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Rational Numbers - Representation on the Number Line

Grade 8ICSE

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

🔑Concepts

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A rational number is any number that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. On a horizontal number line, the point 0 is the origin, with positive rational numbers located to the right and negative rational numbers located to the left.

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To represent a proper fraction ab\frac{a}{b} (where a<ba < b), the unit distance between 0 and 1 (or 0 and -1 if negative) is divided into bb equal parts. The atha^{th} division from the origin represents the fraction. Visually, if the denominator is 4, the segment between 0 and 1 is split by 3 marks into 4 equal lengths.

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Improper fractions, where the numerator is greater than the denominator, are first converted into mixed numbers of the form xabx \frac{a}{b}. The integer part xx indicates that the number lies between the integers xx and x+1x+1. For example, 73=213\frac{7}{3} = 2 \frac{1}{3} lies between 2 and 3 on the number line.

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The denominator of a rational number in its simplest form determines the number of equal sub-divisions required between any two consecutive integers. If representing 58\frac{5}{8}, the space between 0 and 1 is divided into 8 equal sub-intervals using 7 equidistant points.

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Negative rational numbers are represented to the left of zero. To plot −23-\frac{2}{3}, move from 0 towards -1, dividing the unit length into 3 equal parts and selecting the 2nd2^{nd} mark to the left. This point is equidistant from 0 as +23+\frac{2}{3} but in the opposite direction.

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The 'Density Property' states that between any two rational numbers, there are infinitely many other rational numbers. This means the number line is densely populated with points representing rational numbers, though it still contains gaps that are filled by irrational numbers.

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Equivalent rational numbers such as 12\frac{1}{2}, 24\frac{2}{4}, and 510\frac{5}{10} all represent the exact same unique point on the number line. Visually, these fractions simplify to the same position, which is the midpoint between 0 and 1.

📐Formulae

Rational Number Form: r=pq,q≠0r = \frac{p}{q}, q \neq 0

Conversion of Improper Fraction: NumeratorDenominator=QuotientRemainderDenominator\frac{\text{Numerator}}{\text{Denominator}} = \text{Quotient} \frac{\text{Remainder}}{\text{Denominator}}

Midpoint of two numbers aa and bb: m=a+b2m = \frac{a + b}{2}

Finding nn rational numbers between xx and yy: d=y−xn+1d = \frac{y - x}{n + 1}

💡Examples

Problem 1:

Represent the rational number 35\frac{3}{5} on the number line.

Solution:

Step 1: Since 0<35<10 < \frac{3}{5} < 1, the point lies between 0 and 1. Step 2: The denominator is 5, so divide the distance between 0 and 1 into 5 equal parts. Step 3: Starting from 0 and moving to the right, count 3 divisions. Step 4: Label this point as P=35P = \frac{3}{5}.

Explanation:

This is a positive proper fraction. We divide the first unit length into 5 segments because the denominator is 5, and pick the 3rd mark because the numerator is 3.

Problem 2:

Represent −114-\frac{11}{4} on the number line.

Solution:

Step 1: Convert the improper fraction to a mixed number: −114=−234-\frac{11}{4} = -2 \frac{3}{4}. Step 2: This number lies between −2-2 and −3-3 on the left side of zero. Step 3: Divide the interval between −2-2 and −3-3 into 4 equal parts (based on the denominator). Step 4: Starting from −2-2 and moving left towards −3-3, count 3 divisions. Step 5: The resulting point is −234-2 \frac{3}{4} or −114-\frac{11}{4}.

Explanation:

By converting to a mixed number, we identify the specific integer interval where the number resides. The fractional part 34\frac{3}{4} then determines the precise position within that interval.