Real Numbers Projects
Class X Mathematics - Exploring Euclid's Algorithm and Decimal Expansions
Euclid's Algorithm in Action!
Name: K. Arun
Class: X
Roll Number: 10
Lesson: Real Numbers
Subject: Mathematics
Aim
To find the Highest Common Factor (HCF) of given pairs of numbers using Euclid's Division Algorithm and to verify it through a practical activity.
Objectives
- To understand and apply Euclid's Division Lemma
- To find HCF using Euclid's Algorithm
- To verify the algorithm with a practical activity using paper strips
Materials & Tools
Materials Used: Paper strips (two different colours), scale, pen, pencil, sketch pens, scissors
Tools: Mathematical calculation, Practical experiment
Procedure
- Two pairs of numbers were selected: (50, 75) and (32, 96)
- For each pair, Euclid's Division Algorithm was applied step-by-step until the remainder became zero
- To verify the result for (50, 75), two paper strips of lengths 50 cm and 75 cm were cut
- The longer strip (75 cm) was measured using the shorter strip (50 cm)
- The leftover part (25 cm) was used to measure the 50 cm strip
- The process continued until no remainder was left
Observations & Calculations
A) For numbers 50 and 70:
- 70 = 50 × 1 + 20
- 50 = 20 × 2 + 10
- 20 = 10 × 2 + 0
- HCF (50, 70) = 10
B) For numbers 32 and 96:
- 96 = 32 × 3 + 0
- HCF (32, 96) = 32
Application of Euclid's Algorithm
| Numbers (a, b) | Division Steps (a = bq + r) | HCF |
|---|---|---|
| 50, 70 | 70 = 50 × 1 + 20 50 = 20 × 2 + 10 20 = 10 × 2 + 0 |
10 |
| 32, 96 | 96 = 32 × 3 + 0 | 32 |
| 1860, 2015 | 2015 = 1860 × 1 + 155 1860 = 155 × 12 + 0 |
155 |
Conclusion
Euclid's Division Algorithm provides a systematic and efficient method for finding the HCF of two positive integers. The practical activity with paper strips visually confirms that the HCF is the largest common length that can measure both given lengths without any remainder.
Experience of the Student
This project helped me understand that mathematics is not just about calculations but also about visual and practical understanding. Cutting the strips and physically measuring them made the concept of HCF very clear and interesting.
Acknowledgement
1. My sincere thanks to our Mathematics teacher for guiding us through this project
2. I would also like to thank my classmates for their cooperation and support
Reference Books / Resources
- X Class State Board Mathematics Textbook
- IX Class Mathematics Textbook
Project Work-2 Decoding Decimals: Terminating or Non-Terminating?
Name: R.Reema
Class: X
Roll Number: 10
Lesson: Real Numbers
Subject: Mathematics
Aim
To investigate the relationship between the prime factors of the denominator of a rational number and the nature of its decimal expansion (terminating or non-terminating repeating).
Objectives
- Expressing rational numbers in their decimal form
- Expressing the denominator of the rational number in its prime factorized form
- Differentiating between terminating and non-terminating recurring decimals
Materials & Tools
Materials Used: Pen, Pencil, Scale, Chart Paper, Coloured Pens
Tools: Collection of rational numbers, Mathematical calculation, Prime factorization
Procedure
- Collected various rational numbers
- Each rational number was converted into its decimal form by performing division
- Noted whether the decimal was Terminating (T) or Non-Terminating Repeating (NR)
- Each rational number was written in its simplest form p/q
- The denominator (q) was factorized into its prime factors
- The observations were recorded in a table for analysis
Observations
Relation between Denominator and Decimal Expansion
| S.No. | Rational Number | Decimal Form | Type | Simplified Form | Prime Factors of q | Observation |
|---|---|---|---|---|---|---|
| 1 | 3/8 | 0.375 | T | 3/8 | 2³ | Only prime factor 2 |
| 2 | 7/25 | 0.28 | T | 7/25 | 5² | Only prime factor 5 |
| 3 | 13/125 | 0.104 | T | 13/125 | 5³ | Only prime factor 5 |
| 4 | 9/20 | 0.45 | T | 9/20 | 2² × 5¹ | Prime factors 2 and 5 |
| 5 | 1/3 | 0.333... | NR | 1/3 | 3¹ | Prime factor 3 |
| 6 | 5/12 | 0.41666... | NR | 5/12 | 2² × 3¹ | Has prime factor 3 |
| 7 | 7/13 | 0.538461... | NR | 7/13 | 13¹ | Prime factor 13 |
| 8 | 29/343 | 0.08454... | NR | 29/343 | 7³ | Prime factor 7 |
Conclusion
From the observations, it is concluded that the decimal expansion of a rational number p/q (in simplest form) terminates if and only if the prime factorization of the denominator q is of the form 2ⁿ × 5ᵐ, where n and m are non-negative integers. If the denominator has any prime factor other than 2 or 5, the decimal expansion is non-terminating and repeating.
Result
A rational number p/q, where p and q are co-prime, has a terminating decimal expansion if the prime factors of q are only 2 and/or 5.
Experience of the Student
While working on this project, I learned to predict the nature of a decimal just by looking at the denominator of a fraction. Initially, I thought all decimals either stop or show a simple pattern, but I discovered that some repeats after a long sequence, which was fascinating.
Acknowledgement
1. I thank our Mathematics teacher for her invaluable guidance
2. I extend my gratitude to my parents for their support
Reference Books / Resources
- X Class State Board Mathematics Textbook (Chapter 1: Real Numbers)
- IX Class Mathematics Textbook
