Question

Find the multiplicative inverse of \[2,\dfrac{6}{11},\dfrac{-8}{15},\dfrac{19}{18},\dfrac{1}{1000}\].

Answer 1

- Hint: Divide 1 by each of the number or fraction given to get what is asked in the question. In the given question we have to find the multiplicative inverses of \[2,\dfrac{6}{11},\dfrac{-8}{15},\dfrac{19}{18},\dfrac{1}{1000}\].

Complete step-by-step answer:
At first, we will try to understand what we can mean by multiplicative inverse of any number.
In mathematics, a multiplicative inverse or a reciprocal for a number x, denoted by \[{{x}^{-1}}\] is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction \[\dfrac{a}{b}\] is \[\dfrac{b}{a}\]. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (\[\dfrac{1}{5}\] or 0.2) and the reciprocal of 0.25 is 1 divided by 0.25 or 4. The reciprocal function, the function \[f\left( x \right)\] that converts \[x\] to \[\dfrac{1}{x}\].
In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tactically understood. Multiplicative inverse can be defined over many mathematical domains as well as numbers. It can happen that \[ab\ \ne \ ba\]; the inverse typically implies that an element is both left and right inverse.
In the question, the inverse of 2 will be \[\dfrac{1}{2}=\ 0.5\], the inverse of \[\dfrac{6}{11}\]will be \[\dfrac{1}{\left( \dfrac{6}{11} \right)}\ =\ \dfrac{11}{6}\], the inverse of \[-\dfrac{8}{15}\] will be \[\dfrac{1}{\left( -\dfrac{8}{15} \right)}\ =\ \left( -\dfrac{15}{8} \right)\], the inverse of \[\dfrac{19}{18}\] will be \[\dfrac{1}{\left( \dfrac{19}{18} \right)}\ =\ \dfrac{18}{19}\], the inverse of \[\dfrac{1}{1000}\] will be \[\dfrac{1}{\left( \dfrac{1}{1000} \right)}\ =\ 1000\].
Hence the multiplicative inverses are \[0.5,\dfrac{11}{6},\dfrac{-15}{8},\dfrac{18}{19},1000\].

Note: We can also find the multiplicative of fraction or an integer on any number by just dividing it from 1. Just swap the numerator or denominator keeping its positive or negative sign constant.