2 ,


² ,
,
,
𝟐 ,
𝟚 ,
𝟤 ,
𝟮 ,
𝟸 ,
,

🯲 ,
  • 1.1.1 Alternative forms
  • 1.1.2 Derived terms
  • 1.1.3 See also
  • 2.1 Pronunciation
  • 2.2.1 Derived terms
  • 2.2.2 See also
  • 2.4.1 Derived terms
  • 3.1 Pronunciation
  • 3.2 Preposition
  • 3.3 Further reading
  • 4.1 Alternative forms
  • 5.1 Alternative forms
  • 6.1 Alternative forms

Translingual [ edit ]

Symbol [ edit ].

2 ( prev 1 , next 3 )

  • The cardinal number two .
  • A digit in the decimal system of numbering, as well as octal , and hexadecimal .
  • A rotational symmetry element in the Hermann-Mauguin notation, indicating a rotation over π (twofold axis).
  • ( mathematics ) The two-element Boolean algebra , two-point discrete space or a ( canonical ) two-element set .

Alternative forms [ edit ]

The following forms are equivalent as a decimal digit:

  • Arabic (Eastern) numerals : ٢ ‎
  • Arabic (Western) numerals : 2
  • Assamese numerals : ২
  • Balinese numerals : ᭒
  • Bengali numerals : ২
  • Brahmi numerals : 𑁨
  • Chakma numerals : 𑄸
  • Cham numerals : ꩒
  • Devanagari numerals : २
  • Gujarati numerals : ૨
  • Gurmukhi numerals : ੨
  • Hanifi Rohingya numerals : 𐴲 ‎
  • Javanese numerals : ꧒
  • Kannada numerals : ೨
  • Kayah Li numerals : ꤂
  • Khmer numerals : ២
  • Khudawadi numerals : 𑋲
  • Lanna numerals : ( secular ) ᪂ , ( Tham ) ᪒
  • Lao numerals : ໒
  • Lepcha numerals : ᱂
  • Limbu numerals : ᥈
  • Malayalam numerals : ൨
  • Meitei numerals : ꯲
  • Mongolian numerals : ᠒
  • Myanmar numerals : ၂
  • N'Ko numerals : ߂ ‎
  • Ol Chiki numerals : ᱒
  • Oriya numerals : ୨
  • Osmanya numerals : 𐒢
  • Persian numerals : ۲ ‎
  • Saurashtra numerals : ꣒
  • Sharada numerals : 𑇒
  • Sinhalese numerals : ෨
  • Sundanese numerals : ᮲
  • Takri numerals : 𑛂
  • Tamil numerals : ௨
  • Telugu numerals : ౨
  • Thai numerals : ๒
  • Tibetan numerals : ༢
  • Tirhuta numerals : 𑓒
  • Warang Citi numerals : 𑣢

The following additional forms are equivalent as a cardinal number:

  • Armenian numerals : Բ
  • Brahmi numerals : 𑁓
  • Chinese numerals : 貳 / 贰 , 二
  • CJK tally marks : 丅
  • Counting rod numerals : 𝍪 , 𝍡
  • Egyptian numerals : 𓏻
  • Ethiopic numerals : ፪
  • Gothic numerals : 𐌱
  • Greek numerals : Β΄ , β΄
  • Hebrew numerals : ב ‎
  • Old Persian numerals : 𐏒
  • Roman numerals : II , ii
  • Sinhalese numerals : 𑇢
  • Tally marks : 𝍷 𝍷

Derived terms [ edit ]

See also [ edit ].

  • 2 (number) in Wikipedia
  • ( Arabic digits ) : 0 1 2 3 4 5 6 7 8 9

Other representations of 2:

: : : (ICS and NATO):
:

After showing " "

English [ edit ]

Pronunciation [ edit ].

  • ( too, two ) : IPA ( key ) : /tuː/
  • ( to ) : IPA ( key ) : /tuː/ ( stressed ) , IPA ( key ) : /tə/ , /tʊ/ ( unstressed )
Audio (US) ( )
  • Homophones : to , too , two
  • Rhymes: -uː

Preposition [ edit ]

  • ( text messaging , informal ) Abbreviation of to . I have 2 go now. / Send files 2 him.
  • ( computing ) Abbreviation of to ; designating conversion from one format to another pdf 2 txt, i.e. conversion from PDF to plain text

Adverb [ edit ]

2 ( not comparable )

  • ( text messaging , informal , eye dialect ) Abbreviation of too . Can I come 2 ?

Adjective [ edit ]

2 ( comparative more 2 , superlative most 2 )

  • Alternative form of 2S ; Abbreviation of two-spirited .

Noun [ edit ]

2 ( plural 2s )

  • Alternative form of 2S ; Abbreviation of two-spirit .

French [ edit ]

Audio ( )
  • ( text messaging ) Alternative form of de

Further reading [ edit ]

  • “ 2 ”, in Trésor de la langue française informatisé [ Digitized Treasury of the French Language ] , 2012.

