Rational Numbers to Standard Form

Introduction

When a rational number is expressed in its standard form, it signifies that its denominator is a positive integer and that its numerator has no common factors other than 1. Rational numbers are those that can be stated as $\mathrm{\frac{r}{s}}$, where r and s are integers and s is not equal to zero. Therefore, if $\mathrm{\frac{4}{8}}$ is a rational number, its the standard form will be $\mathrm{\frac{1}{2}}$ because we are no longer able to solve $\mathrm{\frac{1}{2}}$. When there is only one common factor between the denominator and the numerator, the result is a rational number. However, as the denominator is always positive, a rational number might be considered to be the standard, if the numerator also has a positive sign. These numbers can be referred to as rational numbers in the standard form if these criteria are met.

Rational Numbers

Rational numbers are of the form $\mathrm{\frac{r}{s}}$, where r and s are integers and s is not equal to zero. Example $\mathrm{\frac{4}{8}}$,2,0 are rational numbers

Standard form of Rational Numbers

The standard form of the rational number is the simplest form of the fraction form of the rational number.

For example:

$\mathrm{\frac{2}{3}}$ is the standard form of the number $\mathrm{\frac{8}{12}}$

How to check if the given form is in the standard form of rational ?

We must first ascertain the H.C.F. of the numerator and denominator in order to determine whether a given rational number is in its standard form. If it is 1, then the provided rational number's numerator and denominator are coprime numbers. We start by dividing the two numbers by their shared component if the numerator and denominator are not coprime. We keep dividing the numerator and denominator by the common factors up until we reach a numerator and denominator with HCF equal to 1.

For example:

The given rational number is $\mathrm{\frac{6}{9}}$, notice that the HCF of 6 and 9 is 3, not 1. so we divide numerator and denominator by HCF so that we get reduced form $\mathrm{\frac{2}{3}}$ which can not be reduced further.

Hence $\mathrm{\frac{2}{3}}$ is the standard form of the rational number $\mathrm{\frac{6}{9}}$.

Conversion in standard form

To convert the given rational number into the standard form we shall follow the following process −

Verify whether the supplied rational number has a positive or negative denominator. If the denominator is negative, multiply the numerator by -1 or divide the denominator by -1 to make it positive. Then, calculate the HCF of the numerator's and denominator's absolute values. Divide the computed HCF value by the specified rational number's numerator and denominator. The rational number's standard form is the obtained resultant.

For example:

The given rational number is $\mathrm{\frac{-12}{36}}$, notice that here the denominator is positive so we not do the multiplication by -1. Now HCF of 12 and 36 is 12 so we divide numerator and denominator by 12 to get the standard form of the given rational number as $\mathrm{\frac{-1}{3}}$.

Algebra of rational numbers

The set of rational numbers is denoted by Q. Let $\mathrm{\frac{a}{b}\: \&\: \frac{c}{d}}$ be two rational numbers such that b≠0,d≠0.

The sum of two rational numbers is defined as $\mathrm{\frac{a}{b} + \frac{c}{d}=\frac{ad+bc}{bd}}$.

The difference of two rational numbers is defined as $\mathrm{\frac{a}{b} - \frac{c}{d}=\frac{ad-bc}{bd}}$.

The product of two rational numbers is defined as $\mathrm{\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}}$

The division of two rational numbers is defined as $\mathrm{\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}=\frac{ad}{bc}}$

Solved Examples

1.What is the standard form of the rational number 24/96? What is the standard form of the rational number $\mathrm{\frac{24}{96}?}$

Solution:

The given rational number is $\mathrm{\frac{24}{96}}$, notice that the HCF of 24 and 96 is 24, it's not 1. so we divide numerator and denominator by HCF 24 ,so that we get reduced form $\mathrm{\frac{1}{4}}$ which can not be reduced further.

Hence $\mathrm{\frac{1}{4}}$ is the standard form of the rational number $\mathrm{\frac{24}{96}}$.

2.Write the following rational numbers in the descending order.

$$\mathrm{\frac{35}{25},\frac{25}{35},\frac{25}{75}}$$

Solution:

Notice that the standard form of the given three rational numbers is $\mathrm{\frac{7}{5},\frac{5}{7},\frac{1}{5}}$ respectively.

