What is in the classic English breakfast blend?

English breakfast tea is a traditional blend of teas originating from Assam, Ceylon, and Kenya. It is one of the most popular blended teas, common in British tea culture.

English breakfast tea

English breakfast tea is a black tea blend usually described as full-bodied, robust, rich and blended to go well with milk and sugar, in a style traditionally associated with a hearty English breakfast.

The black teas included in the blend vary, with Assam, Ceylon and Kenyan teas predominating, and Keemun sometimes included in more expensive blends. Common brands of English breakfast tea include Twinings, Dilmah, Taylors of Harrogate, Ahmad Tea, Tetley and PG Tips.

Accounts of its origins vary. Drinking a blend of black teas for breakfast is indeed a longstanding British custom. The practice of referring to such a blend as "English breakfast tea" appears to have originated not in England but America, as far back as Colonial times. An additional account (referencing a period-era Journal of Commerce article) dates the blend to 1843 and a tea merchant named Richard Davies in New York City. Davies, an English immigrant, started with a base of Congou and added a bit of Pekoe and Pouchong. It sold for 50 cents a pound, and its success led to imitators, helping to popularize the name. An investigation to find this source failed to locate it but did come upon an earlier reference to the same story in an 1876 edition of the Daily Alta California, citing "a New York commercial journal" and dating the tea's origin to 1844. In an 1884 American publication it was noted that "Bohea teas (are) known to trade in this country as "English Breakfast" tea, from its forming the staple shipment to England". Accounts of the emergence of the blend in the UK give its origins in Scotland in 1892, where it was initially known simply as "breakfast tea". It was in part popularised by Queen Victoria, having returned with a supply after tasting the tea at Balmoral and it subsequently acquired the prefix "English".

English breakfast tea

English breakfast tea


What's an irrational number?

A real number that is not rational is called irrational. Irrational numbers include,,, and. 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 number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator. Since may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface (or blackboard bold formula_1, Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano after "quoziente", Italian for "quotient".

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).

Rational numbers can be formally defined as equivalence classes of pairs of integers such that, for the equivalence relation defined by if, and only if. With this formal definition, the fraction becomes the standard notation for the equivalence class of.

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 are called algebraic number fields, and the algebraic closure of 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.

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 (that is 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, for avoiding confusion with "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.

Every rational number may be expressed in a unique way as an irreducible fraction, where and are coprime integers, and. This is often called the canonical form.

Starting from a rational number, its canonical form may be obtained by dividing and by their greatest common divisor, and, if, changing the sign of the resulting numerator and denominator.

Any integer can be expressed as the rational number, which is its canonical form as a rational number.

If both fractions are in canonical form then
If both denominators are positive, and, in particular, if both fractions are in canonical form,

If either denominator is negative, 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.

Two fractions are added as follows:

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

The rule for multiplication is:

Even if both fractions are in canonical form, the result may be a reducible fraction.

Every rational number has an additive inverse, often called its "opposite",
If is in canonical form, the same is true for its opposite.

A nonzero rational number has a multiplicative inverse, also called its "reciprocal",
If is in canonical form, then the canonical form of its reciprocal is either or, depending on the sign of.

If both and are nonzero, the division rule is

Thus, dividing by is equivalent to multiplying by the reciprocal of :

If is a non-negative integer, then
The result is in canonical form if the same is true for.
In particular,

If, then
If is in canonical form, the canonical form of the result is formula_19 if either or is even. Otherwise, the canonical form of the result is formula_20

A finite continued fraction is an expression such as
where "a" are integers. Every rational number "a"/"b" can be represented as a finite continued fraction, whose coefficients "a" can be determined by applying the Euclidean algorithm to ("a","b").are different ways to represent the same rational value.

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

More precisely, let be the set of the pairs of integers such. An equivalence relation is defined on this set by

Addition and multiplication can be defined by the following rules:

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 is the defined as the quotient set by this equivalence relation,, 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 is denoted formula_31
Two pairs and belong to the same equivalence class (that is are equivalent) if and only if formula_32 this means that formula_33 if and only formula_32

Every equivalence class formula_35 may be represented by infinitely many pairs, since
It is often convenient to chose, once for all, in each equivalence class a specific element called the "canonical representative element". This canonical representative is the unique pair in the equivalence class such that and are coprime, and. It is called the representation in lowest terms of the rational number.

The integers may be considered to be rational numbers identifying the integer with the rational number formula_37

A total order may be defined on the rational numbers, that extends the natural order of the integers. One has formula_38 if

The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers Z.

The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime field for characteristic zero.

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

The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.

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
(where formula_41 are positive), we have

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.

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 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 above.

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, where "p" is the highest power of "p" dividing "a".

In addition set For any rational number "a"/"b", we set

Then defines a metric on Q.

The metric space (Q,"d") is not complete, and its completion is the "p"-adic number field Q. 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.

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of 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.

Irrational number

Among irrational numbers are the ratio  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.

It can be shown that irrational numbers, when expressed in a positional numeral system (e.g. as decimal numbers, or with any other natural basis), do not terminate, nor do they repeat, i.e., do not contain a subsequence of digits, the repetition of which makes up the tail of the representation. For example, the decimal representation of the number starts with 3.14159, but no finite number of digits can represent exactly, nor does it repeat. The proof that the decimal expansion of a rational number must terminate or repeat is distinct from the proof that a decimal expansion that terminates or repeats must be a rational number, and although elementary and not lengthy, both proofs take some work. Mathematicians do not generally take "terminating or repeating" to be the definition of the concept of rational number.

Irrational numbers may also be dealt with via non-terminating continued fractions.

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.

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram.
The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

Greek mathematicians termed this ratio of incommensurable magnitudes "alogos", or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.

