Is this number rational or irrational?

How it works

The calculator reads what you type exactly, as a + b·X, where a and b are fractions and X is one irrational part: √r or ∛r of a fraction r, π, 1/π, e or 1/e.

• When b = 0 the number is a fraction a = p/q, so it is rational.
• Otherwise it is irrational. X is irrational: √r of a fraction r in lowest terms is rational only when both its top and bottom are perfect squares (if √(p/q) = m/n, then p n² = q m², and with p/q in lowest terms p = m² and q = n²); ∛r likewise only for perfect cubes; π and e are irrational, and so are 1/π and 1/e. A rational number plus a nonzero rational multiple of an irrational number is irrational: if a + b·X = s were rational, X = (s − a) ÷ b would be rational too.

The calculator turns √r and ∛r into fractions whenever r is a square or a cube: √36 = 6, √(4/9) = 2/3, ∛(−8) = −2.

Any other root is written as the root of a whole number m with no square factor (no cube factor for ∛), so one number has one exact form, in whatever order it was typed. For r = p/q in lowest terms, √r = (1/q)√(pq) and ∛r = (1/q)∛(pq²), ∛(−m) = −∛m, and each square factor k² of the number under √ (cube factor k³ under ∛) comes out of the root as k: √8 = 2√2, √12 = 2√3, √(2/3) = (1/3)√6, ∛(−16) = −2∛2. So √8 − √2 and −√2 + √8 are both √2, and √(2/3) + √6 and √6 + √(2/3) are both (4/3)√6. The factors are found by dividing out every prime up to 100,000 and then taking out what is left when it is a square (a cube); this finds all of them when the number under the root is below 10¹⁵ (10²⁰ for ∛). Above that, a square factor of a prime over 100,000 can stay under the root (√1000250012300171, which is 100,003√100,019, stays as typed), and a sum or product of two such roots keeps the left one's number.

What you can type

• Numbers: whole numbers and decimals (5, 0.75, .5, -3.2), with commas only between groups of three digits (1,234.5; 1,5 is refused). No e-notation: e is the number e.
• Repeating decimals: the repeating digits in brackets right after the decimal digits, 0.(3), 0.58(3), 1.2(34); or the decimal with three dots (... or …), 0.333..., 0.1666…, 0.1212…. With dots, the repeating block is the shortest block of digits at the end that also stands just before itself (in 0.1666… it is 6; in 0.4545… it is 45). With no such block, as in 0.123…, the page asks you to type the repeating digits twice. The value of a decimal with whole part W, digits F after the point and then a repeating block R is (WFR − WF) ÷ (10^|F| × (10^|R| − 1)), where WFR and WF are the digits read as whole numbers: 0.58(3) = (583 − 58) ÷ 900 = 7/12.
• Constants: π or pi, and e.
• Roots: √x or sqrt(x) for the square root, ∛x or cbrt(x) for the cube root, where x is a number, a constant, a root, or a bracket: √2, √(8/9), √0.25, ∛-8 is refused but ∛(-8) = −2. The number under a root must be rational.
• Arithmetic: +, - or −, *, × or ·, / or ÷, and brackets. Multiplication and division go before addition and subtraction, from left to right. A number, constant, root or bracket may be followed by π, e, a root or a bracket with no sign between them, which multiplies: 2π, 3√2, 2(1 + √3). This counts like ×, so 1/2π is (1/2) × π; type 1/(2π) for the other.

What the exact form can hold, and so what the page reads:

• sums and differences of a rational number and multiples of one irrational part: 1 + √2, π − 3, √8 − √2 (√8 = 2√2, since 8 ÷ 2 is a square);
• products and quotients by rational numbers;
• products of two square roots, or two cube roots, with no rational part added (√2 × √8 = 4, ∛2 × ∛4 = 2), and products and quotients of numbers a + b√r with the same √r (up to a square factor);
• division by a + b√r, by b∛r, bπ, b/π, be or b/e;
• π × (1/π) and e × (1/e), and their quotients π/π and e/e.

Rules

No answer, with these messages:

• Anything that is not a number or expression as above: "Type a number such as 3/4, 0.75, 0.(3), √2, 2π or (1 + √5)/2."
• A comma that does not separate groups of three digits: "A comma in a number must separate groups of three digits, as in 1,000."
• Division by 0: "This divides by zero, so it is not a number."
• The square root of a negative number: "The square root of a negative number is not a real number."
• A root of an irrational number, such as √π: "The number under a root must be rational, such as √2 or √(4/9)."
• Two different irrational parts in one sum or product, such as √2 + √3 or π + 1/π: "This calculator handles one kind of irrational part at a time, such as 1 + √2, 3π/4 or ∛2 × ∛4."
• Any sum, product or quotient of π (or 1/π) with e (or 1/e): "No one knows yet whether sums and products of π and e, such as π + e, are rational."
• Three dots after digits that do not repeat: "The digits before the dots do not repeat. Type the repeating digits twice, as in 0.1212…, or in brackets, as in 0.(12)."
• At most 100 characters.

