Reciprocal

Introduction

The term Reciprocal means inversely related. In fraction, the term Reciprocal is taking the inverse, that is the place of numerator and denominator exchanges with each other. To check the Reciprocal value is taken the correct, invert and multiply that inverted number with the original number always equal to one. Or if a number is multiplied by another number which is equal to one is reciprocal to each other. Division of fraction is made only by taking the reciprocal or multiplicative inverse.

Numbers

The term number means a word which represents the count, the measure, the quantity, which is expressed in a figure, word or a symbol. The numbers have different forms to represent them in each place. The numbers are classified as two major types:

$$\mathrm{real\:numbers\:and\:complex\:numbers}$$

Some types of the numbers are as follows:

• Natural numbers are the real numbers that are the counting numbers

Example − 1, 2, 3, 4, 5, .....

• whole numbers are the natural numbers together with zero.

Example − 0, 1, 2, 3, 4, 5….

• Integers are the whole numbers along with the negative numbers.

Example − -3, -2, -1, 0, 1, 2, 3, .....

• Fractions are the numbers represented as Numerator and denominator.

Example − $\mathrm{\frac{4}{5},\:\frac{65}{74},\:\frac{17}{23},\:......}$

• Rational numbers are the numbers represented in the form p/q where q is not equal to 0

Example − $\mathrm{\frac{1}{100},\:\frac{5}{6},\:\frac{13}{2},\:.......}$

• Irrational numbers are all real numbers other than rational numbers

Example − $\mathrm{\sqrt{5},\sqrt{13},\sqrt{9949}}$

• Real numbers, all the numbers explained above come under real numbers.

Example −$ \mathrm{2\:,\:-5\:,\:\frac{4}{17}\:,\sqrt{3}}$

• Complex numbers are real numbers and imaginary numbers added together.

Example −$\mathrm{3\:+\:2i\:,\:5\:-\:4i\:......}$

• Decimal numbers are real numbers that have whole parts and fractional parts separated by dots.

Example −$\mathrm{2.34\:,\:56.7\:.........}$

Reciprocal or multiplicative inverse

A Reciprocal of a number is its inverse of a number is also called a multiplicative inverse. Reciprocals are used in division of fraction, perpendicular lines, inversely proportions and so on. To find an unknown variable, all the other values in that equation must be taken to the other side of the equation by taking its reciprocal. To denote the Reciprocal of x, it can be taken inverse 𝑥−1 which is 1/x. If two numbers are reciprocal to each other, then the multiplication of two numbers results 1. We can represent each type of numbers in a number system to its reciprocal form.

Types of number	Examples	Reciprocals
Natural number	$\mathrm{11,\:52,\:23,\:14}$	$\mathrm{\frac{1}{11},\:\frac{}{52},\:\frac{1}{23},\:\frac{1}{14}}$
Integers	$\mathrm{-1,\:-2,\:-4,\:15}$	$\mathrm{-\frac{1}{1},-\:\frac{1}{2},\:-\frac{1}{4},\:\frac{1}{5}}$
Fraction	$\mathrm{\frac{5}{7},\:\frac{13}{17},\:\frac{24}{27},\:\frac{56}{19}}$	$\mathrm{\frac{7}{5},\:\frac{17}{13},\:\frac{27}{24},\:\frac{19}{56}}$
Rational number	$\mathrm{3.5\:,\:\frac{6}{11}\:,\:-0.25}$	$\mathrm{\frac{1}{3.5}\:or\:\frac{2}{7},\:\frac{11}{6},\:\frac{1}{0.25}\:or\:-4}$
Irrational number	$\mathrm{\sqrt{10}\:\sqrt{8},\:-9.94}$	$\mathrm{\frac{1}{\sqrt{10}},\:\frac{1}{\sqrt{8}},\:-\:\frac{1}{994}}$
real number	$\mathrm{-15,\:1,\:\frac{2}{7},\:\sqrt{5}}$	$\mathrm{\frac{1}{15},\:1\:\frac{17}{2},\:\frac{1}{\sqrt{5}}}$
Complex number	$\mathrm{\frac{3}{2\:-\:3i}}$	$\mathrm{\frac{1}{2\:-\:3i}\:\times\:\frac{2\:+\:3i}{2\:+\:3i}=\frac{2\:+\:3i}{4\:-\:9i^{2}}}$
decimal number	$\mathrm{0.45,\:-2.34}$	$\mathrm{\frac{1}{0.45},\:-\:\frac{1}{2.34}\:or\:-0.427}$

Fraction

Fraction described as a part or portion of a whole or a complete set. Fraction is sharing a whole part or set into small equal parts. There are different types of fractions. They are Like fraction, Unlike fraction, Proper Fraction, Improper fraction, equivalent fraction and mixed fraction. In fraction we can do addition, subtraction, Multiplication and division. Addition and subtraction of same Denominators can be done easily by taking the common denominators and by simply adding the numerator. If not, then by taking LCM of Denominators we can convert them as the same Denominators. To multiply a fraction with same or different denominators, by multiplying the numerator to the other numerator and denominator to the other Denominator.