The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for “whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects.” However Zeno found that in fact “[quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appear….[Q]uantities are, in other words, continuous.” What this means is that, contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. That in fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno’s paradoxes accurately demonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.

The next step was taken by Eudoxus of Cnidus, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude “...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5.” Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. “Eudoxus’ theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.” This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9.

As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of "x" and "x" as "x" squared and "x" cubed instead of "x" to the second power and "x" to the third power. Also crucial to Zeno’s work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory. Out of this necessity, Eudoxus developed his method of exhaustion, a kind of reductio ad absurdum that “…established the deductive organization on the basis of explicit axioms…” as well as “…reinforced the earlier decision to rely on deductive reasoning for proof.” This method of exhaustion is the first step in the creation of calculus.

Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of 17.

It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created.

Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the "Samhitas", "Brahmanas", and the "Shulba Sutras" (800 BC or earlier). (See Bag, Indian Journal of History of Science, 25(1-4), 1990).

It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. However, historian Carl Benjamin Boyer writes that "such claims are not well substantiated and unlikely to be true".

It is also suggested that Aryabhata (5th century AD), in calculating a value of pi to 5 significant figures, used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational).

Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots.

Mathematicians like Brahmagupta (in 628 AD) and Bhaskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century Bhaskara II evaluated some of these formulas and critiqued them, identifying their limitations.

During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as "π" and certain irrational values of trigonometric functions. Jyeṣṭhadeva provided proofs for these infinite series in the "Yuktibhāṣā".

In the Middle ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as "algebraic objects". Middle Eastern mathematicians also merged the concepts of "number" and "magnitude" into a more general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. In his commentary on Book 10 of the "Elements", the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:

In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:

The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots. In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions. Iranian mathematician, Abū Ja'far al-Khāzin (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity is:

Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century. Al-Hassār, a Moroccan mathematician from Fez specializing in Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, formula_1." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.

The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. The completion of the theory of complex numbers in the 19th century entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine ("Crelle's Journal", 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880, and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers, separating them into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Méray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings of Joseph-Louis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

Johann Heinrich Lambert proved (1761) that π cannot be rational, and that "e" is irrational if "n" is rational (unless "n" = 0). While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the Bessel–Clifford function, provided a proof to show that π is irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method, which showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved "e" transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan.

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 and there is a simple proof of its irrationality in its article. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.

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 "k" power of another integer, then that first integer's "k" root is irrational.

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

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

It follows that

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 log 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. log 3 is irrational, and can never be expressed as a quotient of integers "m"/"n" with "n" ≠ 0.

Cases such as log 2 can be treated similarly.

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" and π are irrational for all nonzero rational "r", and, e.g., "e" 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

where the coefficients formula_7 are integers and formula_8. Any rational root of this polynomial equation must be of the form "r" /"s", where "r" is a divisor of "a" and "s" is a divisor of "a". If a real root formula_9 of a polynomial formula_10 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 "x"  = (2 + 1) is an irrational root of a polynomial with integer coefficients: it satisfies ("x" − 1) = 2 and hence "x" − 2"x" − 1 = 0, and this latter polynomial has no rational roots (the only candidates to check are ±1, and "x", being greater than 1, is neither of these), so "x" 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 + 2,  +  and "e" are irrational (and even transcendental).

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:

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:

Now we multiply this equation by 10 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 10:

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

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

Then

is a ratio of integers and therefore a rational number.

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

Consider ; if this is rational, then take "a" = "b" =. Otherwise, take "a" to be the irrational number and "b" =. Then "a" = () = = = 2, which is rational.

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that 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" 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

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, formula_17, is irrational. This is so because, by the formula relating logarithms with different bases,

which we can assume, for the sake of establishing a contradiction, equals a ratio "m/n" of positive integers. Then formula_19 hence formula_20 hence formula_21 hence formula_22, 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 formula_23 can be written either as "a" for some irrational number "a" or as "n" for some natural number "n". Similarly, every positive rational number can be written either as formula_24 for some irrational number "a" or as formula_25 for some natural number "n".

It is not known if formula_26 (or formula_27) is irrational. In fact, there is no pair of non-zero integers formula_28 for which it is known whether formula_29 is irrational. Moreover, it is not known whether or not the set formula_30 is algebraically independent over formula_31.

It is not known if formula_32 Catalan's constant, or the Euler–Mascheroni constant formula_33 are irrational. It is not known if either of the tetrations formula_34 or formula_35 is rational for some integer formula_36

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 have a basis of clopen sets so the space is zero-dimensional.


What is the worlds tallest building?

"Guinness World Records" has called the CN Tower "the world's tallest self-supporting tower" and "the world's tallest free-standing tower". Although Guinness did list this description of the CN Tower under the heading "tallest building" at least once, it has also listed it under "tallest tower", omitting it from its list of "tallest buildings." In 1996, Guinness changed the tower's classification to "World's Tallest Building and Freestanding Structure". Emporis and the Council on Tall Buildings and Urban Habitat both listed the CN Tower as the world's tallest free-standing structure on land, and specifically state that the CN Tower is not a true building, thereby awarding the title of world's tallest building to Taipei 101, which is shorter than the CN Tower. The issue of what was tallest became moot when Burj Khalifa, then under construction, exceeded the height of the CN Tower in 2007 (see below).

CN Tower

The CN Tower () is a concrete communications and observation tower located in Downtown Toronto, Ontario, Canada. Built on the former Railway Lands, it was completed in 1976. Its name "CN" originally referred to Canadian National, the railway company that built the tower. Following the railway's decision to divest non-core freight railway assets prior to the company's privatization in 1995, it transferred the tower to the Canada Lands Company, a federal Crown corporation responsible for real estate development.

