Real Numbers (Rational vs. Irrational Numbers)

Rational number

The rational numbers ($\mathbb {Q}$) are included in the real numbers ($\mathbb {R}$), while themselves including the integers ($\mathbb {Z}$), which in turn include the natural numbers ($\mathbb {N}$)

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ${\frac {p}{q}}$ of two integers, a numerator p and a non-zero denominator q. For example, ${\frac {-3}{7}}$ is a rational number, as is every integer (e.g. $5={\frac {5}{1}}$). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q ; it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient", and first appeared in Bourbaki's Algèbre.

The decimal expansion of a rational number either terminates after a finite number of digits (example: ${\frac {3}{4}}=0.75$), or eventually begins to repeat the same finite sequence of digits over and over (example: ${\frac {9}{44}}=0.20454545...$). Conversely, any repeating or terminating decimal represents a rational number. These statements are true in base 10, and in every other integer base (for example, binary or hexadecimal).

A real number that is not rational is called irrational. Irrational numbers include ${\sqrt {2}}$, $\pi$, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows:

$\left(p_{1},q_{1}\right)\sim \left(p_{2},q_{2}\right)\iff p_{1}q_{2}=p_{2}q_{1}.$

The fraction ${\frac {p}{q}}$ then denotes the equivalence class of (p, q).

Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (for more, see Construction of the real numbers).

Terminology

The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.

Etymology

Although nowadays rational numbers are defined in terms of ratios, the term rational is not a derivation of ratio. On the opposite, it is ratio that is derived from rational: the first use of ratio with its modern meaning was attested in English about 1660, while the use of rational for qualifying numbers appeared almost a century earlier, in 1570. This meaning of rational came from the mathematical meaning of irrational, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of ἄλογος)".

This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers". So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).

This etymology is similar to that of imaginary numbers and real numbers.

Arithmetic

Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction ${\frac {a}{b}}$, where a and b are coprime integers and b > 0. This is often called the canonical form of the rational number.

Starting from a rational number ${\frac {a}{b}}$, its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.

Embedding of integers

Any integer n can be expressed as the rational number ${\frac {n}{1}}$, which is its canonical form as a rational number.

Equality

${\frac {a}{b}}={\frac {c}{d}}$ if and only if $ad=bc$

If both fractions are in canonical form, then:

${\frac {a}{b}}={\frac {c}{d}}$ if and only if $a=c$ and $b=d$

Ordering

If both denominators are positive (particularly if both fractions are in canonical form):

${\frac {a}{b}}<{\frac {c}{d}}$ if and only if $ad<bc.$

On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator—by changing the signs of both its numerator and denominator.

Addition

Two fractions are added as follows:

${\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.$

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.

Subtraction

${\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.$

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.

Multiplication

The rule for multiplication is:

${\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.$

where the result may be a reducible fraction—even if both original fractions are in canonical form.

Inverse

Every rational number ${\frac {a}{b}}$ has an additive inverse, often called its opposite,

$-\left({\frac {a}{b}}\right)={\frac {-a}{b}}.$

If ${\frac {a}{b}}$ is in canonical form, the same is true for its opposite.

A nonzero rational number ${\frac {a}{b}}$ has a multiplicative inverse, also called its reciprocal,

$\left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.$

If ${\frac {a}{b}}$ is in canonical form, then the canonical form of its reciprocal is either ${\frac {b}{a}}$ or ${\frac {-b}{-a}}$ , depending on the sign of a.

Division

If b, c, and d are nonzero, the division rule is

${\frac {\frac {a}{b}}{\frac {c}{d}}}={\frac {ad}{bc}}.$

Thus, dividing ${\frac {a}{b}}$ by ${\frac {c}{d}}$ is equivalent to multiplying ${\frac {a}{b}}$ by the reciprocal of ${\frac {c}{d}}$:

${\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.$

Exponentiation to integer power

If n is a non-negative integer, then

$\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.$

The result is in canonical form if the same is true for ${\frac {a}{b}}$. In particular,

$\left({\frac {a}{b}}\right)^{0}=1.$

If a ≠ 0, then

$\left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.$

If ${\frac {a}{b}}$ is in canonical form, the canonical form of the result is ${\frac {b^{n}}{a^{n}}}$ if a > 0 or n is even. Otherwise, the canonical form of the result is ${\frac {-b^{n}}{-a^{n}}}$ .

Continued fraction representation

A finite continued fraction is an expression such as

$a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},$

where a_n are integers. Every rational number ${\frac {a}{b}}$ can be represented as a finite continued fraction, whose coefficients a_n can be determined by applying the Euclidean algorithm to (a, b).

Other representations

• common fraction: ${\frac {8}{3}}$
• mixed numeral: $2{\frac {2}{3}}$
• repeating decimal using a vinculum: $2.{\overline {6}}$
• repeating decimal using parentheses: 2.(6)
• continued fraction using traditional typography: $2+{\frac {1}{1+{\frac {1}{2}}}}$
• continued fraction in abbreviated notation: [2; 1, 2]
• Egyptian fraction: $2+{\frac {1}{2}}+{\frac {1}{6}}$
• prime power decomposition: 2³ × 3⁻¹
• quote notation: 3'6

are different ways to represent the same rational value.

