Positive Rational Numbers

Any number greater than zero that can be represented as a fraction of two integers is a positive rational number. Also remember that the two integers are always positive and the denominator is not equal to zero. For example, ½, 3, 0.75, etc., are positive rational numbers.

No, 0 is not a positive rational number because it is located at zero on the number line and is neither positive nor negative. Positive rational numbers are defined as numbers greater than zero.
 

Is Every Natural Number a Positive Rational Number?

Yes, every natural number is a positive rational number. Natural numbers such as 1, 2, 3, etc., can be expressed as rational numbers by writing them as fractions with 1 as the denominator (e.g., 2 = \(2 \over 1\)).
 

Positive Rational Numbers Less Than 1

Positive rational numbers less than 1 are fractions where the numerator is smaller than the denominator. Examples include:

• \(1 \over 2\)
• \(3 \over 4\)
• \(7 \over 10\), etc.

These numbers are greater than 0 but less than 1. They are used in measuring; for e.g., cooking often requires measuring of ingredients in fractions. 
 

Positive Rational Numbers: Symbol

There is no unique symbol for positive rational numbers. Since they are a subset of rational numbers (Q), they are represented as Q⁺, meaning all rational numbers greater than zero.

Reciprocal of a Positive Rational Number

The reciprocal of a positive rational number is another positive rational number, obtained by flipping the numerator and denominator. Let us consider \(1 \over 2\) as an example. The reciprocal of \(1 \over 2\) is \(2 \over 1\). Multiplying a number by its reciprocal always yields 1. Every positive rational number has a reciprocal.

There are many differences between positive rational numbers and negative rational numbers, the differences are shown below in the table:

There are various properties of positive rational numbers. Some key properties are listed below:
 

• Numbers to the right of 0 on a number line are positive rational numbers. They are always greater than 0.
   
• These numbers can be expressed as a fraction where the numerator and the denominator will always be positive.
   
• When any two positive rational numbers are added or multiplied, the result will always be a positive rational number.
   
• Positive rational numbers can be represented as recurring or terminating decimals.
   

To be comfortable in working with positive rational numbers, it's helpful to understand what you mean by the terms fractions, decimal, and the relationships between fractions, decimals, and whole numbers. The following tips and tricks will help students develop precision and confidence working with positive rational numbers:
 

• Be Clear with the Definition: Remember that positive rational numbers are numbers that can be written as a ratio comparing two positive integers, where the denominator is never zero.
   
• Connect Fractions and Decimals Together: Using fractions and decimal forms interchangeably will establish flexibility in representations.
   
• Simplify: Always simplify fractions to the lowest terms, which will help to comparison and calculation.
   
• Compare Using Common Denominators: When arranging or comparing values, use equivalent fractions with common denominators in order to keep it accurate.
   
• Use Operations Correctly: Always use proper rules for adding, subtracting, multiplying, and dividing rational numbers will assist in precision in calculations.

Positive rational numbers have numerous applications across various fields. Let us check them out one by one:
 

• Money and Financial Transactions: We use positive rational numbers when we do monetary transactions. Amounts like $5.75 and $28.46 are positive rational numbers. These numbers help in budgeting, calculating interest rates, discounts, taxes, and dividing costs among people. 
   
• Cooking: If we follow a recipe while cooking, we will often come across measurements like “1/2 tablespoon of sugar and \(3 \over 4\) cup of water.” These measurements represent positive rational numbers.
   
• Construction and Architecture: Builders and architects use numbers like 4.5 and 1.25 to measure lengths, widths, and heights. Positive rational numbers play an important role in building edifices.

• Science and Measurements: Positive rational numbers are an important part of data collection by scientists, such as temperature, chemical mass and volume, and distances in experiments, establishing accuracy in their findings.  

• Sports and Athletics: In the sporting world, time, such as in a race (e.g., 9.58 seconds), or averages (e.g., scoring average of 23.75 points per game) are examples of positive rational numbers used to quantify performance and compare record- setting statistics.