The CN Tower held the record for the world's tallest free-standing structure for 32 years until 2007 and was the world's tallest tower until 2009 when it was overtaken by Burj Khalifa and Canton Tower, respectively. It is now the ninth tallest free-standing structure in the world and remains the tallest free-standing structure in the Western Hemisphere. In 1995, the CN Tower was declared one of the modern Seven Wonders of the World by the American Society of Civil Engineers. It also belongs to the World Federation of Great Towers.

It is a signature icon of Toronto's skyline and attracts more than two million international visitors annually.

The original concept of the CN Tower originated in 1968 when the Canadian National Railway wanted to build a large TV and radio communication platform to serve the Toronto area, as well as demonstrate the strength of Canadian industry and CN in particular. These plans evolved over the next few years, and the project became official in 1972.

The tower would have been part of Metro Centre (see CityPlace), a large development south of Front Street on the Railway Lands, a large railway switching yard that was being made redundant by newer yards outside the city. Key project team members were NCK Engineering as structural engineer; John Andrews Architects; Webb, Zerafa, Menkes, Housden Architects; Foundation Building Construction; and Canron (Eastern Structural Division).

As Toronto grew rapidly during the late 1960s and early 1970s, multiple skyscrapers were constructed in the downtown core, most notably First Canadian Place. The reflective nature of the new buildings compromised the quality of broadcast signals necessitating new, higher antennas that were at least tall.

At the time, most data communications took place over point-to-point microwave links, whose dish antennae covered the roofs of large buildings. As each new skyscraper was added to the downtown, former line-of-sight links were no longer possible. CN intended to rent "hub" space for microwave links, visible from almost any building in the Toronto area.

The CN Tower can be seen from at least as far away as Kennedy Street in Aurora, Ontario, approximately to the north. It is also viewable to the naked eye from east of Toronto in Oshawa, several points along the Niagara Escarpment west of Toronto in Hamilton, Ontario, and to the south from Fort Niagara State Park in the U.S. state of New York.

The original plan for the tower envisioned a tripod consisting of three independent cylindrical "pillars" linked at various heights by structural bridges. Had it been built, this design would have been considerably shorter, with the metal antenna located roughly where the concrete section between the main level and the SkyPod lies today. As the design effort continued, it evolved into the current design with a single continuous hexagonal core to the SkyPod, with three support legs blended into the hexagon below the main level, forming a large Y-shape structure at the ground level.

The idea for the main level in its current form evolved around this time, but the Space Deck (now named SkyPod) was not part of the plans until some time later. One engineer in particular felt that visitors would feel the higher observation deck would be worth paying extra for, and the costs in terms of construction were not prohibitive. It was also some time around this point that it was realized that the tower could become the world's tallest structure, and plans were changed to incorporate subtle modifications throughout the structure to this end.

Construction on the CN Tower began on February 6, 1973, with massive excavations at the tower base for the foundation. By the time the foundation was complete, of earth and shale were removed to a depth of in the centre, and a base incorporating of concrete with of rebar and of steel cable had been built to a thickness of. This portion of the construction was fairly rapid, with only four months needed between the start and the foundation being ready for construction on top.

To create the main support pillar, workers constructed a hydraulically raised slipform at the base. This was a fairly unprecedented engineering feat on its own, consisting of a large metal platform that raised itself on jacks at about per day as the concrete below set. Concrete was poured continuously by a team of 1,532 people until February 22, 1974, during which it had already become the tallest structure in Canada, surpassing the recently built Inco Superstack in Sudbury, which was built using similar methods.

In total, the tower contains of concrete, all of which was mixed on-site in order to ensure batch consistency. Through the pour, the vertical accuracy of the tower was maintained by comparing the slip form's location to massive plumb bobs hanging from it, observed by small telescopes from the ground. Over the height of the tower, it varies from true vertical accuracy by only.
In August 1974, construction of the main level commenced. Using 45 hydraulic jacks attached to cables strung from a temporary steel crown anchored to the top of the tower, twelve giant steel and wooden bracket forms were slowly raised, ultimately taking about a week to crawl up to their final position. These forms were used to create the brackets that support the main level, as well as a base for the construction of the main level itself. The Space Deck (currently named SkyPod) was built of concrete poured into a wooden frame attached to rebar at the lower level deck, and then reinforced with a large steel compression band around the outside.

The antenna was originally to be raised by crane as well, but during construction the Sikorsky S-64 Skycrane helicopter became available when the United States Army sold one to civilian operators. The helicopter, named "Olga", was first used to remove the crane, and then flew the antenna up in 36 sections.

The flights of the antenna pieces were a minor tourist attraction of their own, and the schedule was printed in the local newspapers. Use of the helicopter saved months of construction time, with this phase taking only three and a half weeks instead of the planned six months. The tower was topped-off on April 2, 1975, after 26 months of construction, officially capturing the height record from Moscow's Ostankino Tower, and bringing the total mass to.

Two years into the construction, plans for Metro Centre were scrapped, leaving the tower isolated on the Railway Lands in what was then a largely abandoned light-industrial space. This caused serious problems for tourists to access the tower. Ned Baldwin, project architect with John Andrews, wrote at the time that "All of the logic which dictated the design of the lower accommodation has been upset," and that "Under such ludicrous circumstances Canadian National would hardly have chosen this location to build."

The CN Tower opened to the public on June 26, 1976. The construction costs of approximately ($ in dollars) were repaid in fifteen years. Canadian National Railway sold the tower prior to taking the company private in 1995, when it decided to divest all operations not directly related to its core freight shipping businesses.

