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单词 Division
释义

division


Division is the operationMathworldPlanetmath which assigns to every two numbers (or more generally, elements of a field) a and b their quotientPlanetmathPlanetmath or ratio, provided that the latter, b, is distinct from zero.

The quotient (or ratio)  ab  of a and b may be defined as such a number (or element of the field) x that  bx=a.  Thus,

bab=a,

which is the “fundamental property of quotient”.

The quotient of the numbers a and b (0) is auniquely determined number, since if one had

ab=xy=ab,

then we could write

b(x-y)=bx-by=a-a=0

from which the supposition b0 would imply x-y=0, i.e.x=y.

The explicit general expression for ab is

ab=b-1a

where b-1 is the inverse number (the multiplicative inverseMathworldPlanetmath) of a, because

b(b-1a)=(bb-1)a=1a=a.
  • For positive numbers the quotient may be obtained by performing the division algorithmPlanetmathPlanetmath with a and b.  If  a>b>0,  then ab indicates how many times b fits in a.

  • The quotient of a and b does not change if both numbers (elements) are multiplied (or divided, which is called reductionPlanetmathPlanetmath) by any  k0:

    kakb=(kb)-1(ka)=b-1k-1ka=b-1a=ab

    So we have the method for getting the quotient of complex numbersMathworldPlanetmathPlanetmath,

    ab=b¯ab¯b,

    where b¯ is the complex conjugate of b, and the quotient of http://planetmath.org/SquareRootOfSquareRootBinomialsquare root polynomials, e.g.

    15+22=5-22(5-22)(5+22)=5-2225-8=5-2217;

    in the first case one aspires after a real and in the second case after a rational denominator.

  • The division is neither associative nor commutativePlanetmathPlanetmathPlanetmathPlanetmath, but it is right distributive over additionPlanetmathPlanetmath:

    a+bc=ac+bc
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更新时间:2025/5/4 15:26:19