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Unit fractions can also be expressed using negative exponents, as in 2 −1, which represents 1/2, and 2 −2, which represents 1/(2 2) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two , e.g. 1 / 8 = 1 / 2 3 .
For example, 1 / 4 , 5 / 6 , and −101 / 100 are all irreducible fractions. On the other hand, 2 / 4 is reducible since it is equal in value to 1 / 2 , and the numerator of 1 / 2 is less than the numerator of 2 / 4 . A fraction that is reducible can be reduced by dividing both the numerator ...
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 .
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.
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
[6] [7] [a] The parentheses can be omitted if the input is a single numerical variable or constant, [2] as in the case of sin x = sin(x) and sin π = sin(π). [a] Traditionally this convention extends to monomials; thus, sin 3x = sin(3x) and even sin 1 / 2 xy = sin(xy/2), but sin x + y = sin(x) + y, because x + y is not a monomial ...
A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials. Thus 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} is a rational fraction, but not x + 2 x 2 − 3 , {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}},} because the numerator contains a square root function.
6.2 Continued fraction. 6.3 Nested square. 7 Applications. Toggle Applications subsection. 7.1 Paper size. ... The square root of 2 (approximately 1.4142) ...