From the mid-1970s to the mid-1980s, the CN Tower was practically the only development along Front Street West; it was still possible to see Lake Ontario from the foot of the CN Tower due to the expansive parking lots and lack of development in the area at the time. As the area around the tower was developed, particularly with the completion of the Metro Toronto Convention Centre in 1984 and the SkyDome in 1989 (renamed Rogers Centre in 2005), the former Railway Lands were redeveloped and the tower became the centre of a newly developing entertainment area. Access was greatly improved with the construction of the SkyWalk in 1989, which connected the tower and SkyDome to the nearby Union Station railway and subway station, and, in turn, to the city's PATH underground pedestrian system. By the mid-1990s, it was the centre of a thriving tourist district. The entire area continues to be an area of intense building, notably a boom in condominium construction in the first two decades of the 21st century, as well as the 2013 opening of the Ripley's Aquarium by the base of the tower.The CN Tower consists of several substructures. The main portion of the tower is a hollow concrete hexagonal pillar containing the stairwells and power and plumbing connections. The Tower's six elevators are located in the three inverted angles created by the Tower's hexagonal shape (two elevators per angle). Each of the three elevator shafts is lined with glass, allowing for views of the city as the glass-windowed elevators make their way through the Tower. The stairwell was originally located in one of these angles (the one facing north), but was moved into the central hollow of the Tower; the Tower's new fifth and sixth elevators were placed in the hexagonal angle that once contained the stairwell. On top of the main concrete portion of the Tower is a tall metal broadcast antenna, carrying TV and radio signals. There are three visitor areas: the Glass Floor and Outdoor Observation Terrace, which are both located at an elevation of, the Indoor Lookout Level (formerly known as "Indoor Observation Level") located at, and the higher SkyPod (formerly known as "Space Deck") at, just below the metal antenna. The hexagonal shape can be seen between the two areas; however, below the main deck, three large supporting legs give the tower the appearance of a large tripod.

The main deck level is seven storeys, some of which are open to the public. Below the public areas — at — is a large white donut-shaped radome containing the structure's microwave receivers. The glass floor and outdoor observation deck are at. The glass floor has an area of and can withstand a pressure of. The floor's thermal glass units are thick, consisting of a pane of laminated glass, airspace and a pane of laminated glass. Some people experience acrophobia when standing on the glass floor and looking down at the ground below. In 2008, one elevator was upgraded to add a glass floor panel, believed to have the highest vertical rise of any elevator equipped with this feature. The Horizons Cafe and the lookout level are at. The 360 Restaurant, a revolving restaurant that completes a full rotation once every 72 minutes, is at. When the tower first opened, it also featured a disco named Sparkles (at the Indoor Observation Level), billed as the highest disco and dance floor in the world.

The SkyPod was once the highest public observation deck in the world until it was surpassed by the Shanghai World Financial Center in 2008.

A metal staircase reaches the main deck level after 1,776 steps, and the SkyPod above after 2,579 steps; it is the tallest metal staircase on Earth. These stairs are intended for emergency use only and are not open to the public, except for twice per year for charity stair-climb events. The average climber takes approximately 30 minutes to climb to the base of the radome, but the fastest climb on record is 7 minutes and 52 seconds in 1989 by Brendan Keenoy, an Ontario Provincial Police officer. In 2002, Canadian Olympian and Paralympic champion Jeff Adams climbed the stairs of the tower in a specially designed wheelchair. The stairs were originally on one of the three sides of the tower (facing north), with a glass view, but these were later replaced with the third elevator pair and the stairs were moved to the inside of the core. Top climbs on the new, windowless stairwell used since around 2003 have generally been over ten minutes.

On August 1, 2011, the CN Tower opened the EdgeWalk, an amusement in which thrill-seekers can walk on and around the roof of the main pod of the tower at, which is directly above the 360 Restaurant. It is the world's highest full-circle, hands-free walk. Visitors are tethered to an overhead rail system and walk around the edge of the CN Tower's main pod above the 360 Restaurant on a metal floor. The attraction is closed throughout the winter season and during periods of electrical storms and high winds.

One of the notable guests who visited EdgeWalk includes Canadian comedian Rick Mercer, as featured on his CBC Television news satire show, "Rick Mercer Report". There, he was accompanied by Canadian pop singer Jann Arden.

A freezing rain storm on March 2, 2007, resulted in a layer of ice several centimetres thick forming on the side of the tower and other downtown buildings. The sun thawed the ice, and winds of up to blew some of it away from the structure. There were fears that cars and windows of nearby buildings would be smashed by large chunks of ice. In response, police closed some streets surrounding the tower. During morning rush hour on March 5 of the same year, police expanded the area of closed streets to include the Gardiner Expressway away from the tower as increased winds blew the ice farther away, as far north as King Street, away, where a taxicab window was shattered. Subsequently, on March 6, 2007, the Gardiner Expressway reopened after winds abated.

On April 16, 2018, falling ice from the CN Tower punctured the roof of the nearby Rogers Centre, causing the Toronto Blue Jays to postpone the game that day to the following day as a doubleheader. On April 20, the CN Tower reopened.

In August 2000, a fire broke out at the Ostankino Tower in Moscow killing three people and causing extensive damage. The fire was blamed on poor maintenance and outdated equipment. The failure of the fire-suppression systems and the lack of proper equipment for firefighters allowed the fire to destroy most of the interior and spark fears the tower might even collapse.

