IIT Madras
Mathematics 1
Rational Numbers🔥

Rational Numbers

Here is a detailed, emoji-enhanced guide to rational numbers, ordering, GCD, and density, with diagrams and markdown image code for added visuals.

Learning Outcomes:

  1. Define rational numbers.
  2. Order and compare rational numbers.
  3. Find the greatest common divisor of two integers.
  4. Understand why rational numbers are dense on the number line.

🧮 Define Rational Numbers

A rational number is any number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.123

  • Examples: $\frac{1}{2}, -\frac{3}{4}, 2, 0.3$ (which is $\frac{3}{10}$), and $-5$ (which is $\frac{-5}{1}$).24
  • All integers and whole numbers are rational since they can be written as fractions too.45
  • Denoted by the set $\mathbb{Q}$.1

A diagram showing the classification of numbers, including Natural, Whole, Integers, Rational, Irrational, and Real numbers 1

📊 Order and Compare Rational Numbers

To compare rational numbers, write them as decimals or with a common denominator.

  • Example: Compare $\frac{2}{3}$ and $\frac{3}{5}$:
    • $\frac{2}{3} = 0.666…$
    • $\frac{3}{5} = 0.6$
    • $0.666… > 0.6$ so $\frac{2}{3} > \frac{3}{5}$.67
  • When ordering a list, convert all to decimals or use common denominators.

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🏆 Find the Greatest Common Divisor (GCD)

The GCD (greatest common divisor) of two numbers is the largest number that divides both exactly.

  • Example: GCD of 12 and 26:
    • Factors of 12: $1, 2, 3, 4, 6, 12$
    • Factors of 26: $1, 2, 13, 26$
    • Common factors: $1, 2$
    • Greatest: 28910

A diagram explaining how to find the GCD (Greatest Common Divisor) of two numbers using prime factorization

  • Visual method: List factors or use the Euclidean algorithm (keep dividing and taking remainders).

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🌈 Rational Numbers Are Dense on the Number Line

Density means between any two rational numbers, there’s always another rational number!

  • Example: Between $\frac{1}{2}$ and $\frac{3}{4}$, the average $\frac{1}{2} + \frac{3}{4}/2 = \frac{5}{8}$ is in between.
  • No matter how close two rational numbers are, you can ALWAYS find another rational number between them.111213
  • On the number line, rational numbers fill it densely and continuously.

A diagram illustrating the density property of rational numbers

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📝 Summary Table

ConceptDefinition/MethodDiagram/Emoji
Rational NumbersFractions $\frac{p}{q}$, integers, recurring decimals🧮, 1
Compare & OrderConvert to decimals, common denominators📊, 6
GCDLargest shared factor, Euclidean algorithm🏆, 8
Rational DensityAlways a rational between any two rationals🌈, 11

Let these visual tools and emojis help reinforce your understanding of rational numbers and their properties! 1415

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Exercise Questions 🤯

alt text

1. Which of the following option(s) is (are) true?

A) $ \frac{4}{7} > \frac{5}{8} $

B) $ \frac{17}{22} > \frac{13}{18} $

C) $ \frac{11}{7} > \frac{13}{8} $

D) $ \frac{7}{15} < \frac{5}{20} $

Detailed Solution:

A) $ \frac{4}{7} > \frac{5}{8} $

Find a common denominator:

$4/7 = 32/56$, $5/8 = 35/56$

$32/56 < 35/56$

So, $4/7 < 5/8$. Not true.

  • B) $ \frac{17}{22} > \frac{13}{18} $

Find a common denominator:

$17/22 = (17 \times 18) / (22 \times 18) = 306/396$

$13/18 = (13 \times 22) / (18 \times 22) = 286/396$

$306/396 > 286/396$

So, $17/22 > 13/18$. True.

  • C) $ \frac{11}{7} > \frac{13}{8} $

Convert to improper fractions with common denominators:

$11/7 = (11 \times 8)/(7 \times 8) = 88/56$

$13/8 = (13 \times 7)/(8 \times 7) = 91/56$

$88/56 < 91/56$

So, $11/7 < 13/8$. Not true.

  • D) $ \frac{7}{15} < \frac{5}{20} $

$7/15 \approx 0.4667$, $5/20 = 0.25$

$0.4667 > 0.25$

So, $7/15 > 5/20$. Not true.

Correct Answer: Only option B is true.

2. Which of the following option(s) is (are) in reduced form?

A) $ \frac{5}{60} $ B) $ \frac{12}{27} $ C) $ \frac{11}{18} $ D) $ \frac{13}{91} $

Detailed Solution:

  • A) $ \frac{5}{60} $: Both numerator and denominator divisible by 5. $5/60 = 1/12$. Not reduced.
  • B) $ \frac{12}{27} $: Both divisible by 3. $12/27 = 4/9$. Not reduced.
  • C) $ \frac{11}{18} $: 11 and 18 have no common factors other than 1. This is reduced form.
  • D) $ \frac{13}{91} $: 91 = 7 × 13. Both divisible by 13. $13/91 = 1/7$. Not reduced.

Correct Answer: Only C is in reduced form.

3. Let gcd(a,4) = 2 and gcd(a, b) = 1. If 4 > a > b and a, b are natural numbers, then the values of a and b are respectively

A) 2, 1 B) 1, 2 C) 2, 4 D) 4, 2

Detailed Solution:

  • $gcd(a,4) = 2$: $a$ must be even but not divisible by 4. Since $a < 4$, possible $a$ is 2.
  • $gcd(a,b) = 1$: $a$ and $b$ are coprime.
  • Since $a = 2$ and $a > b$, try $b=1$: $gcd(2,1) = 1$ ✓

Hence, values are $a=2$, $b=1$.

Correct Answer: A) 2, 1

Summary Table

QnCorrect OptionExplanation
1B$17/22 > 13/18$ is true; others are false.
2C$11/18$ is the only fraction in reduced form.
3AOnly $a = 2, b = 1$ fit all GCD and ordering conditions.

1

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  1. https://byjus.com/maths/rational-numbers/ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎

  2. https://www.cuemath.com/numbers/rational-numbers/ ↩︎ ↩︎

  3. https://www.begalileo.com/math/numbers/rational-numbers ↩︎

  4. https://en.wikipedia.org/wiki/Rational_number ↩︎ ↩︎

  5. https://www.adda247.com/school/rational-numbers/ ↩︎

  6. https://www.slideshare.net/slideshow/7-lesson-42-compare-and-order-rational-numbers/122114001 ↩︎ ↩︎ ↩︎

  7. https://study.com/academy/lesson/comparing-ordering-rational-numbers.html ↩︎

  8. https://www.splashlearn.com/math-vocabulary/greatest-common-divisor-gcd ↩︎ ↩︎ ↩︎

  9. https://www.cuemath.com/numbers/greatest-common-divisor-gcd/ ↩︎

  10. https://byjus.com/maths/greatest-common-divisor/ ↩︎

  11. https://www.slideshare.net/slideshow/densityof-numberline/29878041 ↩︎ ↩︎ ↩︎

  12. https://www.geeksforgeeks.org/maths/what-is-the-density-property-of-rational-numbers/ ↩︎

  13. https://www.slideshare.net/slideshow/densityof-numberline-30862263/30862263 ↩︎

  14. https://studynlearn.com/what-is-rational-number ↩︎

  15. https://www.nagwa.com/en/explainers/570164325210/ ↩︎

Natural Numbers and their operations🔥Real and Complex Numbers