Indonesian [ edit ]

  • ( dated or informal ) An iteration mark used as an abbreviation for the second part of a reduplicated compound word.

Malay [ edit ]

  • An iteration mark used as an abbreviation for the second part of a reduplicated compound word.

Tagalog [ edit ]

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Square root of 2

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Square root of 2
of an with legs of length 1.
Representations
Decimal1.4142135623730950488...
Continued fraction
Binary1.01101010000010011110...
Hexadecimal1.6A09E667F3BCC908B2F...

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Sequence A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: [3]

History [ edit ]

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The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of √ 2 in four sexagesimal figures, 1 24 51 10 , which is accurate to about six decimal digits, [4] and is the closest possible three-place sexagesimal representation of √ 2 :

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Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC), as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. [5] That is,

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This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers , which can be derived from the continued fraction expansion of √ 2 . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.

Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational . Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. [1] [6] [7] [8] [ excessive citations ] The square root of two is occasionally called Pythagoras's number or Pythagoras's constant , for example by Conway & Guy (1996) . [9]

Ancient Roman architecture [ edit ]

In ancient Roman architecture , Vitruvius describes the use of the square root of 2 progression or ad quadratum technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato . The system was employed to build pavements by creating a square tangent to the corners of the original square at 45 degrees of it. The proportion was also used to design atria by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width. [10]

Decimal value [ edit ]

Computation algorithms [ edit ].

There are a number of algorithms for approximating √ 2 as a ratio of integers or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the Babylonian method [11] for computing square roots. It goes as follows:

First, pick a guess, a 0 > 0 ; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:

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The more iterations through the algorithm (that is, the more computations performed and the greater " n "), the better the approximation. Each iteration roughly doubles the number of correct digits. Starting with a 0 = 1 , the results of the algorithm are as follows:

  • 3 / 2 = 1 .5 ( a 1 )
  • 17 / 12 = 1.41 6... ( a 2 )
  • 577 / 408 = 1.41421 5... ( a 3 )
  • 665857 / 470832 = 1.41421356237 46... ( a 4 )

Rational approximations [ edit ]

A simple rational approximation 99 / 70 (≈ 1.4142 857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than 1 / 10,000 (approx. +0.72 × 10 −4 ).

The next two better rational approximations are 140 / 99 (≈ 1.414 1414...) with a marginally smaller error (approx. −0.72 × 10 −4 ), and 239 / 169 (≈ 1.4142 012) with an error of approx −0.12 × 10 −4 .

The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a 0 = 1 ( 665,857 / 470,832 ) is too large by about 1.6 × 10 −12 ; its square is ≈  2.000 000 000 0045 .

Records in computation [ edit ]

In 1997 the value of √ 2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada 's team. In February 2006 the record for the calculation of √ 2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010. [12] Among mathematical constants with computationally challenging decimal expansions, only π , e , and the golden ratio have been calculated more precisely as of March 2022. [13] Such computations aim to check empirically whether such numbers are normal .

This is a table of recent records in calculating the digits of √ 2 . [13]

Date Name Number of digits
January 5, 2022 Tizian Hanselmann 10000000001000
June 28, 2016 Ron Watkins 10000000000000
April 3, 2016 Ron Watkins 5000000000000
January 20, 2016 Ron Watkins 2000000000100
February 9, 2012 Alexander Yee 2000000000050
March 22, 2010 Shigeru Kondo 1000000000000

Proofs of irrationality [ edit ]

A short proof of the irrationality of √ 2 can be obtained from the rational root theorem , that is, if p ( x ) is a monic polynomial with integer coefficients , then any rational root of p ( x ) is necessarily an integer. Applying this to the polynomial p ( x ) = x 2 − 2 , it follows that √ 2 is either an integer or irrational. Because √ 2 is not an integer (2 is not a perfect square ), √ 2 must therefore be irrational. This proof can be generalized to show that any square root of any natural number that is not a perfect square is irrational.

For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent .

Proof by infinite descent [ edit ]

One proof of the number's irrationality is the following proof by infinite descent . It is also a proof of a negation by refutation : it proves the statement " √ 2 is not rational" by assuming that it is rational and then deriving a falsehood.

  • Assume that √ 2 is a rational number, meaning that there exists a pair of integers whose ratio is exactly √ 2 .
  • If the two integers have a common factor , it can be eliminated using the Euclidean algorithm .
  • Then √ 2 can be written as an irreducible fraction a / b such that a and b are coprime integers (having no common factor) which additionally means that at least one of a or b must be odd .
  • It follows that a 2 / b 2 = 2 and a 2 = 2 b 2 .   (  ( a / b ) n = a n / b n  )   ( a 2 and b 2 are integers)
  • Therefore, a 2 is even because it is equal to 2 b 2 . ( 2 b 2 is necessarily even because it is 2 times another whole number.)
  • It follows that a must be even (as squares of odd integers are never even).
  • Because a is even, there exists an integer k that fulfills a = 2 k .
  • Substituting 2 k from step 7 for a in the second equation of step 4: 2 b 2 = a 2 = (2 k ) 2 = 4 k 2 , which is equivalent to b 2 = 2 k 2 .
  • Because 2 k 2 is divisible by two and therefore even, and because 2 k 2 = b 2 , it follows that b 2 is also even which means that b is even.
  • By steps 5 and 8 a and b are both even, which contradicts that a / b is irreducible as stated in step 3.

