“There can be very little of present-day science and technology that is not dependent on complex numbers in one way or another.” ~Keith Devlin

“I tell you, with complex numbers you can do anything.” ~ John Derbyshire

 The LUMS Math Circle, held on 19 May, 2023, led by Dr. Momin Uppal and Dr. Imran Cheema, was an engaging and enlightening session that explored various aspects of  mathematics in a dynamic and interactive environment. The circle began with Dr. Momin Uppal delving into the intriguing history of numbers and highlighting their natural occurrence of natural numbers in our daily lives. He captivated the students' attention as he explained the significance of zero and seamlessly integrated it into the number system.  

However, Dr. Momin didn't stop there. He posed a thought-provoking question about potential real-life scenarios involving money loss, which triggered a flurry of student discussions. Sensing the curiosity in the room, Dr. Momin and Dr. Imran Cheema expanded the number system to include negative numbers, thus introducing integers based on intuition and practical applications. This addition broadened the mathematical landscape, allowing students to explore and understand the concept of numbers more comprehensively. 

Dr. Imran Cheema, known for his knack for puzzles, took the opportunity to further engage the students. He presented them with intriguing challenges, such as connecting nine dots using only four straight lines and dividing a cake into three equal pieces with only three cuts. Initially, most students believed these tasks to be impossible, but soon "Eureka" moments ensued as a few individuals showcased their "Out of the box thinking" abilities by coming up with innovative solutions. This episode emphasized the power of creativity and unconventional approaches to problem-solving. 

Building upon this momentum, Dr. Momin and Dr. Cheema introduced the concept of imaginary numbers. They guided the students through the addition of an imaginary axis to the existing number system and explained how "iota" could be interpreted as a counterclockwise rotation of 90 degrees in the plane. The students were fascinated by this new realm of numbers and the possibilities it offered. 

To provide a visual representation of complex numbers, the instructors employed GeoGebra software. Through this tool, they showcased the shadows of these numbers and astutely connected them to the concept of waves. The room buzzed with anticipation when Dr. Cheema posed the question, "Can these complex numbers be observed in the real world?" The students were on the edge of their seats, eager to uncover the answer. In a compelling demonstration, Dr. Momin and Dr. Cheema employed an oscilloscope to reveal that the observed waveforms were, in fact, the shadows of these complex numbers. 

The LUMS Math Circle was a blend of fun-filled moments and valuable insights into mathematical and engineering concepts. The students expressed their gratitude to Dr. Momin Uppal and Dr. Imran Cheema for their exceptional efforts in creating an enjoyable and enlightening learning environment. Through their expertise and engaging teaching methods, they fostered a deep appreciation for mathematics and encouraged the students to think critically, creatively, and "outside the box."

Applause for the dedicated efforts of Ms. Noreen Sohail, Mr. Qamar Hussain, and all volunteers for their organizational support.

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