The Ostankino Tower was completed nine years before the CN Tower, and is only shorter. The parallels between the towers led to some concern that the CN Tower could be at risk of a similar tragedy. However, Canadian officials subsequently stated that it is "highly unlikely" that a similar disaster could occur at the CN Tower, as it has important safeguards that were not present in the Ostankino Tower. Specifically, officials cited:Officials also noted that the CN Tower has an excellent safety record, although there was an electrical fire in the antennae on August 16, 2017 — the tower's first fire. Moreover, other supertall structures built between 1967 and 1976 — such as the Willis Tower (formerly the Sears Tower), the World Trade Center (until its destruction on September 11, 2001), the Fernsehturm Berlin, the Aon Center, 875 North Michigan Avenue (formerly the John Hancock Center), and First Canadian Place — also have excellent safety records, which suggests that the Ostankino Tower accident was a rare safety failure, and that the likelihood of similar events occurring at other supertall structures is extremely low.

The CN Tower was originally lit at night with incandescent lights, which were removed in 1997, because they were inefficient and expensive to repair. In June 2007, the tower was outfitted with 1,330 super-bright LED lights inside the elevator shafts, shooting over the main pod and upward to the top of the tower's mast to light the tower from dusk until 2 a.m. The official opening ceremony took place on June 28 before the Canada Day holiday weekend.

The tower changes its lighting scheme on holidays and to commemorate major events. After the 95th Grey Cup in Toronto, the tower was lit in green and white to represent the colours of the Grey Cup champion Saskatchewan Roughriders. From sundown on August 27, 2011, to sunrise the following day, the tower was lit in orange, the official colour of the New Democratic Party (NDP), to commemorate the death of federal NDP leader and leader of the official opposition Jack Layton. When former South African president Nelson Mandela died, the tower was lit in the colours of the South African flag. When former federal finance minister under Stephen Harper's Conservatives Jim Flaherty died, the tower was lit in green to reflect his Irish Canadian heritage. On the night of the attacks on Paris on November 13, 2015, the tower displayed the colours of the French flag.

Programmed from a desktop computer with a wireless network interface controller in Burlington, Ontario, the LEDs use less energy to light than the previous incandescent lights (10% less energy than the dimly lit version and 60% less than the brightly lit version). The estimated cost to use the LEDs is $1,000 per month.

During the spring and autumn bird migration seasons, the lights would be turned off to comply with the voluntary Fatal Light Awareness Program, which "encourages buildings to dim unnecessary exterior lighting to mitigate bird mortality during spring and summer migration."

The CN Tower is the tallest freestanding structure in the Western Hemisphere. As of 2013, there are only two other freestanding structures in the Western Hemisphere which exceed in height; the Willis Tower in Chicago, which stands at when measured to its pinnacle; and the topped-out One World Trade Center in New York City, which has a pinnacle height of, or approximately shorter than the CN Tower. Due to the symbolism of the number 1776 (the year of the signing of the United States Declaration of Independence), the height of One World Trade Center is unlikely to be increased. The proposed Chicago Spire was expected to exceed the height of the CN Tower, but its construction was halted early due to financial difficulties amid the Great Recession, and was eventually cancelled in 2010.

Although the CN Tower contains a restaurant, a gift shop and multiple observation levels, it does not have floors continuously from the ground, and therefore it is not considered a building by the Council on Tall Buildings and Urban Habitat (CTBUH) or Emporis. CTBUH defines a building as "a structure that is designed for residential, business, or manufacturing purposes. An essential characteristic of a building is that it has floors." The CN Tower and other similar structures—such as the Ostankino Tower in Moscow, Russia; the Oriental Pearl Tower in Shanghai, China; the Stratosphere Tower in Las Vegas, Nevada, USA; and the Eiffel Tower in Paris, France—are categorized as "towers", which are free-standing structures that may have observation decks and a few other habitable levels, but do not have floors from the ground up. The CN Tower was the tallest tower by this definition until 2010 (see below).

Taller than the CN Tower are numerous radio masts and towers, which are held in place by guy-wires, the tallest being the KVLY-TV mast in Blanchard, North Dakota, USA at tall, leading to a distinction between these and "free-standing" structures. Additionally, the Petronius Platform stands above its base on the bottom of the Gulf of Mexico, but only the top of this oil and natural gas platform are above water, and the structure is thus partially supported by its buoyancy. Like the CN Tower, none of these taller structures are commonly considered buildings.

On September 12, 2007, Burj Khalifa, which is a hotel, residential and commercial building in Dubai, United Arab Emirates, and was formerly known as Burj Dubai before opening, passed the CN Tower's 553.33-metre height. The CN Tower held the record of tallest free-standing structure on land for over 30 years.

After Burj Khalifa had been formally recognized by the Guinness World Records as the world's tallest freestanding structure, Guinness re-certified CN Tower as the world's tallest freestanding tower. The tower definition used by Guinness was defined by the Council on Tall Buildings and Urban Habitat as 'a building in which less than 50% of the construction is usable floor space'. "Guinness World Records" editor-in-chief Craig Glenday announced that Burj Khalifa was not classified as a tower because it has too much usable floor space to be considered to be a tower. CN Tower still held world records for highest above ground wine cellar (in 360 Restaurant) at 351 metres, highest above ground restaurant at 346 metres (Horizons Restaurant), and tallest free-standing concrete tower during Guinness's recertification. The CN Tower was surpassed in 2009 by the Canton Tower in Guangzhou, China, which stands at tall, as the world's tallest tower; which in turn was surpassed by the Tokyo Skytree in 2011, which currently is the tallest tower at in height. The CN Tower, as of 2018, stands as the ninth-tallest free-standing structure on land, remains the tallest free-standing structure in the Western Hemisphere, and is the third-tallest tower.

Since the construction of the tower had been completed, it has gained the following world height records:

The CN Tower has been and continues to be used as a communications tower for a number of different media and by numerous companies.

There is no AM broadcasting on the CN Tower. The FM antennas are situated above ground.
The CN Tower has been featured in numerous films, television shows, and video games, as well as having its own official mascot, which resembles the tower itself.