Formal construction

A diagram showing a representation of the equivalent classes of pairs of integers

The rational numbers may be built as equivalence classes of ordered pairs of integers.

More precisely, let (Z × (Z \ {0})) be the set of the pairs (m, n) of integers such n ≠ 0. An equivalence relation is defined on this set by

$\left(m_{1},n_{1}\right)\sim \left(m_{2},n_{2}\right)\iff m_{1}n_{2}=m_{2}n_{1}.$

Addition and multiplication can be defined by the following rules:

$\left(m_{1},n_{1}\right)+\left(m_{2},n_{2}\right)\equiv \left(m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}\right),$

$\left(m_{1},n_{1}\right)\times \left(m_{2},n_{2}\right)\equiv \left(m_{1}m_{2},n_{1}n_{2}\right).$

This equivalence relation is a congruence relation, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers Q is the defined as the quotient set by this equivalence relation, (Z × (Z \ {0})) / ~, equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any integral domain and produces its field of fractions.)

The equivalence class of a pair (m, n) is denoted ${\frac {m}{n}}$. Two pairs (m₁, n₁) and (m₂, n₂) belong to the same equivalence class (that is are equivalent) if and only if m₁n₂ = m₂n₁. This means that ${\frac {m_{1}}{n_{1})}}={\frac {m_{2}}{n_{2}}}$ if and only m₁n₂ = m₂n₁.

Every equivalence class ${\frac {m}{n}}$ may be represented by infinitely many pairs, since

$\cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .$

Each equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (m, n) in the equivalence class such that m and n are coprime, and n > 0. It is called the representation in lowest terms of the rational number.

The integers may be considered to be rational numbers identifying the integer n with the rational number ${\frac {n}{1}}$.

A total order may be defined on the rational numbers, that extends the natural order of the integers. One has

${\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}$

if

$(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\qquad {\text{or}}\qquad (n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).$

Properties

Illustration of the countability of the positive rationals

The set Q of all rational numbers, together with the addition and multiplication operations shown above, forms a field.

Q has no field automorphism other than the identity.

With the order defined above, Q is an ordered field that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield isomorphic to Q.

Q is a prime field, which is a field that has no subfield other than itself. The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to Q.

Q is the field of fractions of the integers Z. The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the field of algebraic numbers.

The set of all rational numbers is countable (see the figure), while the set of all real numbers (as well as the set of irrational numbers) is uncountable. Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that

${\frac {a}{b}}<{\frac {c}{d}}$

(where $b,d$ are positive), we have

${\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.$

Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

Real numbers and topological properties

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.

By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric d(x, y) = |x − y|, and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric d(x, y) = |x − y| above.

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field:

Let p be a prime number and for any non-zero integer a, let |a|_p = p⁻ⁿ, where pⁿ is the highest power of p dividing a.

In addition set |0|_p = 0. For any rational number ${\frac {a}{b}}$, we set $|{\frac {a}{b}}|_{p}={\frac {|a|_{p}}{|b|_{p}}}$.

Then d_p(x, y) = |x − y|_p defines a metric on Q.

The metric space (Q, d_p) is not complete, and its completion is the p-adic number field Q_p. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.

Irrational number

The number ${\sqrt {2}}$ is irrational.

In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio $\pi$ of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of $\pi$ starts with 3.14159, but no finite number of digits can represent $\pi$ exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.

Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.

As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.

Examples

Square roots

The square root of 2 was the first number proved irrational, and that article contains a number of proofs. The golden ratio is another famous quadratic irrational number. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.

General roots

The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact kth power of another integer, then that first integer's kth root is irrational.

Logarithms

Perhaps the numbers most easy to prove irrational are certain logarithms. Here is a proof by contradiction that log₂ 3 is irrational (log₂ 3 ≈ 1.58 > 0).

Assume log₂ 3 is rational. For some positive integers m and n, we have

$\log _{2}3={\frac {m}{n}}.$

It follows that

$2^{m/n}=3$

$(2^{m/n})^{n}=3^{n}$

$2^{m}=3^{n}.$

However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log₂ 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0). The contradiction means that this assumption must be false, i.e. log₂ 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.

Cases such as log₁₀ 2 can be treated similarly.

Types

• number theoretic distinction : transcendental/algebraic
• normal/ abnormal (non-normal)

Transcendental/algebraic

Almost all irrational numbers are transcendental and all real transcendental numbers are irrational (there are also complex transcendental numbers): the article on transcendental numbers lists several examples. So e ^r and π ^r are irrational for all nonzero rational r, and, e.g., e^π is irrational, too.

