Properties of Rational Numbers
We will learn some useful properties of rational numbers.
Property 1:
If a/b is a rational number and m is a nonzero integer, then
\(\frac{a}{b}\) = \(\frac{a × m}{b × m}\)
In other words, a rational number remains unchanged, if we multiply its numerator and denominator by the same non-zero integer.
For examples:
\(\frac{-2}{5}\) = \(\frac{(-2) × 2}{5 × 2}\) = \(\frac{-4}{10}\), \(\frac{(-2) × 3}{5 × 3}\) = \(\frac{-6}{15}\), \(\frac{(-2) × 4}{5 × 4}\) = \(\frac{-8}{20}\) and so on ……
Therefore, \(\frac{-2}{5}\) = \(\frac{(-2) × 2}{5 × 2}\) = \(\frac{(-2) × 3}{5 × 3}\) = \(\frac{(-2) × 4}{5 × 4}\) and so on ……
Property 2:
If \(\frac{a}{b}\) is a rational number and m is a common divisor of a and b, then
\(\frac{a}{b}\) = \(\frac{a ÷ m}{a ÷ m}\)
In other words, if we divide the numerator and denominator of a rational number by a common divisor of both, the rational number remains unchanged.
For examples:
\(\frac{-32}{40}\) = \(\frac{-32 ÷ 8}{40 ÷ 8}\) = \(\frac{-4}{5}\)
Property 3:
Let \(\frac{a}{b}\) and \(\frac{c}{d}\) be two rational numbers.
Then \(\frac{a}{b}\) = \(\frac{c}{d}\) ⇔ \(\frac{a × d}{b × c}\).
a × d = b × c
For examples:
If \(\frac{2}{3}\) and \(\frac{4}{6}\) are the two rational numbers then, \(\frac{2}{3}\) = \(\frac{4}{6}\) ⇔ (2 × 6) = (3 × 4).
Note:
Except zero every rational number is either positive or negative.
Every pair of rational numbers can be compared.
Property 4:
For each rational number m, exactly one of the following is true:
(i) m > 0 (ii) m = 0 (iii) m < 0
For examples:
The rational number \(\frac{2}{3}\) is greater than 0.
The rational number \(\frac{0}{3}\) is equal to 0.
The rational number \(\frac{-2}{3}\) is less than 0.
Property 5:
For any two rational numbers a and b, exactly one of the following is true:
(i) a > b (ii) a = b (iii) a < b
For examples:
If \(\frac{1}{3}\) and \(\frac{1}{5}\) are the two rational numbers then, \(\frac{1}{3}\) is greater than \(\frac{1}{5}\).
If \(\frac{2}{3}\) and \(\frac{6}{9}\) are the two rational numbers then, \(\frac{2}{3}\) is equal to \(\frac{6}{9}\).
If \(\frac{-2}{7}\) and \(\frac{3}{8}\) are the two rational numbers then, \(\frac{-2}{7}\) is less than \(\frac{3}{8}\).
Property 6:
If a, b and c be rational numbers such that a > b and b > c, then a > c.
For examples:
If \(\frac{3}{5}\), \(\frac{17}{30}\) and \(\frac{-8}{15}\) are the three rational numbers where \(\frac{3}{5}\) is greater than \(\frac{17}{30}\) and \(\frac{17}{30}\) is greater than \(\frac{-8}{15}\), then \(\frac{3}{5}\) is also greater than \(\frac{-8}{15}\).
So, the above explanations with examples help us to understand the useful properties of rational numbers.
● Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
Standard form of a Rational Number
Equality of Rational Numbers using Standard Form
Equality of Rational Numbers with Common Denominator
Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
Rational Numbers in Descending Order
Representation of Rational Numbers on the Number Line
Rational Numbers on the Number Line
Addition of Rational Number with Same Denominator
Addition of Rational Number with Different Denominator
Properties of Addition of Rational Numbers
Subtraction of Rational Number with Same Denominator
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Rational Expressions Involving Addition and Subtraction
Simplify Rational Expressions Involving the Sum or Difference
Multiplication of Rational Numbers
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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