The tallest building in the world, as of 2022, is Burj Khalifa. The title of "world's tallest building" has been borne by various buildings, such as the Rouen Cathedral and the Empire State Building.

History of the world's tallest buildings

The skyscraper was invented in Chicago in 1884 when the Home Insurance Building was constructed using a steel-frame with curtain walls instead of load-bearing walls. For the next hundred years, the world's tallest building was always in the United States with New York City accumulating 86 years, and Chicago accumulating 30 years. After just over a century (1885–1998), the distinction moved to the Eastern Hemisphere. Malaysia was the first country to break the United States' record of constructing the tallest buildings in the world when the Petronas Twin Towers were completed in 1998. Taiwan's Taipei 101 was the next building to hold the record, beginning in 2004.

Before the modern skyscraper era, many Christian churches and cathedrals, built mainly in England and Germanic territories between 1250–1894, comprised the tallest buildings in the world. Before the 13th century, the tallest buildings in the world cannot be conclusively determined. For instance, the Lighthouse of Alexandria (completed about 280 BC) has been estimated to be tall, but its true height is not known. For thousands of years, the Great Pyramid in Egypt was the tallest "structure" in the world, but the Great Pyramid is not considered a building since it is not habitable.

The earliest structures now known to be the tallest in the world were the Egyptian pyramids, with the Great Pyramid of Giza, at an original height of, being the tallest man-made structure in the world for over 3,800 years, until the construction of Lincoln Cathedral in 1300. From then until the completion of the Washington Monument (capped in 1884) the world's tallest buildings were churches or cathedrals. Later, the Eiffel Tower and, still later, some radio masts and television towers were the world's tallest structures.

However, though all of these are "structures", some are not "buildings" in the sense of being regularly inhabited or occupied. It is in this sense of being regularly inhabited or occupied that the term "building" is generally understood to mean when determining what is the world's tallest building. The non-profit international organization Council on Tall Buildings and Urban Habitat (CTBUH), which maintains a set of criteria for determining the height of tall buildings, defines a "building" as "(A) structure that is designed for residential, business or manufacturing purposes" and "has floors".

Tall churches and cathedrals occupy a middle ground: their lower areas are regularly occupied, but much of their height is in bell towers and spires which are not. Whether a church or cathedral is a "building" or merely a "structure" for the purposes of determining the title of "world's tallest building" is a subjective matter of definition (this article treats churches and cathedrals as buildings).

The Council on Tall Buildings and Urban Habitat based in Chicago uses three different criteria for determining the height of a tall building, each of which may give a different result. "Height of the highest floor" is one criterion, and "height to the top of any part of the building" is another, but the default criterion used by the CTBUH is "height of the architectural top of the building", which includes spires but not antennae, masts or flag poles.

Hwangnyongsa, or Hwangnyong Temple (also spelled Hwangryongsa) is the name of a former Buddhist temple in the city of Gyeongju, South Korea. Completed in the 7th century, the enormous 9-story structure was built entirely with wood with interlocking design with no iron nails. It had a standing total height of 68 m (223 ft) or 80 m (262 ft), making it the tallest structure in East Asia and the tallest wooden structure in the world at the time of its construction.

From the 13th century until 1894, the world's tallest building was always a church or cathedral. Old St Paul's Cathedral with its spire was completed in the 13th century. The central spire of Lincoln Cathedral surpassed Old St Paul's in the early 14th century. The Lincoln Cathedral's spire collapsed in 1549, beginning a long interval where the status of world's tallest building was borne by shorter buildings. St. Mary's Church in Stralsund became the world's tallest building after the collapse of Lincoln Cathedral's spire. The central tower of St. Pierre's Cathedral was tallest from 1569 until it collapsed in 1573, making St. Mary's the tallest once again. In 1647, the bell tower of St. Mary's burned down, making the shorter Strasbourg Cathedral the world's tallest building.

It was not until the completion of the Ulm Minster in 1890 that the world's tallest building was again also the tallest building ever constructed, surpassing the original configuration of Lincoln Cathedral.
* – Also set record at time of completion as tallest building ever built.

The height of Lincoln Cathedral is disputed by some, but accepted by most sources. The completion date for the spire is given as 1311 rather than 1300 by some sources. Also the height of the spire of Old St Paul's Cathedral, destroyed by lightning in 1561, is disputed, for example Christopher Wren (1632–1723) judged that an overestimate and gave a height of.

The spire of Mole Antonelliana in Turin, completed in 1889, is claimed to have been tall; however, the upper part of the structure was destroyed by a 1953 tornado and rebuilt. If the original spire was as tall as is claimed, then the Mole Antonelliana was the world's tallest building from 1889 to 1908, surpassing Philadelphia City Hall by.

The buildings that were the tallest secular skyscrapers, which excludes religious buildings, were:
The list of tallest buildings is based on the default metric of the Council on Tall Buildings and Urban Habitat (CTBUH), that of measuring to the highest architectural element. Other criteria would generate a different list. Shanghai World Financial Center is not on the above list, but it surpassed Taipei 101 in 2008 to become the building with the highest occupied floor. Using the criteria of highest tip (including antennae), the World Trade Center in New York City was the world's tallest building from 1972 to 2000, until the Sears Tower in Chicago (which already had a higher occupied floor than the World Trade Center) had its antenna extended to give that building the world's tallest tip; a title it held until the 2010 completion of Burj Khalifa. Petronas Towers and Taipei 101 were never the world's tallest buildings by the highest–tip criteria.

Since 2010, Burj Khalifa has been the tallest building by any criteria. It has the highest architectural element, tip and occupied floor, and is indeed the tallest structure of any kind ever built, surpassing the (now destroyed) Warsaw radio mast.

