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The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the casus irreducibilis , in which all three real solutions are written in terms of cube roots of complex numbers.
A non-nested radical expression is said to be in simplified form if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.
Distribution of the number of entries per radical in the Kangxi Dictionary. The Kangxi dictionary lists a total of 47,035 characters divided among the 214 radicals, for an average of 220 characters per radical; however, the distribution is unequal, with the median number of characters per radical being 64, the maximum number being 1,902 (for radical 140 艸), and the minimum being 5 (for ...
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Some simplified characters have added new components of Chinese characters. For example, ' 书农长 ' and so on. Although the traditional character 農 has more strokes, it is very clear to say: ' 曲 + 辰 農 '. When we simplify Chinese characters, we should avoid new unspeakable and uncommon components.
A radical (Chinese: 部首; pinyin: bùshǒu; lit. 'section header'), or indexing component, is a visually prominent component of a Chinese character under which the character is traditionally listed in a Chinese dictionary.
Simplified Cangjie, also known as quick, 簡易; jiǎnyì or 速成; sùchéng, has the same radicals, auxiliary shapes, decomposition rules, and short list of exceptions as Cangjie, but only the first and last codes are used if more than two codes are required in Cangjie.