Since we have derived a falsehood, the assumption (1) that √ 2 is a rational number must be false. This means that √ 2 is not a rational number; that is to say, √ 2 is irrational.

This proof was hinted at by Aristotle , in his Analytica Priora , §I.23. [14] It appeared first as a full proof in Euclid 's Elements , as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an interpolation and not attributable to Euclid. [15]

Proof by unique factorization [ edit ]

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Geometric proof [ edit ]

A simple proof is attributed by John Horton Conway to Stanley Tennenbaum when the latter was a student in the early 1950s [16] and whose most recent appearance is in an article by Noson Yanofsky in the May–June 2016 issue of American Scientist . [17] Given two squares with integer sides respectively a and b , one of which has twice the area of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle ( (2 b − a ) 2 ) must equal the sum of the two uncovered squares ( 2( a − b ) 2 ). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1.

Another geometric reductio ad absurdum argument showing that √ 2 is irrational appeared in 2000 in the American Mathematical Monthly . [18] It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the same algebraic proof as in the previous paragraph, viewed geometrically in another way.

Let △  ABC be a right isosceles triangle with hypotenuse length m and legs n as shown in Figure 2. By the Pythagorean theorem , m / n = √ 2 . Suppose m and n are integers. Let m : n be a ratio given in its lowest terms .

Draw the arcs BD and CE with centre A . Join DE . It follows that AB = AD , AC = AE and ∠ BAC and ∠ DAE coincide. Therefore, the triangles ABC and ADE are congruent by SAS .

Because ∠ EBF is a right angle and ∠ BEF is half a right angle, △  BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n . By symmetry, DF = m − n , and △  FDC is also a right isosceles triangle. It also follows that FC = n − ( m − n ) = 2 n − m .

Hence, there is an even smaller right isosceles triangle, with hypotenuse length 2 n − m and legs m − n . These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m : n is in lowest terms. Therefore, m and n cannot be both integers, hence √ 2 is irrational.

Constructive proof [ edit ]

While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let a and b be positive integers such that 1< a / b < 3/2 (as 1<2< 9/4 satisfies these bounds). Now 2 b 2 and a 2 cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus | 2 b 2 − a 2 | ≥ 1 . Multiplying the absolute difference | √ 2 − a / b | by b 2 ( √ 2 + a / b ) in the numerator and denominator, we get [19]

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the latter inequality being true because it is assumed that 1< a / b < 3/2 , giving a / b + √ 2 ≤ 3 (otherwise the quantitative apartness can be trivially established). This gives a lower bound of 1 / 3 b 2 for the difference | √ 2 − a / b | , yielding a direct proof of irrationality in its constructively stronger form, not relying on the law of excluded middle ; see Errett Bishop (1985, p. 18). This proof constructively exhibits an explicit discrepancy between √ 2 and any rational.

Proof by Pythagorean triples [ edit ]

This proof uses the following property of primitive Pythagorean triples :

This lemma can be used to show that two identical perfect squares can never be added to produce another perfect square.

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Here, (b, b, a) is a primitive Pythagorean triple, and from the lemma a is never even. However, this contradicts the equation 2 b 2 = a 2 which implies that a must be even.

Multiplicative inverse [ edit ]

The multiplicative inverse (reciprocal) of the square root of two (i.e., the square root of 1 / 2 ) is a widely used constant .

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One-half of √ 2 , also the reciprocal of √ 2 , is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates

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This number satisfies

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Properties [ edit ]

One interesting property of √ 2 is

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This is related to the property of silver ratios .

√ 2 can also be expressed in terms of copies of the imaginary unit i using only the square root and arithmetic operations , if the square root symbol is interpreted suitably for the complex numbers i and − i :

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√ 2 is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1 , x 1 = c and x n +1 = c x n for n > 1 , the limit of x n as n → ∞ will be called (if this limit exists) f ( c ) . Then √ 2 is the only number c > 1 for which f ( c ) = c 2 . Or symbolically:

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√ 2 appears in Viète's formula for π :

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for m square roots and only one minus sign. [21]

Similar in appearance but with a finite number of terms, √ 2 appears in various trigonometric constants : [22]

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It is not known whether √ 2 is a normal number , which is a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two . [23]

Representations [ edit ]

Series and product [ edit ].

