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This is a common procedure in mathematics, used to reduce fractions or calculate a value for a given variable in a fraction. If we have an equation =, where x is a variable we are interested in solving for, we can use cross-multiplication to determine that =.
This method was an improvement compared to other methods because it started from the beginning of the continued fraction rather than the tail, had a built-in check for convergence, and was numerically stable. The original algorithm uses algebra to bypass a zero in either the numerator or denominator. [5]
A continued fraction is an expression of the form = + + + + + where the a n (n > 0) are the partial numerators, the b n are the partial denominators, and the leading term b 0 is called the integer part of the continued fraction.
For instance, if the one solving the math word problem has a limited understanding of the language (English, Spanish, etc.) they are more likely to not understand what the problem is even asking. In Example 1 (above), if one does not comprehend the definition of the word "spent," they will misunderstand the entire purpose of the word problem.
(For example, 2 / 5 and 3 / 5 are both read as a number of fifths.) Exceptions include the denominator 2, which is always read half or halves, the denominator 4, which may be alternatively expressed as quarter/quarters or as fourth/fourths, and the denominator 100, which may be alternatively expressed as hundredth/hundredths or ...
The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0 , a mathematical truth. But the same substitution applied to the original equation results in x /6 + 0/0 = 1 , which is mathematically meaningless .
An obvious necessary condition is that the starting fraction x / y have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known [20] that every x / y with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm.
Horner's method can be used to convert between different positional numeral systems – in which case x is the base of the number system, and the a i coefficients are the digits of the base-x representation of a given number – and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.