: mark each interval with its own line and read off the common part or the whole coverage.

An important property of the real numbers is density: between any two different numbers there are infinitely many rationals and infinitely many irrationals. For instance, the mean (a + b)/2 lies between a and b.

Worked example

Find the intersection and union of A = [−2; 5) and B = (1; 7].

1. Mark them on the line: A from −2 (included) to 5 (excluded), B from 1 (excluded) to 7 (included).
2. Common part: from 1 to 5, both ends excluded — A ∩ B = (1; 5).
3. Whole coverage: from −2 to 7 — A ∪ B = [−2; 7].

Worked example

Find a rational number between √2 and √3.

1. Work with squares: 2 < x² < 3.
2. For x = 1.5, x² = 2.25, which lies between 2 and 3.
3. So √2 < 1.5 < √3.

Check yourself

1. Which of √16, √5, 0.(3), π are rational?

2. Compare 3√2 and 2√5.

3. Between which consecutive integers is √50?

4. Write 0.000 32 in standard form.

5. Is the sum of √2 and 2 − √2 rational?

6. Write −2 < x ≤ 5 as an interval.

7. Convert 0.1(6) to a fraction.

8. Does 7/40 give a terminating decimal?

Common mistakes

• Treating every infinite decimal as irrational — a recurring decimal is rational.
• Claiming that the sum of two irrationals must be irrational.
• Moving a factor under a root without squaring it: 2√10 ≠ √20; it is √40.
• Using a square bracket at infinity: not [a; +∞] but [a; +∞).
• Converting a mixed recurring decimal as a pure one: 0.2(3) ≠ 23/99.
• Comparing negative numbers by squaring — order is preserved only for non-negative numbers.