The identity cos  π / 4 = sin  π / 4 = 1 / √ 2 , along with the infinite product representations for the sine and cosine , leads to products such as

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or equivalently,

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The number can also be expressed by taking the Taylor series of a trigonometric function . For example, the series for cos  π / 4 gives

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The Taylor series of √ 1 + x with x = 1 and using the double factorial n !! gives

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The convergence of this series can be accelerated with an Euler transform , producing

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It is not known whether √ 2 can be represented with a BBP-type formula . BBP-type formulas are known for π √ 2 and √ 2   ln (1+ √ 2 ) , however. [24]

The number can be represented by an infinite series of Egyptian fractions , with denominators defined by 2 n  th terms of a Fibonacci -like recurrence relation a ( n ) = 34 a ( n −1) − a ( n −2), a (0) = 0, a (1) = 6. [25]

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Continued fraction [ edit ]

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The square root of two has the following continued fraction representation:

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The convergents p / q formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (i.e., p 2 − 2 q 2 = ±1 ). The first convergents are: 1 / 1 , 3 / 2 , 7 / 5 , 17 / 12 , 41 / 29 , 99 / 70 , 239 / 169 , 577 / 408 and the convergent following p / q is p + 2 q / p + q . The convergent p / q differs from √ 2 by almost exactly 1 / 2 √ 2 q 2 , which follows from:

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Nested square [ edit ]

The following nested square expressions converge to √ 2 :

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Applications [ edit ]

Paper size [ edit ].

In 1786, German physics professor Georg Christoph Lichtenberg [26] found that any sheet of paper whose long edge is √ 2 times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standardised paper sizes at the beginning of the 20th century, they used Lichtenberg's ratio to create the "A" series of paper sizes. [26] Today, the (approximate) aspect ratio of paper sizes under ISO 216 (A4, A0, etc.) is 1: √ 2 .

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Physical sciences [ edit ]

There are some interesting properties involving the square root of 2 in the physical sciences :

  • The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music.
  • The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of areas between two successive apertures is 2.
  • The celestial latitude (declination) of the Sun during a planet's astronomical cross-quarter day points equals the tilt of the planet's axis divided by √ 2 .

See also [ edit ]

  • List of mathematical constants
  • Square root of 3 , √ 3
  • Square root of 5 , √ 5
  • Gelfond–Schneider constant , 2 √ 2
  • Silver ratio , 1 + √ 2

Notes [ edit ]

  • ^ Fowler, David H. (2001), "The story of the discovery of incommensurability, revisited", Neusis (10): 45–61, MR   1891736
  • ^ "A002193 - OEIS" . oeis.org . Retrieved 2020-08-10 .
  • ^ Fowler and Robson, p. 368. Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection Archived 2012-08-13 at the Wayback Machine High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
  • ^ Henderson.
  • ^ Stephanie J. Morris, "The Pythagorean Theorem" Archived 2013-05-30 at the Wayback Machine , Dept. of Math. Ed., University of Georgia .
  • ^ Brian Clegg, "The Dangerous Ratio ..." Archived 2013-06-27 at the Wayback Machine , Nrich.org, November 2004.
  • ^ Kurt von Fritz, "The discovery of incommensurability by Hippasus of Metapontum" , Annals of Mathematics, 1945.
  • ^ Conway, John H. ; Guy, Richard K. (1996), The Book of Numbers , Copernicus, p. 25
  • ^ Williams, Kim ; Ostwald, Michael (2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s . Birkhäuser. p. 204. ISBN   9783319001371 .
  • ^ Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of √ 2 seen on tablet YBC 7289. Fowler and Robson offer informed and detailed conjectures. Fowler and Robson, p. 376. Flannery, p. 32, 158.
  • ^ "Constants and Records of Computation" . Numbers.computation.free.fr. 2010-08-12. Archived from the original on 2012-03-01 . Retrieved 2012-09-07 .
  • ^ a b "Records set by y-cruncher" . Archived from the original on 2022-04-07 . Retrieved 2022-04-07 .
  • ^ All that Aristotle says, while writing about proofs by contradiction , is that “the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate”.
  • ^ The edition of the Greek text of the Elements published by E. F. August in Berlin in 1826–1829 already relegates this proof to an Appendix. The same thing occurs with J. L. Heiberg's edition (1883–1888).
  • ^ Proof 8‴ Archived 2016-04-22 at the Wayback Machine
  • ^ Yanofsky, N. (2016). "Paradoxes, Contradictions, and the Limits of Science" . Archived from the original on 2016-06-30.
  • ^ Tom M. Apostol (Nov 2000), "Irrationality of The Square Root of Two -- A Geometric Proof", The American Mathematical Monthly , 107 (9): 841–842, doi : 10.2307/2695741 , JSTOR   2695741
  • ^ See Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?", Intellectica , 56 (2): 223–302 (see esp. Section 2.3, footnote 15), arXiv : 1110.5456 , Bibcode : 2011arXiv1110.5456U
  • ^ Sierpiński, Wacław (2003), Pythagorean Triangles , Dover, pp. 4–6, ISBN   978-0-486-43278-6
  • ^ Courant, Richard; Robbins, Herbert (1941), What is mathematics? An Elementary Approach to Ideas and Methods , London: Oxford University Press, p. 124
  • ^ Julian D. A. Wiseman Sin and cos in surds Archived 2009-05-06 at the Wayback Machine
  • ^ Good & Gover (1967) .
  • ^ Bailey, David H. (13 February 2011). "A Compendium of BBP-Type Formulas for Mathematical Constants" (PDF) . Archived (PDF) from the original on 2011-06-10 . Retrieved 2010-04-30 .
  • ^ Sloane, N. J. A. (ed.). "Sequence A082405 (a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . Retrieved 2016-09-05 .
  • ^ a b Houston, Keith (2016). The Book: A Cover-to-Cover Exploration of the Most Powerful Object of Our Time . W. W. Norton & Company. p. 324. ISBN   978-0393244809 .

References [ edit ]

  • Apostol, Tom M. (2000), "Irrationality of the square root of two – A geometric proof", American Mathematical Monthly , 107 (9): 841–842, doi : 10.2307/2695741 , JSTOR   2695741 .
  • Aristotle (2007), Analytica priora , [email protected]
  • Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.
  • Flannery, David (2005), The Square Root of Two , Springer-Verlag, ISBN   0-387-20220-X .
  • Fowler, David ; Robson, Eleanor (1998), "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context", Historia Mathematica , 25 (4): 366–378, doi : 10.1006/hmat.1998.2209 .
  • Good, I. J. ; Gover, T. N. (1967), "The generalized serial test and the binary expansion of √ 2 ", Journal of the Royal Statistical Society, Series A , 130 (1): 102–107, doi : 10.2307/2344040 , JSTOR   2344040 .
  • Henderson, David W. (2000), "Square roots in the Śulba Sūtras", in Gorini, Catherine A. (ed.), Geometry At Work: Papers in Applied Geometry , Cambridge University Press, pp. 39–45, ISBN   978-0-88385-164-7 .

External links [ edit ]

  • Gourdon, X.; Sebah, P. (2001), "Pythagoras' Constant: √ 2 ", Numbers, Constants and Computation .
  • The Square Root of Two to 5 million digits by Jerry Bonnell and Robert J. Nemiroff . May, 1994.
  • Square root of 2 is irrational , a collection of proofs
  • Grime, James; Bowley, Roger. "The Square Root √ 2 of Two" . Numberphile . Brady Haran .
  • √ 2 Search Engine 2 billion searchable digits of √ 2 , π and e
) )
 
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Type 2 Diabetes

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Healthy eating is your recipe for managing diabetes.

More than 37 million Americans have diabetes (about 1 in 10), and approximately 90-95% of them have type 2 diabetes. Type 2 diabetes most often develops in people over age 45, but more and more children, teens , and young adults are also developing it.

What Causes Type 2 Diabetes?

Insulin is a hormone made by your pancreas that acts like a key to let blood sugar into the cells in your body for use as energy. If you have type 2 diabetes, cells don’t respond normally to insulin; this is called insulin resistance . Your pancreas makes more insulin to try to get cells to respond. Eventually your pancreas can’t keep up, and your blood sugar rises, setting the stage for prediabetes and type 2 diabetes. High blood sugar is damaging to the body and can cause other serious health problems, such as heart disease ,  vision loss , and kidney disease .

Symptoms and Risk Factors

Type 2 diabetes  symptoms often develop over several years and can go on for a long time without being noticed (sometimes there aren’t any noticeable symptoms at all). Because symptoms can be hard to spot, it’s important to know the  risk factors and to see your doctor to get your blood sugar tested if you have any of them.

Testing for Type 2 Diabetes

A simple blood test will let you know if you have diabetes. If you’ve gotten your blood sugar tested at a health fair or pharmacy, follow up at a clinic or doctor’s office to make sure the results are accurate.

Managing Diabetes

Unlike many health conditions, diabetes is managed mostly by you, with support from your health care team (including your primary care doctor, foot doctor, dentist, eye doctor, registered dietitian nutritionist, diabetes educator, and pharmacist), family, and other important people in your life. Managing diabetes can be challenging, but everything you do to improve your health is worth it!

You may be able to manage your diabetes with healthy eating and being active, or your doctor may prescribe insulin, other injectable medications, or oral diabetes medicines to help manage your blood sugar and avoid complications . You’ll still need to eat healthy and be active if you take insulin or other medicines. It’s also important to keep your blood pressure and cholesterol close to the targets your doctor sets for you and get necessary screening tests.

You’ll need to check your blood sugar  regularly. Ask your doctor how often you should check it and what your target blood sugar levels should be. Keeping your blood sugar levels as close to target as possible will help you prevent or delay diabetes-related complications.

Stress is a part of life, but it can make managing diabetes harder, including managing your blood sugar levels and dealing with daily diabetes care. Regular physical activity, getting enough sleep, and relaxation exercises can help. Talk to your doctor and diabetes educator about these and other ways you can manage stress.

Make regular appointments with your health care team to be sure you’re on track with your treatment plan and to get help with new ideas and strategies if needed.

