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Positive Rational Numbers: Definition and Examples | EDU.COM

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Positive Rational Numbers: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Positive Rational NumbersPositive Rational Numbers: Definition and ExamplesTable of ContentsPositive Rational Numbers Definition of Positive Rational Numbers

Positive rational numbers are numbers that can be written in the form pq\frac{p}{q}qp​, where ppp and qqq are integers having the same sign (both positive or both negative) and qqq ≠ 0. These numbers are always greater than zero and appear to the right of zero on the number line. There are two cases: when both ppp and qqq are positive integers (like 67\frac{6}{7}76​ or 1825\frac{18}{25}2518​), and when both ppp and qqq are negative integers (like −6−7\frac{-6}{-7}−7−6​ which equals 67\frac{6}{7}76​).

Positive rational numbers have several important properties. They are closed under addition and multiplication, meaning the sum and product of two positive rational numbers always yields another positive rational number. Every natural number is a positive rational number since any natural number "nnn" can be written as n1\frac{n}{1}1n​. Unlike rational numbers in general, positive rational numbers are closed under division because dividing one positive rational number by another always gives a positive rational number (since zero is not included in positive rational numbers). However, they are not closed under subtraction since subtracting two positive rational numbers might not result in a positive rational number.

Examples of Positive Rational Numbers Example 1: Determining if All Whole Numbers are Positive Rational Numbers Problem:

Are all whole numbers positive rational numbers?

Step-by-step solution:

Step 1, List what whole numbers include. Whole numbers = {0, 1, 2, 3, 4, ...}

Step 2, Check if 0 is a positive rational number. We know that 0 is not a positive rational number.

Step 3, Make a conclusion. Not all whole numbers are positive rational numbers. Whole numbers except 0 (which are called natural numbers) are positive rational numbers because they can be written as fractions with denominator 1, and they are greater than zero.

Example 2: Classifying a Rational Number as Positive or Negative Problem:

Is −22−25\frac{-22}{-25}−25−22​ a positive or negative rational number?

Step-by-step solution:

Step 1, Check the signs of numerator and denominator. In −22−25\frac{-22}{-25}−25−22​, both numerator and denominator have negative signs.

Step 2, Apply the rule about signs. When both numerator and denominator have the same sign (both positive or both negative), the rational number is positive.

Step 3, Convert to an equivalent form to verify. We can multiply and divide both numerator and denominator by -1: −22×(−1)−25×(−1)=2225\frac{-22 \times (-1)}{-25 \times (-1)} = \frac{22}{25}−25×(−1)−22×(−1)​=2522​

Step 4, Make a conclusion. Thus, −22−25\frac{-22}{-25}−25−22​ is a positive rational number.

Example 3: Finding the Reciprocal of a Positive Rational Number Problem:

Find the reciprocal of 1825\frac{18}{25}2518​.

Step-by-step solution:

Step 1, Remember the definition of reciprocal. The reciprocal of ab\frac{a}{b}ba​ (where bbb ≠ 000) is ba\frac{b}{a}ab​.

Step 2, Apply the reciprocal formula. For 1825\frac{18}{25}2518​, we can write the reciprocal as: 11825=2518\frac{1}{\frac{18}{25}} = \frac{25}{18}2518​1​=1825​

Step 3, Verify the result. The product of a number and its reciprocal should equal 1: 1825×2518=18×2525×18=450450=1\frac{18}{25} \times \frac{25}{18} = \frac{18 \times 25}{25 \times 18} = \frac{450}{450} = 12518​×1825​=25×1818×25​=450450​=1

Comments(2)VViolinistNinaNovember 4, 2025This definition of positive rational numbers is super helpful! I've used it to explain to my students, and it made the concept much clearer.

NNatureLover85September 17, 2025I’ve been using this page to help my kids grasp positive rational numbers, and the examples really clicked for them! It’s clear and easy to follow—great resource for both parents and teachers.

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