The LUMS Math Circle session on May 14, 2026 took students on a journey through the history and evolution of number systems. Led by Dr. Naveed Bhatti, the session explored a simple but profound question: how do humans represent quantities? Beginning with everyday examples of numbers around us, students discovered that number systems are not merely collections of symbols but carefully designed languages that help us count, communicate, calculate, and store information.

Session Structure and Content

The session began by encouraging students to think about numbers in daily life and to distinguish between a quantity and the symbol used to represent it. This naturally led to a discussion about why we use the digits 0–9 and whether other systems might work just as well.

Students were then introduced to tally marks, one of humanity’s earliest counting methods. Through examples and activities, they experienced both the usefulness and limitations of tally-based counting, particularly when representing larger quantities.

The first major worksheet activity challenged students to become inventors of their own number systems. Working in groups, they created symbols, designed grouping rules, and explored what makes a number system efficient and easy to use.

The discussion then moved to Roman numerals, where students learned additive and subtractive notation and compared Roman numerals with their own invented systems. This comparison highlighted both the creativity and limitations of historical number systems.

Having explored earlier approaches, students were introduced to the decimal positional system. Through place-value examples, they discovered how the position of a digit determines its value and why positional notation is one of the most powerful ideas in mathematics.

The final part of the session focused on binary numbers. Students practiced converting between decimal and binary representations and explored why computers rely on binary systems. By connecting binary digits to ON/OFF electronic states, they saw how mathematical ideas underpin modern computing technology.

The session concluded with an introduction to bits, bytes, kilobytes, megabytes, gigabytes, and terabytes, relating these concepts to familiar digital objects such as text messages, photographs, songs, movies, and hard drives.

Learning Outcomes

By the end of the session, students understood the distinction between quantities and symbols, explored historical and modern number systems, designed their own number systems, learned place value, converted numbers between decimal and binary representations, and gained insight into how digital information is stored and processed.

Conclusion

The session showed students that number systems are much more than symbols written on a page. From tally marks and Roman numerals to binary code and digital storage, participants traced humanity’s search for better ways to represent and communicate information. By the end of the session, students gained a deeper appreciation for the mathematical ideas that power both ancient civilizations and modern technology.

Acknowledgments

The LUMS Math Circle team gratefully acknowledges Dr. Naveed Bhatti for conducting this engaging and insightful session. Special thanks are extended to the Mathematics Department staff for their organizational support and to all participating students for their enthusiasm, curiosity, and active involvement throughout the session.

Certificates

At the conclusion of the session, certificates of participation were distributed among the students in recognition of their active engagement and successful completion of the activities.

Here are some highlights from the event: