What Is The Meaning Of Square Root – The square root of 2 is equal to lg of an imaginary isosceles rectangle with leg lg 1 .
Geometrically, the square root of 2 is equal to the length of the diagonal of a square whose sides are each unit length;
What Is The Meaning Of Square Root
The fraction 99/70 (≈ 1.4142857) is sometimes used as a good rational approximation with sufficiently small dominance.
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A002193 of the Online Encyclopedia of Integer Sequences consists of digits in the decimal expansion of the square root of 2, where the decimal places are reduced to 65;
Annotated Babylonian clay tablet YBC 7289. In addition to showing the square root of 2 in gender (1245110), the table also provides an example of a square with one side 30 and the third diagonal 422535. The hex 30 can also represent 030 =
Babylonian clay tablet YBC 7289 (1800–1600 BC) gives an approximate value of √2 in four sex digits: 1 24 51 10, correct to six decimal places.
Another early approximation of the ancient Indian mathematical texts Salbasutra (800-200 BCE) is given below. Multiply the length [of the side] by its third, subtract that third from its fourth, thirty-four. It is the fourth in length.
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1 + 1 3 + 1 3 × 4 – 1 3 × 4 × 34 = 577408 = 1.41421 56862745098039 ¯ . }+}-}=}=1.41421}.}
This approximation is the seventh in a series of more accurate approximations based on a sequence of Pell numbers obtained from the continued fractional expansion of √2. Despite the small dominance, it is slightly less accurate than the Babylonian approximation.
The Pythagoreans discovered that the diagonal of a square is inconsistent with its side, or in modern parlance, that the square root of both is irrational. Little is known for certain about the time or circumstances of this discovery, but the name Hippasus of Metapontum is often mentioned. The Pythagoreans’ discovery of the irrationality of the square root of two was kept an official secret for a time and, according to legend, Hippasus was killed for discovering it.
The square root of two is sometimes called the Pythagorean number or the Pythagorean constant, as in Conway & Guy (1996).
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In ancient Roman architecture, Vitruvius describes the use of the square root of 2 progression or square adatum technique. It basically consists of a geometric rather than an arithmetic method of doubling a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes this idea to Plato. The system was used to construct pavements by creating square tangents at 45 degree angles to the corners of the original square. Proportion was also used to design the atrium, giving a length equal to the diagonal of a square whose sides are equal to the desired width of the atrium.
There are a number of algorithms for approximating √2 as an integer or decimal ratio. The most common algorithm for this, used as the basis of many computers and calculators, is the Babylonian method.
First choose a guess, a0 > 0; The guess value only affects how many iterations are required to get an approximation with a certain accuracy. Using this assumption, perform the following iterative calculations:
The more iterations of the algorithm (ie, more calculations and more “n”), the better the approximation. Each iteration roughly doubles the number of correct digits. Starting with a0 = 1, the algorithm results in:
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Sometimes 99/70 (≈ 1.4142857) is used. Although only 70 dominators, it varies by less than the correct value
A rational approximation to the square root of two obtained by four iterations of the Babylonian method after starting with a0 = 1 is (
665, 857/470, 832) is extremely large at about 1.6×10−12; Its area is ≈ 2.000000 000 0045.
In 1997, Yasumasa’s group in Canada calculated the value of √2 to the decimal digits 137, 438, 953, 444. In February 2006, the record for calculating √2 using a home computer was broken. Shigeru Kondo calculated 1 trillion decimal places in 2010.
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As of March 2022, only π, e, and the golden ratio have been calculated more accurately among mathematical constants with difficult-to-calculate decimal expansions.
A short proof of the irrationality of √2 can be obtained from the Rational Root Theorem, which states that if p(x) is a unique polynomial with integer coefficients, then any rational root of p(x) is necessarily an integer. Applying this to the polynomial p(x) = x2 − 2 we get that √2 is an integer or an irrational number. Since √2 is not an integer (2 is not a perfect square), √2 must 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 Irrationality or infinity description of a square.
One proof of the irrationality of a number using an infinite set is the following proof. It is a proof by contradiction, also known as an indirect proof, that proves a proposition by assuming its opposite to be true and then showing that this proposition is false, thereby implying that the proposition must be true.
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As there is a contradiction, the assumption (1) that √2 is a rational number must be false. This means that √2 is not a rational number. That is, √2 is irrational.
It first appeared as a complete proof as Proposition 117 in Book X of Euclid’s Elements. However, since the early 19th century, historians have agreed that this proof is anecdotal and cannot be attributed to Euclid.
A simple proof John Horton Conway attributes to Stanley Tanbaum in the early 1950s.
His most direct appearance is in an article by Noson Janofsky in the May-June 2016 issue of American Scientist.
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Two squares with sides a and b respectively, one of which is twice the area of the other, and place two copies of the smaller square in the larger one as shown in the figure. 1. The area of the middle overlapping squares ((2b – a)2) must be equal to the sum of the two uncovered squares (2(a – b)2). However, these squares on the diagonal have smaller positive integer sides than the original squares. By repeating this process, you can get arbitrarily small squares, one with twice the area of the other, but both with positive integers, which is impossible because no positive integer can be less than 1.
Another geometric deduction and absurd argument showing that √2 is irrational appeared in the 2000 American Journal of Mathematics.
This is an example of a proof that uses an infinite description. He uses the classical structure of the compass and direction, proving the theorem as in ancient Greek geometry. In summary, this is the same algebraic proof as in the previous paragraph, only from a different geometric point of view.
△ ABC is a hypothetical right isosceles triangle of length m and base n as shown in Figure 2. According to the Pythagorean theorem, m/n = √2. Suppose m and n are integers. Let m:n be the ratio given in the lowest expression.
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Through the vertex A draw arcs BD and CE. Turn on the DE. This implies that AB = AD, AC = AE, and ∠BAC and ∠DAE are congruent. Hence, triangles ABC and ADE meet in SAS.
Since ∠EBF is a right angle and ∠BEF is a half right angle, △ BEF is a right isosceles triangle. So BE = m – n means BF = m – n. By symmetry, DF = m − n and △ FDC is also a right isosceles triangle. This also implies that FC = n – (m – n) = 2n – m.
Hence, there exists a hypothetical small right-angled isosceles triangle on ev with lgth 2n – m and m – n legs. These values are integers, less than m and n, in the same ratio, contradicting the assumption that m:n is the smallest expression. Hence, both m and n cannot be integers, so √2 is irrational.
In a constructivist approach, one can distinguish between irrationality on the one hand and irrationality on the other (i.e., given positive integers a and b such that 1<a / b <3/2 (since √2 satisfies these constraints), evaluation (i.e., the number highest power of division 2) Since 2b2 is odd and the evaluation of a2 is ev, they must be distinct integers, so |2b2 − a2 | ≥ 1. th
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The last inequality is true because 1 < a / b < 3/2 is assumed, which gives a / b + √ 2 ≤ 3 (otherwise the quantitative difference can be trivially established). This gives