To arrange the numbers in the descending order first of all we make their denominator the same by taking their LCM.

Note that the LCM of 5 and 7 is 35,

$$\mathrm{so\: \frac{7}{5}=\frac{49}{35},\frac{5}{7}=\frac{25}{35},\frac{1}{5}=\frac{7}{35}}$$

$$\mathrm{Now,\: \frac{49}{35}>\frac{25}{35}>\frac{7}{35}.}$$

Thus the required descending order of the given rational numbers is $\mathrm{\frac{7}{5},\frac{5}{7},\frac{1}{5}}$

3.What number should be added to $\mathrm{\frac{3}{5}}$ to get the result $\mathrm{\frac{1}{5}}$?

Solution:

Let x be added to the rational number $\mathrm{\frac{3}{5}}$ to get the result $\mathrm{\frac{1}{5}}$.

Therefore,

$$\mathrm{x+\frac{3}{5}=\frac{1}{5}}$$

$$\mathrm{x=\frac{1}{5}-\frac{3}{5}=\frac{1-3}{5}=\frac{-2}{5}}$$

Therefore, $\mathrm{\frac{-2}{5}}$ to be added to the rational number $\mathrm{\frac{3}{5}}$ to get the result $\mathrm{\frac{1}{5}}$.

4.What number should be subtracted from $\mathrm{\frac{2}{7}}$ to get the result $\mathrm{\frac{1}{3}}$?

Solution:

Let x be subtracted from the rational number $\mathrm{\frac{2}{7}}$ to get the result $\mathrm{\frac{1}{3}}$.

Therefore,

$$\mathrm{\frac{2}{7}-x=\frac{1}{3}}$$

Therefore, $\mathrm{\frac{-1}{21}}$ to be subtracted from the rational number $\mathrm{\frac{2}{7}}$ to get the result $\mathrm{\frac{1}{3}}$.

5.Evaluate the following:

• $\mathrm{\frac{6}{7} \times 3}$

• $\mathrm{\frac{8}{9}÷\frac{16}{81}}$

Solution:

• $\mathrm{\frac{6}{7}×3=\frac{6}{7}\times \frac{3}{1}=\frac{6×3}{7×1}=\frac{18}{7}}$

• $\mathrm{\frac{8}{9}÷\frac{16}{81}=\frac{8}{9 }\times \frac{81}{16}=1×\frac{9}{2}=\frac{9}{2}}$

Conclusion

In this article we learned about Rational Numbers, and their standard form. When the common factor between the denominator and the numerator is only 1, and the denominator is always positive, the rational number is said to be in its standard form. Additionally, when the numerator has a positive sign, the standard form of a rational number is satisfied. We refer to these numbers as rational numbers in the standard form.

FAQs

1.What type of number is zero?

Yes, the number zero can be written in the form of $\mathrm{\frac{p}{q}=\frac{0}{q}}$, with q≠0, hence zero is a rational number.

2.Are whole numbers rational?

Whole numbers can be written in the form of $\mathrm{\frac{p}{q}}$,q≠0. For example the whole number 100 can be written as $\mathrm{\frac{100}{1}}$, which is the standard form of rational number, hence every whole number is rational.

3.Write four numbers that are rational, that are equivalent to the given number $\mathrm{\frac{3}{4}}$.

To find the numbers equivalent to a given rational number in standard form, we multiply the numerator and denominator of the standard form of the given rational number with an integer other than 1 and 0. The four rational numbers that are equivalent to the given number $\mathrm{\frac{3}{4}\: are\: \frac{9}{12},\frac{6}{8},\frac{15}{20},\frac{18}{24}}$

4.What 3 characteristics do rational numbers have?

Any number that meets all three of the following requirements is considered rational: It can be written as a simple fraction $\mathrm{\frac{p}{q}}$,q≠0. The numerator and denominator must both be authentic ordinary integers or coprime to each other and q can’t be zero

5.Is the number infinite rational?

By the definition of Real numbers, i.e., the real numbers are the numbers that may be rational or irrational, between the interval [-∞,∞]. Thus, by this definition infinity is not a real number and by definition not a rational number itself.