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Number System - Rational Numbers: Definition, Representation on Number Line, Standard Form

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A rational number is defined as a number that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. This includes natural numbers, whole numbers, and integers as they can all be written with a denominator of 11.

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Positive and Negative Rational Numbers: A rational number is positive if both its numerator and denominator are either positive or negative (e.g., 23\frac{2}{3} or −5−7\frac{-5}{-7}). It is negative if the numerator and denominator have opposite signs (e.g., −49\frac{-4}{9} or 1−5\frac{1}{-5}).

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Standard Form: A rational number pq\frac{p}{q} is said to be in standard form if the denominator qq is a positive integer, and the numerator pp and denominator qq have no common factor other than 11. If a rational number has a negative denominator, multiply both numerator and denominator by −1-1 to standardize it.

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Equivalent Rational Numbers: By multiplying or dividing the numerator and the denominator of a rational number by the same non-zero integer, we obtain another rational number that is equivalent to the given one. For example, −23=−2×23×2=−46\frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6}.

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Representation on the Number Line: Rational numbers can be visualized as points on a horizontal line. Positive numbers are placed to the right of 00, and negative numbers to the left. To plot a number like 34\frac{3}{4}, the distance between 00 and 11 is divided into 44 equal intervals (based on the denominator), and the point is placed at the 3rd3^{rd} mark (the numerator) from 00.

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Density Property: Between any two rational numbers, there exist an infinite number of rational numbers. This distinguishes rational numbers from integers, where there are a fixed number of integers between any two given integers.

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Comparison of Rational Numbers: To compare two rational numbers, they should first be converted to a common positive denominator using the Least Common Multiple (LCM). Once the denominators are equal, the rational number with the greater numerator is the larger value.

📐Formulae

General Form: pq where p,q∈Z,q≠0\text{General Form: } \frac{p}{q} \text{ where } p, q \in \mathbb{Z}, q \neq 0

Equivalent Form: pq=p×mq×m for any integer m≠0\text{Equivalent Form: } \frac{p}{q} = \frac{p \times m}{q \times m} \text{ for any integer } m \neq 0

Simplification: pq=p÷extHCF(p,q)q÷extHCF(p,q)\text{Simplification: } \frac{p}{q} = \frac{p \div ext{HCF}(p, q)}{q \div ext{HCF}(p, q)}

Standard Form Condition: q>0 and extHCF(∣p∣,q)=1\text{Standard Form Condition: } q > 0 \text{ and } ext{HCF}(|p|, q) = 1

💡Examples

Problem 1:

Express the rational number 45−75\frac{45}{-75} in its standard form.

Solution:

Step 1: Make the denominator positive by multiplying both numerator and denominator by −1-1: 45×(−1)−75×(−1)=−4575\frac{45 \times (-1)}{-75 \times (-1)} = \frac{-45}{75} Step 2: Find the Highest Common Factor (HCF) of 4545 and 7575. The HCF of 4545 and 7575 is 1515. Step 3: Divide both the numerator and denominator by their HCF: −45÷1575÷15=−35\frac{-45 \div 15}{75 \div 15} = \frac{-3}{5} Therefore, the standard form is −35\frac{-3}{5}.

Explanation:

To reach standard form, we ensure the denominator is positive and the fraction is in its simplest terms by dividing by the HCF.

Problem 2:

Represent −23\frac{-2}{3} on a number line.

Solution:

Step 1: Draw a horizontal line and mark the origin as 00. Step 2: Since the number −23\frac{-2}{3} is negative, it will lie to the left of 00. Step 3: The denominator is 33, so divide the unit length between 00 and −1-1 into 33 equal parts. Step 4: The numerator is −2-2, so count 22 divisions to the left from 00. Step 5: Mark this point as AA. Point AA represents the rational number −23\frac{-2}{3}.

Explanation:

The denominator determines how many equal segments each unit space is divided into, and the numerator indicates how many of those segments to count from zero.