Irrational numbers can also be found within the countable set of real algebraic numbers (essentially defined as the real roots of polynomials with integer coefficients), i.e., as real solutions of polynomial equations

$p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=0\;,$

where the coefficients $a_{i}$ are integers and $a_{n}\neq 0$. Any rational root of this polynomial equation must be of the form r /s, where r is a divisor of a₀ and s is a divisor of a_n. If a real root $x_{0}$ of a polynomial $p$ is not among these finitely many possibilities, it must be an irrational algebraic number. An exemplary proof for the existence of such algebraic irrationals is by showing that x₀  = (2^1/2 + 1)^1/3 is an irrational root of a polynomial with integer coefficients: it satisfies (x³ − 1)² = 2 and hence x⁶ − 2x³ − 1 = 0, and this latter polynomial has no rational roots (the only candidates to check are ±1, and x₀, being greater than 1, is neither of these), so x₀ is an irrational algebraic number.

Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, $3\pi +2$, $\pi +{\sqrt {2}}$ and $e{\sqrt {3}}$ are irrational (and even transcendental).

Decimal expansions

The decimal expansion of an irrational number never repeats or terminates (the latter being equivalent to repeating zeroes), unlike any rational number. The same is true for binary, octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases.

To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m, only m remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats.

Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers. For example, consider:

$A=0.7\,162\,162\,162\,\ldots$

Here the repetend is 162 and the length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain:

$10A=7.162\,162\,162\,\ldots$

Now we multiply this equation by 10^r where r is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10³:

$10,000A=7\,162.162\,162\,\ldots$

The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000A matches the tail end of 10A exactly. Here, both 10,000A and 10A have .162 162 162 ... after the decimal point.

Therefore, when we subtract the 10A equation from the 10,000A equation, the tail end of 10A cancels out the tail end of 10,000A leaving us with:

$9990A=7155.$

Then

$A={\frac {7155}{9990}}={\frac {53}{74}}$

is a ratio of integers and therefore a rational number.

Irrational powers

Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a^b is rational:

Consider ${\sqrt {2}}^{\sqrt {2}}$; if this is rational, then take a = b = ${\sqrt {2}}$. Otherwise, take a to be the irrational number ${\sqrt {2}}^{\sqrt {2}}$ and b = ${\sqrt {2}}$. Then $a^{b}={({\sqrt {2}}^{\sqrt {2}})}^{\sqrt {2}}={\sqrt {2}}^{{\sqrt {2}}\cdot {\sqrt {2}}}={\sqrt {2}}^{2}=2$, which is rational.

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that \sqrt{2}^\sqrt{2} is transcendental, hence irrational. This theorem states that if a and b are both algebraic numbers, and a is not equal to 0 or 1, and b is not a rational number, then any value of a^b is a transcendental number (there can be more than one value if complex number exponentiation is used).

An example that provides a simple constructive proof is

$\left({\sqrt {2}}\right)^{\log _{\sqrt {2}}3}=3.$

The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, $\log _{\sqrt {2}}3$, is irrational. This is so because, by the formula relating logarithms with different bases,

$\log _{\sqrt {2}}3={\frac {\log _{2}3}{\log _{2}{\sqrt {2}}}}={\frac {\log _{2}3}{1/2}}=2\log _{2}3$

which we can assume, for the sake of establishing a contradiction, equals a ratio m/n of positive integers. Then $\log _{2}3=m/2n$ hence $2^{\log _{2}3}=2^{m/2n}$ hence $3=2^{m/2n}$ hence $3^{2n}=2^{m}$, which is a contradictory pair of prime factorizations and hence violates the fundamental theorem of arithmetic (unique prime factorization).

A stronger result is the following: Every rational number in the interval $((1/e)^{1/e},\infty )$ can be written either as a^a for some irrational number a or as nⁿ for some natural number n. Similarly, every positive rational number can be written either as $a^{a^{a}}$ for some irrational number a or as $n^{n^{n}}$ for some natural number n.

Open questions

It is not known if $\pi +e$ (or $\pi -e$) is irrational. In fact, there is no pair of non-zero integers $m,n$ for which it is known whether $m\pi +ne$ is irrational. Moreover, it is not known if the set $\{\pi ,e\}$ is algebraically independent over $\mathbb {Q}$.

It is not known if $\pi e,\ \pi /e,\ 2^{e},\ \pi ^{e},\ \pi ^{\sqrt {2}},\ \ln \pi ,$ Catalan's constant, or the Euler–Mascheroni constant $\gamma$ are irrational. It is not known if either of the tetrations $^{n}\pi$ or $^{n}e$ is rational for some integer $n>1.${

Set of all irrationals

Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.

Under the usual (Euclidean) distance function d(x, y) = |x − y|, the real numbers are a metric space and hence also a topological space. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed, the induced metric is not complete. However, being a G-delta set—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is completely metrizable: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.

Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen sets so the space is zero-dimensional.

Licensing

Content obtained and/or adapted from:

• Rational number, Wikipedia under a CC BY-SA license
• Irrational number, Wikipedia under a CC BY-SA license