Whether you were just diagnosed with diabetes or have had it for some time, meeting with a diabetes educator is a great way to get support and guidance, including how to:

  • Develop a healthy eating and activity plan
  • Test your blood sugar and keep a record of the results
  • Recognize the signs of high or low blood sugar and what to do about it
  • If needed, give yourself insulin by syringe, pen, or pump
  • Monitor your feet, skin, and eyes to catch problems early
  • Buy diabetes supplies and store them properly
  • Manage stress and deal with daily diabetes care

Ask your doctor about diabetes self-management education and support services and to recommend a diabetes educator, or search the Association of Diabetes Care & Education Specialists’ (ADCES) nationwide directory for a list of programs in your community.

Type 2 Diabetes in Children and Teens

Childhood obesity rates are rising, and so are the rates of type 2 diabetes in youth. More than 75% of children with type 2 diabetes have a close relative who has it, too. But it’s not always because family members are related; it can also be because they share certain habits that can increase their risk. Parents can help prevent or delay type 2 diabetes by developing a plan for the whole family:

  • Drinking more water and fewer sugary drinks
  • Eating more fruits and vegetables
  • Making favorite foods healthier
  • Making physical activity more fun

Healthy changes become habits more easily when everyone makes them together. Find out how to take charge family style with these healthy tips .

Get Support

Tap into online diabetes communities for encouragement, insights, and support. The American Diabetes Association’s Community page and ADCES’s Peer Support Resources  are great ways to connect with others who share your experience.

  • Living With Diabetes
  • Managing Diabetes: Medicare Coverage and Resources [PDF - 1 MB]
  • Diabetes: What Is It? Your Health with Joan Lunden and CDC
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Penalty Shooters 2

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Description

Penalty Shooters 2  is a fun soccer game where you can pick your favorite team, try to win each match as you climb up the ranks, and win the League.

In the game Penalty Shooters 2, you are given the opportunity to represent your favorite team and even the football team of a country. So be prepared mentally to make perfect shots. No one dares to say that being a footballer is easy. especially when you're going through a rough patch to get to this risky and mind-boggling penalty shootout. A long road is about to come to an end. And now is the time to show your resilience and strong will along with your superior level. Only one final match and victory will go to the most deserving.

Penalty kicks are always risky. But it is that nature that has brought the most attractiveness that other forms of competition never have. In the penalty shootout, the team that scores the most goals after the first five innings will be the winner. However, if one side successfully takes more penalties than the other side can achieve with all of its remaining kicks, the shootout will end immediately, regardless of the number of kicks remaining.

The goal of this game is to win every match. If you do that, you'll be the champion of the year. The first thing you need to do is choose which team you want to be a member of? The game has up to 11 tournaments on the world's top football platforms and the World Cup with a number of 32 teams in each tournament. A quantity too complete for you to choose from.

And now, you need to be with your team that has gone through three matches in the group stage. The pressure isn't high, but don't let it go even for a second. After a series of three games in the group stage, you are now with your team into the real finals. Up to this point, every match is a final for the whole team. The media is just waiting to put you and your team on the front page tomorrow. Millions of passionate spectators poured into the field, filling the stands and anxiously awaiting the performances of the best players in the world. In each match, you have at least five shots to beat your opponents before going to the final match. A special feature of this game is that you are both a goalkeeper and a goalkeeper in a wooden frame.

In the role of a free-kicker, to score a goal, when the target red dot is running around the goal, choose the appropriate time and hold the mouse button, wait for the red dot to run to the position you choose to shoot. Release the mouse quickly. Immediately after, the ball will be kicked and the result will be displayed by a beacon in the lower right corner of the screen. Note that the opponent's goalkeeper is smart enough to remember where you shot before. Therefore, you should move the shooting position continuously through the shots.

In the role of the goalkeeper, you have to be really focused. While your team player is about to kick the ball, pay attention to wait for the red dot to appear. You have to quickly click on the closest approximate position to control the goalkeeper rushing to catch the ball there. In addition to the accuracy, the time of clicking the mouse to let the goalkeeper jump out to catch the ball is also an important factor.

All in all, Penalty Shooters 2 brings us one of the most popular online soccer games of all time. It completely conquers every player with its eye-catching graphics, immersive sound, and extremely rich team options. Try to score as many goals as possible and win the tournament.

Features of Penalty Shooters 2

  • Fun and attractive gameplay
  • Various opponents
  • Fantasy league

Developer: This game was made by Vladeta Marinkovic and released in April 2016.

How to play

Use your mouse to play this game.

  • Concentrate
  • Press anywhere, hold and release to shoot

Goalkeeping:

  • Wait to see the target
  • Press and hold to move
  • Release to dive

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  • Wordle Unlimited
  • Heardle Unlimited

If you’re looking for an addictive word game - Wordle 2 - you’ve come to the right place! This word game is both simple and challenging. In it, you must try to make words using a big, colorful word cloud. But that’s not all. There are power-ups that you can use to help you get through the levels. You can also add letters so that you can create words of your choice. You can even play Wordle with new friends or family! If you’re ready to start playing, then check out the link below!

What is Wordle 2 Game?

