Multiplicative Inverse

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A Multiplicative Inverse is a form of the inverse relation in which a number when multiplied by x yields the multiplicative identity, 1.



  • (Wikipedia, 2020) ⇒ Retrieved:2020-3-5.
    • In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).

      The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements. [1]

      In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ; then "inverse" typically implies that an element is both a left and right inverse.

      The notation f −1 is sometimes also used for the inverse function of the function f, which is not in general equal to the multiplicative inverse. For example, the multiplicative inverse is the cosecant of x, and not the inverse sine of x denoted by or . Only for linear maps are they strongly related (see below). The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called bijection réciproque).


The multiplicative inverse of a nonzero number [math]\displaystyle{ z }[/math] is its reciprocal [math]\displaystyle{ 1/z }[/math] (zero is not invertible). For complex [math]\displaystyle{ z=x+iy\neq 0 }[/math],
[math]\displaystyle{ 1/z=1/(x+iy)=x/(x^2+y^2)-iy/(x^2+y^2). }[/math]
The inverse of a nonzero real quaternion [math]\displaystyle{ h=x+yi+vj+wk }[/math] (where [math]\displaystyle{ x,y,v,w }[/math] are real numbers, and not all of them are zero) is its reciprocal
[math]\displaystyle{ \frac{1}{h}=\frac{x}{\alpha}-\frac{y}{\alpha} i-\frac{v}{\alpha}j-\frac{w}{\alpha} k \quad \textrm{where} \quad \alpha=x^2+y^2+v^2+w^2. }[/math]
The multiplicative inverse of a nonsingular matrix is its matrix inverse.

  1. " In equall Parallelipipedons the bases are reciprokall to their altitudes". OED "Reciprocal" §3a. Sir Henry Billingsley translation of Elements XI, 34.