Output format

• Rational or irrational: Rational or Irrational.
• Why: for a rational number, It equals p/q, a ratio of two integers. with p/q in lowest terms (q = 1 for a whole number, so 6 is 6/1). Otherwise It is F, and m is not the square of a fraction, so its square root is irrational. (or … not the cube of a fraction, so its cube root is irrational.), It is F, and π is irrational., It is F, and 1/π is irrational because π is., It is F, and e is irrational. or It is F, and 1/e is irrational because e is., where F is the exact value and m the whole number under its root (√44 gives It is 2√11, and 11 is not the square of a fraction, …).
• Exact value: a fraction is written p/q in lowest terms or as a whole number. Otherwise a + bX or a − |b|X, or bX alone when a = 0: the coefficient is left out when it is 1, written as a whole number, or as a fraction in brackets, (1/2)√5, with − before it when negative; X is √m, ∛m, π or e, with m the whole number under the root from "How it works" (√(2/3) is (1/3)√6, ∛(−2) is −∛2); 1/π and 1/e are written p/π or p/(qπ). Every minus sign is the true minus (−).
• As a fraction: rational numbers only, p/q in lowest terms.
• Decimal: for a rational number, its exact decimal by long division: every digit when it ends within 60 decimal places; when a remainder repeats within 60 places, the repeating block with a combining overline (U+0305) after each of its digits, as in 0.16̅ or 3.1̅4̅2̅8̅5̅7̅; otherwise the first 60 places and …. For an irrational number, ≈ and the approximate value below to 12 significant figures (shortest form, e-notation from 10²¹ or below 10⁻⁶).
• Decimal type: ends, repeats, or neither ends nor repeats (irrational). A fraction's decimal ends when its bottom in lowest terms has no prime factor but 2 and 5, also when it is longer than the 60 places shown (1/2⁶¹ ends after 61 places); otherwise it repeats.
• Approximate value: the nearest 64-bit float to a + b·X, worked exactly with X to 120 decimal places (rounded down): √m and ∛m as whole-number roots, π and e from their first 120 decimal places. So typed digits that cancel the leading figures of X do not spoil the figures shown, unless they match X to more than about 100 decimal places: √2 − 1.41421356237 ≈ 3.09504880169e−12. Shown to 12 significant figures.
• Number sets, from OpenStax Example 7.4: a positive whole number is Counting number, whole number, integer, rational number, real number; 0 is Whole number, integer, rational number, real number; a negative whole number Integer, rational number, real number; another fraction Rational number, real number; an irrational number Irrational number, real number.
• The answer sentence reads Is x rational or irrational? R. W, with x as typed, R the verdict and W the reason.

Worked examples by hand

√2. 2 is not a perfect square, so √2 is irrational (≈ 1.41421356237).

√36 and √44 (OpenStax Example 7.3). 36 = 6², so √36 = 6, rational: a counting number, whole number, integer, rational and real. 44 is not a perfect square, so √44 is irrational; 44 = 2² × 11, so the page writes it 2√11.

0.583 with the 3 repeating (OpenStax Example 7.2). Typed 0.58(3): W = 0, F = 58, R = 3, so the value is (583 − 58) ÷ (100 × 9) = 525 ÷ 900 = 7/12, rational, decimal 0.583̅. 0.475 = 475/1,000 = 19/40, rational; its decimal ends.

The golden ratio. (1 + √5)/2 = 1/2 + (1/2)√5: 5 is not a square, so it is irrational, ≈ 1.61803398875.

√2 × √8. Both are pure square roots, so √2 × √8 = √16 = 4, rational.

2π/3. 2 × π ÷ 3 = (2/3)π, a nonzero rational multiple of π: irrational.

0.1666…. The last digit 6 equals the digit before it, so the block is 6: 0.16̅ = (16 − 1) ÷ (10 × 9) = 15/90 = 1/6.

∛27/8. ∛27 = 3 because 27 = 3³; ÷ 8 gives 3/8, rational.

−22/7. Rational, decimal −3.142857 with the six digits 142857 repeating: −3.1̅4̅2̅8̅5̅7̅. Close to −π, but a fraction.