Since the completion of the Washington Monument in 1884, the world's tallest building has not usually also been the world's tallest "structure". The exceptions are 1930–1954, when the Chrysler Building and then the Empire State building surpassed the Eiffel Tower (to be surpassed in turn by a succession of broadcast masts, starting with the Griffin Television Tower in Oklahoma), and from 2010 with the completion of Burj Khalifa.

In this chart, time progresses from right to left. Note the early buildings that lost the title as their spires collapsed.
After the construction of the Home Insurance Building in Chicago in the 19th century, the incrementation in the height of skyscrapers began with the construction of the Chrysler Building, followed by the Empire State Building, in New York City. The Chrysler Building was the first building in the world to break the barrier, and the Empire State Building was the first building to have more than 100 floors. It stands at and has 102 floors. The next tallest skyscraper was the World Trade Center, which was completed in 1971. The north tower was and the south tall. It surpassed the height of the Empire State Building by. Two years later the Sears Tower was built in Chicago, standing at with 110 floors, surpassing the height of the World Trade Center by. The Petronas Towers rose 10 meters above the Sears Tower, standing at a height of and each having 88 floors.

In 2004, the construction of Taipei 101 brought the height of skyscrapers to a new level, standing at with 101 floors. It is taller than the previous record holders, the Petronas Towers. Burj Khalifa surpassed the height of Taipei 101 by in 2009, making it 60% taller. It has broken several skyscraper records, and it is almost twice as tall as the Empire State Building. Burj Khalifa has also broken the record of the world's tallest structure.Since the early skyscraper boom that took place in North America, the significant number of skyscrapers in North America have dominated the 100 tallest buildings in the world. In 1930, 99 of the 100 tallest buildings in the world were located in North America. In the future, this percentage is expected to decline to only 22 percent. The predominance of skyscrapers in North America is decreasing due to skyscraper construction in other parts of the world, especially in Asia.

In Asia, there has been an increase in the number of supertall skyscrapers beginning with the construction of Petronas Twin Towers. There are currently sixty buildings in the world's 100 tallest that are located in Asia (including the Middle East).

Since the skyscraper era began, the great majority of skyscrapers were used predominantly as office space. From 1930 to 2000, the percentage of office towers never fell below 86 percent, but in the future it is expected to be as low as 46 percent. By 2010, less than half of the 100 tallest buildings in the world were office towers with the majority utilized as residential and mixed use. Presently, only four out of the ten tallest buildings in the world, and twenty eight out of the fifty tallest in the world, are used primarily as offices.

A mixed-use tall building contains two or more functions (uses), where each of the functions occupies a significant proportion of the tower's total space. Support areas such as car parks and mechanical plant space do not contribute towards mixed-use status.

Skyscrapers used primarily or exclusively as hotels or residential space are generally shorter than office and mixed-use buildings, with only a few supertall buildings of the residential or hotel types among the 100 tallest skyscrapers. The tallest completed residential building (minimum 85% residential) is 432 Park Avenue in New York City, followed by the Princess Tower and 23 Marina, both in Dubai. The tallest completed hotels (primarily hotel space) are the Gevora Hotel, the JW Marriott Marquis Dubai twin towers, the Rose Tower, and the Burj Al Arab, all located in Dubai.

The tallest building in the world, as of 2022, is Burj Khalifa. The title of "world's tallest building" has been borne by various buildings, such as the Rouen Cathedral and the Empire State Building.

It was not until the completion of the Ulm Minster in 1890 that the world's tallest building was again also the tallest building ever constructed, surpassing the original configuration of Lincoln Cathedral.
* – Also set record at time of completion as tallest building ever built.

The height of Lincoln Cathedral is disputed by some, but accepted by most sources. The completion date for the spire is given as 1311 rather than 1300 by some sources. Also the height of the spire of Old St Paul's Cathedral, destroyed by lightning in 1561, is disputed, for example Christopher Wren (1632–1723) judged that an overestimate and gave a height of.

The spire of Mole Antonelliana in Turin, completed in 1889, is claimed to have been tall; however, the upper part of the structure was destroyed by a 1953 tornado and rebuilt. If the original spire was as tall as is claimed, then the Mole Antonelliana was the world's tallest building from 1889 to 1908, surpassing Philadelphia City Hall by.

Since 2010, Burj Khalifa has been the tallest building by any criteria. It has the highest architectural element, tip and occupied floor, and is indeed the tallest structure of any kind ever built, surpassing the (now destroyed) Warsaw radio mast.

In Asia, there has been an increase in the number of supertall skyscrapers beginning with the construction of Petronas Twin Towers. There are currently sixty buildings in the world's 100 tallest that are located in Asia (including the Middle East).

A mixed-use tall building contains two or more functions (uses), where each of the functions occupies a significant proportion of the tower's total space. Support areas such as car parks and mechanical plant space do not contribute towards mixed-use status.

Dubai, the largest city in the United Arab Emirates, is home to many modern high-rises, 88 of which stand taller than. The tallest building in Dubai is the Burj Khalifa, which rises and contains 161 floors. The tower has stood as both the tallest building in the world and the tallest man-made structure of any kind in the world since its completion in January 2010. The second-tallest building in Dubai is the Marina 101, which also stands as the world's tallest residential skyscraper. The skyscrapers of Dubai are, for the most part, clustered in three different locations. The land along Sheikh Zayed Road was the first to develop, followed by the Dubai Marina neighborhood and the Business Bay district.

List of tallest buildings in Dubai

Overall, Dubai has 18 completed and topped-out buildings that rise at least in height, which is more than any other city in the world. Dubai has 73 completed and topped-out buildings that rise at least in height, again which is more than any other city in the world. Based on the average height of the ten tallest completed buildings, Dubai has the tallest skyline in the Middle East and the world., the skyline of Dubai is ranked sixth in the world with 248 buildings rising at least in height.