Do you like word games? Puzzle games are a great way to exercise your brain and have fun for free. In this wordle 2 game, we will teach you how to play word games online for free. You don't need to download any software. All you need is a computer, internet connection and some time. If you are looking for more ways to exercise your brain, try experiencing similar brain games on this website. There you can learn more about Sudoku, crossword puzzles, and other games that can help you improve your logical thinking.

Interesting things in Wordle 2

Wordle 2 is a simple game based on word search. It is a word search game in which the player has to find the correct word to fill in the grid. In this game, you can create as many words as you want and you can also make them as difficult as you want. In this game, you need a word that starts with a certain letter and is as long as a grid. These options help you to make the game more interesting.

You can also choose the grid size and number of colors for the cells. You can easily choose a grid size from 4×6 to 11×6. You can fill the grid by selecting the checkboxes on the left side of the grid. You can also choose the speed at which the grid fills in the letters. You can do this by selecting the checkboxes on the left side of the grid. The game features an auto-save of your words as you type them.

Take note of the first word that comes to your mind after you start this wordle 2 game. You can write down as many words as you want, but you can only use one word per line. If you don't know the meaning of a word, leave it out. The more words you write down, the better you understand how the game works!

How to play Wordle 2 game.

Find the last correct word in the crossword with different topics. By choosing the right word green and matching it in the right place to become a meaningful word. Note that you only have 6 entries to find the most correct answer. Wordle 2 always changes the answer to the question so there will not be a single play the same in this word guessing game.

If you are a newbie, my advice is to start with a simple 4x6 mode. After you get used to the gameplay and draw your own experience, you can completely try your hand at more difficult modes like 10x6 or 11x6.

Tips and tricks to conquer the Wordle 2 game.

  • Don't try to guess all the words in the first line correctly: There aren't any detailed bases or hints for you about the word in the first line. So it's impossible to guess correctly. Think of your first keyword entry as a kind of sacrifice to find clues to the words in the row below.
  • Use popular words in the first line: This helps to increase accuracy to get more suggestions for you later. For example, TEARS with consecutive 3 common words are T, E, A. From there you will get more strength each for the letter below. This tip will lead you to victory faster.
  • Use the correct word on the first turn for all cells in the same row below: This is a perfect trick to speed up you're finding the final answer. It's entirely possible for a letter to appear twice: BANAL is an example of a recent Wordle result. They have the same letter A appearing twice. So even if you have a green A in your answer, that doesn't mean there won't be another A in hidden cells.
  • Use clues from the previously guessed answer: You can change the position of the colored words in the answer above to find their correct position. Or exclude the brown word because it's not in the answer at all.

Top Word Games

If you are a fan of word games, you should not miss the following games. Let's explore and enjoy it!

Foodle players start the game by inputting a five-letter word of their choice like Wordle. The word must, of course, be connected to food in some way in order to be acceptable. To get you started, consider terms like salad, sauce, and berry.

2. Numberle

Do you have a knack for solving arithmetic problems quickly? It's now your turn to show off your skills and see if you can correctly predict the correct answer after six trials. Numberle is a brand-new number game that will test your logic and math skills. In this fun and fast-paced game, you must use your entire mental capacity.

3. Canuckle

Canuckle   is a game based on the popular word game wordle, except this time using Canadian terms. It's a clever game that's also known as the canuckle word game. It entails correctly guessing a Canadian word six times. The Canuckle game gives you the chance to post your solution to this enigmatic term on social media.

Similar to Wordle, Phrazle releases a new puzzle with a different phrase or sentence every 24 hours. Players have six chances to correctly identify a phrase in the free to play online word game the Phrazle. Let's try it!

Lewdle is the game for you if you prefer engaging in verbal combat and don't mind telling others about it. This tough game allows you to demonstrate your command of the English language while simultaneously letting everyone know exactly how much of a potty-mouthed badass you are.

6. Sedecordle

You solve 16 wordles at simultaneously, similar to wordle. To answer every puzzle, you are given 21 guesses. Every day a different Sedecordle and an endless supply of free puzzles!

If you think you have a terrific memory and can memorize all the music you've heard, then Heardle  was made for you. Anything from mainstream ballads to classical music or lively tunes will be played. The game gives off the impression of playing a calming melody before abruptly switching to exhilarating blended audio. You'll get six fantastic chances because most relaxing games lose appeal after a few trials.

How to play Wordle 2

  • Green means a letter is correct in this spot
  • yellow means a letter is elsewhere in the target word
  • Gray means a letter is not in the target word at all.

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Discuss wordle 2.

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  • Natural Language

2 Thessalonians 2 New International Version

The man of lawlessness.

