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割圆术 (刘徽) Algoritmo de Liu Hui para π خوارزمية ليو هوي لحساب π Liu Hui's π algorithm
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三国时代数学家刘徽的割圆术是中国古代数学中“一个十分精彩的算法”。在此之前，圆周率采用“径一周三”的实验数据。东汉科学家张衡采用和。刘徽认为过大。。东汉天文学家王蕃采用。这些圆周率都是实验值，都只准确到二位数字。刘徽是中国数学史上最先创造了一个从数学上计算圆周率到任意精确度的迭代程序的数学家。他自己通过分割圆为192边形，计算出圆周率在3.141024 与 3.142704之间，取其近似，并以 表示。这个数值准确到三位数字，比前人的圆周率数值都准，但他自己次承认这个数值偏小。后来刘徽发明一种快捷算法，可以只用96边形得到和1536边形同等的精确度，从而得令他自己满意的。 刘徽割圆术简单而又严谨，富于程序性，可以继续分割下去，求得更精确的圆周率。南北朝时期著名数学家祖冲之用刘徽割圆术计算11次，分割圆为12288边形，得圆周率=3.1415926，成为此后千年世界上最准确的圆周率。 刘徽在圆周率领域的贡献，不仅在于求得 和，更重要的在于他创造了一世界数学史上最精彩的割圆术：阿基米德割圆术和刘徽割圆术一样用双向迫近，因而同样严谨完备，但远不如刘徽简洁；阿基米德用双归谬法推证圆面积，不如刘徽用极限论先进；托勒密割圆术和割圆术只是单向迫近，不如刘徽严谨；割圆术和日本关孝和割圆术从正方开割，属于刘徽割圆术的变化，而且也是单向迫近。刘徽割圆术虽然不是世界最早，却是数学史上最严谨完备简洁的割圆术。 خوارزمية ليو هوي لحساب π هي طريقة ابتدعها العالم الصيني ليو هوي (القرن 3م) لحساب قيمة العدد π. حيث قبل هذا الاكتشاف، كان الصينيون يقربون قيمة العدد π بالعدد 3, ثم ابتدع (78-139) تقريبة 3.1724 للعدد π. لكن ليو هوي هو أول من استعمل مضلعا عدد أضلاعه 96 لتحديد قيمة π=3.1416 Liu Hui's π algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of the state of Cao Wei. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth, 92/29) or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits (i.e. one decimal place). Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy o O algoritmo de Liu Hui para π foi inventado por Liu Hui (fl. século III), um matemático do império de Cao Wei. Antes de sua época, a razão entre o perímetro de uma circunferência e seu diâmetro era muitas vezes tomada experimentalmente como três na China, enquanto Zhang Heng (78-139) a representava como sendo 3,1724 (da proporção do círculo celeste com o diâmetro da Terra, 92/29) ou como . Liu Hui não ficou satisfeito com este valor. Ele comentou que ele era muito grande e ultrapassava a marca. Outro matemático, Wang Fan (228-266), forneceu . Todos esses valores empíricos para π eram precisos para dois dígitos (ou seja, uma casa decimal). Liu Hui foi o primeiro matemático chinês a fornecer um algoritmo rigoroso para o cálculo de π para qualquer precisão. O próprio cálculo de Liu Hui com um
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خوارزمية ليو هوي لحساب π هي طريقة ابتدعها العالم الصيني ليو هوي (القرن 3م) لحساب قيمة العدد π. حيث قبل هذا الاكتشاف، كان الصينيون يقربون قيمة العدد π بالعدد 3, ثم ابتدع (78-139) تقريبة 3.1724 للعدد π. لكن ليو هوي هو أول من استعمل مضلعا عدد أضلاعه 96 لتحديد قيمة π=3.1416 三国时代数学家刘徽的割圆术是中国古代数学中“一个十分精彩的算法”。在此之前，圆周率采用“径一周三”的实验数据。东汉科学家张衡采用和。刘徽认为过大。。东汉天文学家王蕃采用。这些圆周率都是实验值，都只准确到二位数字。刘徽是中国数学史上最先创造了一个从数学上计算圆周率到任意精确度的迭代程序的数学家。他自己通过分割圆为192边形，计算出圆周率在3.141024 与 3.142704之间，取其近似，并以 表示。这个数值准确到三位数字，比前人的圆周率数值都准，但他自己次承认这个数值偏小。后来刘徽发明一种快捷算法，可以只用96边形得到和1536边形同等的精确度，从而得令他自己满意的。 刘徽割圆术简单而又严谨，富于程序性，可以继续分割下去，求得更精确的圆周率。南北朝时期著名数学家祖冲之用刘徽割圆术计算11次，分割圆为12288边形，得圆周率=3.1415926，成为此后千年世界上最准确的圆周率。 刘徽在圆周率领域的贡献，不仅在于求得 和，更重要的在于他创造了一世界数学史上最精彩的割圆术：阿基米德割圆术和刘徽割圆术一样用双向迫近，因而同样严谨完备，但远不如刘徽简洁；阿基米德用双归谬法推证圆面积，不如刘徽用极限论先进；托勒密割圆术和割圆术只是单向迫近，不如刘徽严谨；割圆术和日本关孝和割圆术从正方开割，属于刘徽割圆术的变化，而且也是单向迫近。