Rationality | Irrational |
---|---|
Representations | |
Decimal | 2.645751311064590590..._10 |
Algebraic form | |
Continued fraction | |
Binary | 10.10100101010011111111..._2 |
Hexadecimal | 2.A54FF53A5F1D..._16 |
The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in surd form as: [1]
and in exponent form as:
It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:
which can be rounded up to 2.646 to within about 99.99% accuracy (about 1 part in 10000); that is, it differs from the correct value by about 1/4,000. The approximation 127/48 (≈ 2.645833...) is better: despite having a denominator of only 48, it differs from the correct value by less than 1/12,000, or less than one part in 33,000.
More than a million decimal digits of the square root of seven have been published. [3]
The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773 [4] and 1852, [5] 3 in 1835, [6] 6 in 1808, [7] and 7 in 1797. [8] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth". [9]
For a family of good rational approximations, the square root of 7 can be expressed as the continued fraction
The successive partial evaluations of the continued fraction, which are called its convergents, approach :
Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…(sequence A041008 in the OEIS ) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequence A041009 in the OEIS ).
Each convergent is a best rational approximation of ; in other words, it is closer to than any rational with a smaller denominator. Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step:
Every fourth convergent, starting with 8/3, expressed as x/y, satisfies the Pell's equation [10]
When is approximated with the Babylonian method, starting with x1 = 3 and using xn+1 = 1/2(xn + 7/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction:
All but the first of these satisfy the Pell's equation above.
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial . The Newton's method update, is equal to when . The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).
In plane geometry, the square root of 7 can be constructed via a sequence of dynamic rectangles, that is, as the largest diagonal of those rectangles illustrated here. [11] [12] [13]
The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7. [14]
Due to the Pythagorean theorem and Legendre's three-square theorem, is the smallest square root of a natural number that cannot be the distance between any two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). is the next smallest such number. [15]
On the reverse of the current US one-dollar bill, the "large inner box" has a length-to-width ratio of the square root of 7, and a diagonal of 6.0 inches, to within measurement accuracy. [16]
In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square is x. For example, 4 and −4 are square roots of 16 because .
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction, the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction.
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In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as , where x is the radicand and n is the index. This is pronounced as "the nth root of x". The definition then of an nth root of a number x is a number r which, when raised to the power of the positive integer n, yields x:
In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other cube roots of 8 are and . The three cube roots of −27i are
The square root of 2 is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or . It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.
25 (twenty-five) is the natural number following 24 and preceding 26.
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e raised to the power π. Like both e and π, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that
In mathematics, the plastic numberρ is a mathematical constant which is the unique real solution of the cubic equation
Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root of a real number. Arithmetically, it means given , a procedure for finding a number which when multiplied by itself, yields ; algebraically, it means a procedure for finding the non-negative root of the equation ; geometrically, it means given two line segments, a procedure for constructing their geometric mean.
Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as or . It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
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In mathematics, two quantities are in the supergolden ratio if their quotient equals the unique real solution to the equation This solution is commonly denoted The name supergolden ratio results of a analogy with the golden ratio , which is the positive root of the equation
The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as:
Dynamic Symmetry root rectangles.