Division of fractions

Fraction is itself a form of division. The division of fraction is breaking up or splitting up fraction into further small parts. There are three ways to divide a fraction. They are by area model, number line and algorithm. The algorithm method follows. There are four steps required to divide a fraction. The steps are as follows.

• A fraction can be divided by taking the multiplicative inverse of the given number.

• The numerator is multiplied to the inverted numerator.

• The denominator is multiplied with the other Denominator.

• Take HCF to find the simplest form.

Consider $\mathrm{\frac{1}{2}}$ portion of cake to be divided between 4 children. $\mathrm{\frac{1}{2}}$ divided to 4.

i.e.., $\mathrm{\frac{1}{2}\:\div\:4\:=\:\frac{1}{2}\:\times\:\frac{1}{4}\:=\:\frac{1}{8}}$

Here the cake of $\mathrm{\frac{1}{2}}$ Portion can be divided to $\mathrm{\frac{1}{8}}$ to share equally to the 4 children.

Solved examples

Divide the fraction $\mathrm{\frac{5}{6}\:\div\:\frac{7}{2}}$

Solution

By taking the multiplicative inverse,

$$\mathrm{\frac{5}{6}\:\div\:\frac{7}{2}\:=\:\frac{5}{6}\:\times\:\frac{2}{7}}$$

By multiplying the numerators and denominators,

$\mathrm{=\:\frac{5\:\times\:2}{6\:\times\:7}\:=\:\frac{10}{42}}$

HCF of $\mathrm{(10\:,\:42)\:=\:2}$

$$\mathrm{\frac{10}{42}\:\div\:\frac{2}{2}\:=\:\frac{5}{21}}$$

2. Divide the mixed fraction $\mathrm{7\:\frac{3}{5}\:\div\:6\frac{4}{2}}$

Solution

$$\mathrm{7\:\frac{3}{5}\:\div\:6\frac{4}{2}\:=\:\frac{38}{5}\:+\:\frac{16}{2}}$$

By taking the multiplicative inverse

$$\mathrm{\frac{38}{5}\:\div\:\frac{16}{2}\:\frac{38}{5}\:\times\:\frac{2}{16}}$$

By multiplying the numerators and denominators

$\mathrm{=\:\frac{38\:\times\:2}{5\:\times\:16}\:=\:\frac{76}{80}}$

HCF of $\mathrm{(76,\:80)\:=\:4}$

$$\mathrm{\frac{76}{80}\:\div\:\frac{4}{4}\:=\:\frac{19}{20}}$$

3. Heidi has $\mathrm{\frac{13}{21}}$ portion of watermelon with her. She wants to divide them and share $\mathrm{\frac{2}{3}}$ portion with her family. How pieces can Heidi share with her family?

Solution

By multiplicative inverse

$$\mathrm{\frac{13}{21}\:\div\:\frac{2}{3}\:=\:\frac{13}{21}\:\times\:\frac{3}{2}}$$

By multiplying the numerators and denominators,

$$\mathrm{\frac{13}{21}\:\times\:\frac{3}{2}\:=\:\frac{13\:\times\:2}{21\:\times\:2}\:=\:\frac{39}{42}}$$

HCF of $\mathrm{(39\:,\:42)\:=\:3}$

$$\mathrm{\frac{39}{42}\:\div\:\frac{3}{3}\:=\:\frac{13}{14}}$$

Heidi can share $\mathrm{\frac{13}{14}}$ pieces of watermelon to her family.

Conclusion

Reciprocal of a number is multiplied with the given number equals 1. Reciprocal of different types of numbers such as natural, integers, fraction, rational, irrational and complex numbers are obtained by taking the multiplicative inverse. To divide a mixed fraction, the mixed fraction must be converted to improper fraction and divided with the second fraction by taking a multiplicative inverse. To divide a fraction, the first step is to take the multiplicative inverse of the divisor and multiply it with the dividend, by taking HCF of the numerator and denominator we can get the fraction as a solution.

FAQs

1. Can we divide a mixed fraction with different denominators?

Yes, a mixed fraction can be divided by converting them to improper fractions by taking multiplicative inverse. Then by multiplying the numerators and denominators and taking HCF we can get the simplest form of the mixed fraction. As the mixed fraction will have at least one whole in it, it cannot convert to the proper fraction.

2. Is it possible to divide two improper fractions?

Yes, to divide a number the first step is to take the multiplicative inverse and multiply the numerators and denominators. Only for addition and subtraction the denominators need to be the same.

3. What is the Reciprocal of $\mathrm{3\:\frac{4}{5}}$ ?

To find the reciprocal of mixed fraction, convert the mixed fraction to improper fraction.

i.e., $\mathrm{3\:\frac{4}{5}\:=\:\frac{19}{5}}$ ,then the reciprocal of $\mathrm{\frac{19}{5}\:is\:\frac{5}{19}}$

4. Can we take the Reciprocal of 0?

No, we cannot take the inverse of 0 that is 0/1 which gives indefinite values. We can take reciprocal of every real value other than 0.

5. How to express a negative exponent into a positive fraction?

To convert a negative exponent, by taking a multiplicative inverse the negative exponent converts to a positive fraction