刘徽割圆术虽然不是世界最早，却是数学史上最严谨完备简洁的割圆术。 Liu Hui's π algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of the state of Cao Wei. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth, 92/29) or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits (i.e. one decimal place). Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: π ≈ 3.1416. Liu Hui remarked in his commentary to The Nine Chapters on the Mathematical Art, that the ratio of the circumference of an inscribed hexagon to the diameter of the circle was three, hence π must be greater than three. He went on to provide a detailed step-by-step description of an iterative algorithm to calculate π to any required accuracy based on bisecting polygons; he calculated π to between 3.141024 and 3.142708 with a 96-gon; he suggested that 3.14 was a good enough approximation, and expressed π as 157/50; he admitted that this number was a bit small. Later he invented an ingenious to improve on it, and obtained π ≈ 3.1416 with only a 96-gon, with an accuracy comparable to that from a 1536-gon. His most important contribution in this area was his simple iterative π algorithm. O algoritmo de Liu Hui para π foi inventado por Liu Hui (fl. século III), um matemático do império de Cao Wei. Antes de sua época, a razão entre o perímetro de uma circunferência e seu diâmetro era muitas vezes tomada experimentalmente como três na China, enquanto Zhang Heng (78-139) a representava como sendo 3,1724 (da proporção do círculo celeste com o diâmetro da Terra, 92/29) ou como . Liu Hui não ficou satisfeito com este valor. Ele comentou que ele era muito grande e ultrapassava a marca. Outro matemático, Wang Fan (228-266), forneceu . Todos esses valores empíricos para π eram precisos para dois dígitos (ou seja, uma casa decimal). Liu Hui foi o primeiro matemático chinês a fornecer um algoritmo rigoroso para o cálculo de π para qualquer precisão. O próprio cálculo de Liu Hui com um polígono de 96 lados forneceu uma precisão de cinco dígitos: π ≈ 3,1416. Liu Hui observou em seu comentário no Os nove capítulos da arte matemática, que a relação do perímetro de um hexágono inscrito com o diâmetro da circunferência era três, então π deve ser maior que três. Ele passou a fornecer uma descrição detalhada passo-a-passo de um algoritmo iterativo para calcular π com qualquer precisão estabelecida com base em bisseção de polígonos; ele calculou π entre 3,141024 e 3,142708 com um polígono de 96 lados; ele sugeriu que 3,14 era uma aproximação boa o suficiente, e expressou π como 157/50; ele admitiu que este número era um pouco pequeno. Inventou mais tarde um engenhoso para melhorá-lo, e obteve π ≈ 3,1416 com apenas um polígono de 96 lados, com uma precisão comparável à de um polígono de 1536 lados. Sua contribuição mais importante nessa área foi seu algoritmo simples iterativo para π.
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