# Properties of Rational Numbers

In this chapter we will discuss some important properties of rational number with examples.

To understand the properties, you should have basic understanding of concept of rational number.

## Rational numbers properties

Given below are properties of rational number with example;

(a) Rational number are generally expressed in form of P / Q

Where P & Q are integers.

Numbers like \mathtt{\frac{2}{3} ,\ \frac{-7}{6} \ \&\ \frac{\ 10}{9}} are some of the examples of rational numbers.

(b) Decimal number are also rational numbers

Decimal numbers can be easily converted in the form of P / Q. Hence they are also rational numbers.

For example;
The decimal 10.25 can be converted into fraction as follows.

\mathtt{10.25\ \Longrightarrow \ \frac{1025}{100}}

Hence, the above decimal is a rational number.

(c) If you multiply number m on both numerator & denominator of rational number, you will get the rational number of same value.

Let \mathtt{\frac{a}{b}} be the rational number.

Multiply m on both numerator and denominator, we get;

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\times m}{b\times m}}\\\ \\ \mathtt{\Longrightarrow \ \frac{am}{bm}}

Here the resultant rational number \mathtt{\frac{am}{bm}} has the same value as the original number \mathtt{\frac{a}{b}}

\mathtt{\frac{a}{b} \Longrightarrow \ \frac{a\times m}{b\times m}}

For example;
Consider the rational number 2 / 3

Multiply number 5 on both numerator and denominator.

\mathtt{\Longrightarrow \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 5}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{15}}

Here both 10/15 and 2/3 have same value.

\mathtt{\frac{10}{15} \Longrightarrow \ 0.67}\\\ \\ \mathtt{\frac{2}{3} \ \Longrightarrow \ 0.67}

Conclusion
The multiplication of same number on numerator and denominator will produce rational number of same value.

(d) If you divide numerator and denominator of rational number by number m, you will get rational number of same value.

Let \mathtt{\frac{a}{b}} be the rational number.

Divide numerator and denominator by m.

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\div m}{b\div m}}

This new rational number value is same as initial value a / b.

\mathtt{\frac{a}{b} =\ \frac{a\div m}{b\div m}}

For example
Consider the rational number \mathtt{\frac{10}{16}}

Divide numerator and denominator by 2

\mathtt{\Longrightarrow \ \frac{10\div 2}{16\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{8}}

According to the property of rational number, both the number 10/16 and 5/8 have same values.

\mathtt{\frac{10}{16} \Longrightarrow \ 0.625}\\\ \\ \mathtt{\frac{5}{8} \ \Longrightarrow \ 0.625 }

(e) If two rational numbers are equal, then the cross multiplication of rational number will also be equal.

Let a/b & c/d be the two rational numbers.

\mathtt{\frac{a}{b} =\ \frac{c}{d}}

Doing the cross multiplication we get;

\mathtt{a\times d\ =\ b\times c}

For example;
Let 2/6 and 8/24 be the equal rational numbers.

\mathtt{\frac{2}{6} =\ \frac{8}{24}}

On doing the cross multiplication, we get the equal numbers.

\mathtt{2\times 24\ =6\ \times 8}\\\ \\ \mathtt{48\ =\ 48}

(f) The value of rational number a/b depend on the values of a & b.

(i) if a > b, the value of rational number will be greater than 1

(ii) if a < b, the value if rational number is less than 1

(iii) if a = b ,the value of rational number is equal to 1

For example;
Consider the rational number 5/4

Here numerator > denominator, so value of rational number is greater than 1.

\mathtt{\frac{5}{4} \Longrightarrow \ 1.25}

Example 02
Consider rational number 4/5

Here numerator is less than denominator, so value of rational number is less than 1.

latex] \mathtt{\frac{4}{5} \Longrightarrow \ 0.8} [/latex]

(g) Suppose there are three rational numbers X, Y & Z such that, X > Y and Y > Z, then it automatically implies that X > Z.

For example;
Let the three rational numbers are \mathtt{\frac{2}{3} ,\ \frac{2}{4} ,\ \frac{2}{6}} .

Its given that;
\mathtt{\frac{2}{3} >\ \frac{2}{4}} and \mathtt{\frac{2}{4} >\ \frac{2}{6}} .

From the above information, we can automatically infer \mathtt{\frac{2}{3} >\ \frac{2}{6}}