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Number System - Prove density of rational numbers and generate rationals between any two numbers

Grade 9CBSE

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

🔑Concepts

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The Density Property of rational numbers states that between any two distinct rational numbers, there are infinitely many rational numbers.

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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.

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There are two primary methods to find rational numbers between two given numbers aa and bb (a<ba < b):

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Method 1 (The Mean Method): Find the average of the two numbers, i.e., a+b2\frac{a + b}{2}. This result will always lie between aa and bb. By repeating this process, you can find as many numbers as required.

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Method 2 (Equivalent Fractions Method): To find nn rational numbers between aa and bb, convert them into fractions with the same denominator. If the gap between numerators is small, multiply both numerator and denominator of both numbers by (n+1)(n+1) to create a sufficient range.

📐Formulae

Rational Number=pq, where p,q∈Z and q≠0\text{Rational Number} = \frac{p}{q}, \text{ where } p, q \in \mathbb{Z} \text{ and } q \neq 0

Midpoint between a and b=a+b2\text{Midpoint between } a \text{ and } b = \frac{a + b}{2}

Condition for density: If r<s, then r<r+s2<s\text{Condition for density: If } r < s, \text{ then } r < \frac{r + s}{2} < s

💡Examples

Problem 1:

Find five rational numbers between 11 and 22.

Solution:

To find n=5n = 5 rational numbers, we write 11 and 22 as rational numbers with a denominator of (n+1)=6(n + 1) = 6.

1=1×66=661 = \frac{1 \times 6}{6} = \frac{6}{6} 2=2×66=1262 = \frac{2 \times 6}{6} = \frac{12}{6}

Now, we can pick any five integers between the numerators 66 and 1212, which are 7,8,9,10,117, 8, 9, 10, 11.

Therefore, five rational numbers between 11 and 22 are: 76,86,96,106, and 116\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \text{ and } \frac{11}{6}

Explanation:

By making the denominator n+1n+1, we ensure there are exactly nn integers available between the new numerators.

Problem 2:

Find 33 rational numbers between 35\frac{3}{5} and 45\frac{4}{5}.

Solution:

Here, n=3n = 3. We multiply the numerator and denominator of both fractions by (n+1)=4(n+1) = 4.

35=3×45×4=1220\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20} 45=4×45×4=1620\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}

The numbers between 1220\frac{12}{20} and 1620\frac{16}{20} are: 1320,1420,1520\frac{13}{20}, \frac{14}{20}, \frac{15}{20}

Simplified forms: 1320,710,34\frac{13}{20}, \frac{7}{10}, \frac{3}{4}.

Explanation:

Since the original denominators were the same but the numerators were consecutive, multiplying by 44 created a gap of 44 in the numerators, allowing us to find 33 rational numbers.

Problem 3:

Find a rational number between 14\frac{1}{4} and 12\frac{1}{2} using the Mean Method.

Solution:

Let a=14a = \frac{1}{4} and b=12b = \frac{1}{2}.

The rational number between them is: a+b2=14+122\frac{a + b}{2} = \frac{\frac{1}{4} + \frac{1}{2}}{2}

First, find the common denominator for the numerator: 14+24=34\frac{1}{4} + \frac{2}{4} = \frac{3}{4}

Now divide by 22: 342=34×12=38\frac{\frac{3}{4}}{2} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}

Thus, 38\frac{3}{8} is a rational number between 14\frac{1}{4} and 12\frac{1}{2}.

Explanation:

The Mean Method is useful for finding a single rational number precisely in the middle of two given values.