Кіріспе
Сандардың арақатынасы, шамамен 1:2.4.
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Бұл арақатынас күміс арақатынасы деп те аталады. Математикада екі шама күміс арақатынасында (немесе күміс орташасында) болады, егер кіші шаманың үлкен шамаға қатынасы, үлкен шаманың кіші шама мен екі есе үлкен шаманың қосындысына қатынасына тең болса (төменде қараңыз). Бұл күміс арақатынасын иррационал математикалық тұрақты ретінде анықтайды, оның мәні бір плюс түбір 2-нің квадраты шамамен 2.4142135623-ке тең. Оның атауы алтын арақатынасқа сілтеме жасайды; алтын арақатынас тікелей Фибоначчи сандарының лимиттік арақатынасы болса, күміс арақатынас тікелей Пелл сандарының лимиттік арақатынасы болып табылады. Күміс арақатынасы кейде δS арқылы белгіленеді, бірақ ол λ-дан σ-ға дейін өзгеруі мүмкін.
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Математиктер күміс арақатынасын гректер заманынан бері зерттеп келеді (дегенмен, соңғы уақытқа дейін оған ерекше атау бермеген болуы мүмкін), себебі оның түбір 2-нің квадратына, конвергенттеріне, квадратты үшбұрышты сандарға, Пелл сандарына, сегізбұрыштарға және т.б. байланысы бар. Жоғарыда сипатталған қатынасты алгебралық түрде, a > b үшін былай көрсетуге болады:
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
немесе, баламалы түрде:
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Күміс арақатынасын қарапайым тізбекті бөлшек түрінде [2; 2, 2, 2, ...] деп те анықтауға болады:
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Бұл тізбекті бөлшектің конвергенттері (, , , , , ...) тікелей Пелл сандарының арақатынастарын құрайды. Бұл бөлшектер күміс арақатынасының дәл рационал жуықтамаларын ұсынады, алтын арақатынасты тікелей Фибоначчи сандарының арақатынастары арқылы жуықтауға ұқсас. Күміс тіктөртбұрыш қалыпты сегізбұрышпен байланысты. Егер қалыпты сегізбұрышты екі теңбүйек трапеция мен тіктөртбұрышқа бөлсек, онда тіктөртбұрыш 1:δS арақатынасы бар күміс тіктөртбұрыш болады, ал трапециялардың 4 қабырғасы 1:1:1:δS арақатынасында болады. Егер қалыпты сегізбұрыштың қабырғасының ұзындығы t болса, онда сегізбұрыштың ені (қарсы қабырғалары арасындағы қашықтық) δSt-ге тең, ал сегізбұрыштың ауданы 2δSt^(2)-ге тең.
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Сандық теория қасиеттері
Күміс қатынасы Писот-Виджаярагхаван саны (PV саны) болып табылады, себебі оның түйіндес шамасының абсолюттік мәні 1-ден кіші. Шындығында, ол алтын қатынастан кейінгі екінші кіші квадраттық PV саны. Бұл, күміс қатынасының ең жақын бүтін санға дейінгі арақашықтығын білдіреді. Осылайша, күміс қатынасының бөлшек бөліктерінің тізбегі (торус элементтері ретінде қарастырылғанда) жинақталады. Атап айтқанда, бұл тізбек 1-ге қарай тең бөлінбейді.
The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from to the nearest integer is Thus, the sequence of fractional parts of , (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.