Multiplicative inverse

Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).


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 ab ≠ ba; 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 for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)⁻¹ is the cosecant of x, and not the inverse sine of x denoted by sin⁻¹ x or arcsin x. 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 the bijection réciproque).

Reciprocals of irrational numbers[edit]

Every real or complex number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of e (≈ 0.367879) and the golden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; ${\displaystyle f(1/e)}$ is the global minimum of ${\displaystyle f(x)=x^{x}}$. The second number is the only positive number that is equal to its reciprocal plus one:${\displaystyle \varphi =1/\varphi +1}$. Its additive inverse is the only negative number that is equal to its reciprocal minus one:${\displaystyle -\varphi =-1/\varphi -1}$.


The function ${\textstyle f(n)=n+{\sqrt {n^{2}+1}},n\in \mathbb {N} ,n>0}$ gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, ${\displaystyle f(2)}$ is the irrational ${\displaystyle 2+{\sqrt {5}}}$. Its reciprocal ${\displaystyle 1/(2+{\sqrt {5}})}$ is ${\displaystyle -2+{\sqrt {5}}}$, exactly ${\displaystyle 4}$ less. Such irrational numbers share an evident property: they have the same fractional part as their reciprocal, since these numbers differ by an integer.


The reciprocal function plays an important role in continued fractions, which have a number of remarkable properties relating to the representation of (both rational and) irrational numbers.