The history of skyscrapers in Dubai began with the construction of Dubai World Trade Centre in 1979, which is usually regarded as the first high-rise in the city. At the time of its completion, it also stood as the tallest building in the Middle East. Since 1999, and especially from 2005 onwards, Dubai has been the site of an extremely large skyscraper building boom, with all 73 of its buildings over tall completed after 1999. In less than ten years, the city has amassed one of the largest skylines in the world; it is now home to the world's tallest building, the world's tallest residence, and the world's second tallest hotel., 363 new skyscrapers are under construction in Dubai; additionally, there are over 640 active high-rise developments that have been proposed for construction in the city.

This list ranks completed and topped out Dubai skyscrapers that stand at least tall, based on standard height measurement. This includes spires and architectural details but does not include antenna masts. An equal sign (=) following a rank indicates the same height between two or more buildings. The "Year" column indicates the year in which a building was completed.
* indicates building is still under construction, but has been topped out

This is the list of buildings that are currently under construction in Dubai and are expected to rise to a height of at least. Buildings under construction that have already been topped out are also included, as are those whose construction has been suspended.
*Table entries with dashes (—) indicate that information regarding building dates of completion has not yet been released.

This table lists buildings that were at one time under construction in Dubai and were expected to rise at least in height, but are now on hold. While not officially cancelled, construction has been suspended on each development.
This table lists buildings that are approved for construction in Dubai and are expected to rise at least in height.
*Table entries with dashes (—) indicate that information regarding building dates of completion has not yet been released.
This table lists buildings that are proposed for construction in Dubai and are expected to rise at least in height.
*Table entries with dashes (—) indicate that information regarding building heights and/or dates of completion has not yet been released.

This is a list of the buildings that once were the tallest in Dubai. Despite Dubai's recent major skyscraper boom, there are only seven buildings on the list.

The international non-profit organization Council on Tall Buildings and Urban Habitat (CTBUH) was formed in 1969 and announces the title of "The World's Tallest Building" and sets the standards by which buildings are measured. It maintains a list of the 100 tallest completed buildings in the world. The organization currently ranks Burj Khalifa in Dubai as the tallest at. The CTBUH only recognizes buildings that are complete however, and some buildings included within the lists in this article are not considered finished by the CTBUH.

List of tallest buildings

This list of tallest buildings includes skyscrapers with continuously occupiable floors and a height of at least. Non-building structures, such as towers, are not included in this list (see list of tallest buildings and structures).

In 1996, as a response to the dispute as to whether the Petronas Towers or the Sears Tower was taller, the council listed and ranked buildings in four categories:

Spires are considered integral parts of the architectural design of buildings, to which changes would substantially change the appearance and design of the building, whereas antennas may be added or removed without such consequences. The Petronas Towers, with their spires, are thus ranked higher than the Willis Tower (formerly the Sears Tower) with its antennas, despite the Petronas Towers' lower roofs and lower highest point.

Until 1996, the world's tallest building was defined by the height to the top of the tallest architectural element, including spires but not antennae. This led to a rivalry between the Bank of Manhattan Building and the Chrysler Building. The Bank of Manhattan Building (aka 40 Wall Street) employed only a short spire and was tall and had a much higher top occupied floor (the second category in the 1996 criteria for tallest building). In contrast, the Chrysler Building employed a very large spire secretly assembled inside the building to claim the title of world's tallest building with a total height of, although it had a lower top occupied floor and a shorter height when both buildings' spires were excluded.

Upset by Chrysler's victory, Shreve & Lamb, the consulting architects of the Bank of Manhattan Building, wrote a newspaper article claiming that their building was actually the tallest, since it contained the world's highest usable floor, at. They pointed out that the observation deck in the Bank of Manhattan Building was nearly above the top floor in the Chrysler Building, whose surpassing spire was strictly ornamental and inaccessible.

At present, the Burj Khalifa tops the list by some margin, regardless of which criterion is applied.

, this list includes all 64 buildings (completed and architecturally topped out) which reach a height of or more, as assessed by their highest architectural feature. Of these, 27 (48%) are in China. Six of the last seven buildings to have held the record as 'tallest building' are still found in the list, with the exception being the North Tower of the original World Trade Center at after its destruction in the September 11 attacks of 2001. If the twin towers were still standing today they would occupy numbers 24 and 25 on the list (or 23 and 24 since it can be assumed the rebuilt One World Trade Center would have never been built).

This measurement disregards distinctions between architectural and non-architectural extensions, and simply measures to the highest point. This measurement is useful for air traffic obstacle determinations, and is also a wholly objective measure. However, this measurement includes extensions that are easily added, removed, and modified from a building and are independent of the overall structure.

This measurement only recently came into use, when the Petronas Towers passed the Sears Tower (now called Willis Tower) in height. The former was considered taller because its spires were considered architectural, while the latter's antennae were not. This led to the split of definitions, with the Sears Tower claiming the lead in this and the height-to-roof (now highest occupied floor) categories, and with the Petronas claiming the lead in the architectural height category.

If the World Trade Center towers were still standing today they would occupy numbers 11 (North Tower at ) and 31 on the list (or 10 and 30 since it can be assumed the rebuilt One World Trade Center would have never been built).

This height is measured from the level of the lowest, significant, open-air, pedestrian entrance to the highest occupied floor within the building.

This is a list of buildings taller than that are currently under construction. More than half of the buildings are located in China.

The following list shows the tallest completed buildings located in each continent listed by greatest to least height (click on name of continent for continent-specific list):