2  Concerning the coming of our Lord Jesus Christ ( A ) and our being gathered to him, ( B ) we ask you, brothers and sisters, 2  not to become easily unsettled or alarmed by the teaching allegedly from us—whether by a prophecy or by word of mouth or by letter ( C ) —asserting that the day of the Lord ( D ) has already come. ( E ) 3  Don’t let anyone deceive you ( F ) in any way, for that day will not come until the rebellion ( G ) occurs and the man of lawlessness [ a ] is revealed, ( H ) the man doomed to destruction. 4  He will oppose and will exalt himself over everything that is called God ( I ) or is worshiped, so that he sets himself up in God’s temple, proclaiming himself to be God. ( J )

5  Don’t you remember that when I was with you I used to tell you these things? ( K ) 6  And now you know what is holding him back, ( L ) so that he may be revealed at the proper time. 7  For the secret power of lawlessness is already at work; but the one who now holds it back ( M ) will continue to do so till he is taken out of the way. 8  And then the lawless one will be revealed, ( N ) whom the Lord Jesus will overthrow with the breath of his mouth ( O ) and destroy by the splendor of his coming. ( P ) 9  The coming of the lawless one will be in accordance with how Satan ( Q ) works. He will use all sorts of displays of power through signs and wonders ( R ) that serve the lie, 10  and all the ways that wickedness deceives those who are perishing. ( S ) They perish because they refused to love the truth and so be saved. ( T ) 11  For this reason God sends them ( U ) a powerful delusion ( V ) so that they will believe the lie ( W ) 12  and so that all will be condemned who have not believed the truth but have delighted in wickedness. ( X )

13  But we ought always to thank God for you, ( Y ) brothers and sisters loved by the Lord, because God chose you as firstfruits [ b ] ( Z ) to be saved ( AA ) through the sanctifying work of the Spirit ( AB ) and through belief in the truth. 14  He called you ( AC ) to this through our gospel, ( AD ) that you might share in the glory of our Lord Jesus Christ.

15  So then, brothers and sisters, stand firm ( AE ) and hold fast to the teachings [ c ] we passed on to you, ( AF ) whether by word of mouth or by letter.

16  May our Lord Jesus Christ himself and God our Father, ( AG ) who loved us ( AH ) and by his grace gave us eternal encouragement and good hope, 17  encourage ( AI ) your hearts and strengthen ( AJ ) you in every good deed and word.

  • 2 Thessalonians 2:3 Some manuscripts sin
  • 2 Thessalonians 2:13 Some manuscripts because from the beginning God chose you
  • 2 Thessalonians 2:15 Or traditions

Cross references

  • 2 Thessalonians 2:1 : S 1Th 2:19
  • 2 Thessalonians 2:1 : Mk 13:27; 1Th 4:15-17
  • 2 Thessalonians 2:2 : ver 15; 2Th 3:17
  • 2 Thessalonians 2:2 : S 1Co 1:8
  • 2 Thessalonians 2:2 : 2Ti 2:18
  • 2 Thessalonians 2:3 : S Mk 13:5
  • 2 Thessalonians 2:3 : Mt 24:10-12
  • 2 Thessalonians 2:3 : ver 8; Da 7:25; 8:25; 11:36; Rev 13:5, 6
  • 2 Thessalonians 2:4 : 1Co 8:5
  • 2 Thessalonians 2:4 : Isa 14:13, 14; Eze 28:2
  • 2 Thessalonians 2:5 : 1Th 3:4
  • 2 Thessalonians 2:6 : ver 7
  • 2 Thessalonians 2:7 : ver 6
  • 2 Thessalonians 2:8 : S ver 3
  • 2 Thessalonians 2:8 : Isa 11:4; Rev 2:16; 19:15
  • 2 Thessalonians 2:8 : S Lk 17:30
  • 2 Thessalonians 2:9 : S Mt 4:10
  • 2 Thessalonians 2:9 : Mt 24:24; Rev 13:13; S Jn 4:48
  • 2 Thessalonians 2:10 : S 1Co 1:18
  • 2 Thessalonians 2:10 : Pr 4:6; Jn 3:17-19
  • 2 Thessalonians 2:11 : Ro 1:28
  • 2 Thessalonians 2:11 : Mt 24:5; S Mk 13:5
  • 2 Thessalonians 2:11 : Ro 1:25
  • 2 Thessalonians 2:12 : Ro 1:32; 2:8
  • 2 Thessalonians 2:13 : S Ro 1:8
  • 2 Thessalonians 2:13 : Eph 1:4
  • 2 Thessalonians 2:13 : 1Th 5:9
  • 2 Thessalonians 2:13 : 1Pe 1:2
  • 2 Thessalonians 2:14 : S Ro 8:28; S 11:29
  • 2 Thessalonians 2:14 : 1Th 1:5
  • 2 Thessalonians 2:15 : S 1Co 16:13
  • 2 Thessalonians 2:15 : S 1Co 11:2
  • 2 Thessalonians 2:16 : S Php 4:20
  • 2 Thessalonians 2:16 : S Jn 3:16
  • 2 Thessalonians 2:17 : 1Th 3:2
  • 2 Thessalonians 2:17 : 2Th 3:3

Holy Bible, New International Version®, NIV® Copyright ©1973, 1978, 1984, 2011 by Biblica, Inc.® Used by permission. All rights reserved worldwide.

NIV Reverse Interlinear Bible: English to Hebrew and English to Greek. Copyright © 2019 by Zondervan.

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