Science Stories Department Of Mathematics Page | SBASSE

A Circle That Keeps Getting Bigger!

Here’s a circle that keeps getting bigger and better; the LUMS Math Circle! What started off as a novel idea emanating from the halls of the Department of Mathematics, in the Syed Babar Ali School of Science and Engineering, as grown into a torrent of curious thoughts and engaging activities, all centered around the beautiful and visual world of Math. This growth is made possible by the combined effort of organizers, parents, and sponsors. What went down in the latest Math Circle? Let’s ‘loop’ in with the latest updates.

Dr. Waqas Ali Azhar brought an enticing topic of exposing school kids to the charisma of infinities beyond our limitations. Yes, you read that right, infinities; plural). This math circle was conducted on February 18 and saw participation from as far as Nankana Sahib, with 13 students coming from that district to attend the event. In total, around 35 students from various school and colleges attended the circle. It is probably the only forum where students from various school get an opportunity to interact with each other and engage in informal ways of learning.

So, how did it go? Dr. Waqas started the circle with a task of finding sum of first 100 number (integers) (i.e 1 + 2 + 3 + . . .+100). Students drilled hard to find a suitable answer where they were even allowed to use the calculators. Then he explained the genius of a famous mathematician Carl F. Gauss, who did it without using any calculator using a beautiful method.

He then gave a challenge to all participants, to fold a sheet of paper 10-time! An impossible task, since the thickness of the sheet increases exponentially. Students were amazed to know that 45 folds will result in a thickness that is equivalent to the distance between the Earth and the Moon! He kept on blowing minds by sharing such mind boggling facts, one after another. Factoids like Zeno's paradox, that yields a surprising infinite sum whose answer is a finite number. This is one of the most spell binding paradoxes, challenging the core concept of space and time. Dr. Waqas continued with introduction of exciting series of paradoxes.

A game was played between students to come-up with the largest possible number they can write on the page within 10 seconds. As part of this activity, participants learned the role of exponents. Dr. Waqas revealed astronomically large numbers like the googol, googol-plex and Graham number. Students were amazed to learn about the fact that Graham number is so large that the observable universe is far too small to contain an ordinary digital representation of it, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. Concluding the session, Dr. Waqas wisely gave the counter-concept of infinitesimal, through the Archimede's way of approximating the value of Pi through inscribed regular polygons.

“Dr. Waqas Ali Azhar did a remarkable job and deserve huge applaud for bringing excitingly important concepts in a highly interactive manner.”, said the chair of the department, Dr. Imran Anwar. “I applaud the efforts of Ms. Noreen Sohail as well, for her dedicated organizational support.”, Dr. Imran also commented. The Chair also expressed his gratitude to all sponsors, who made the event possible and have committed to support endeavors of this wonderful initiative, including the Pak Alliance for Maths and Science and the LUMS Students Mathematics Society.

To find out more about future Math Circle events including an upcoming mega event, please visit this link.

Learning Math from Relation

I remember when I was in 7th grade, just starting out algebra. It was chaos. The x’s and y’s made no sense. My classmates and I just couldn’t wrap our heads around why was this even something to study. And it was an astonishing realization that math will get tougher along the years.

Recently, Dr. Hassan Mohy Ud Din, Assistant Professor at the Electrical Engineering Department at SBASSE LUMS, hosted his Math Circle, “Reconstruction of the Unknown”. The seminar made me wonder if assigning physical context to theoretical mathematics would make it easier for students to find x. Math Circles are hosted by the Math Department at LUMS for middle school and high school students, belonging to different kinds of schools and school systems. Dr. Mohy Ud Din’s was the third in the series. He decided to teach them matrices but in a way that all could understand, regardless of their background. What was the result? Simultaneous equations made easy!

He explained this concept by ensuring that students were able to visualize a matrix and its operations using their experience. At first, the professor explained how really, while computing only two math operations are required. He testified that subtraction is just addition of a negative number, and that division is just multiplication of an inverse of the number. Presenting this idea, Dr. Mohy Ud Din was able to make his audience realize how

Next students were shown graphics of a person walking, forward and backward, what the professor called was sweeping the x-axis whilst performing horizontal motion. Then, similarly, he showed students how a person climbing, up and down, was sweeping the y-axis whilst doing vertical motion. He showed that these two motions each represented one-dimension motion by presenting them an image of a Cartesian graph. Students realized that using both the x-axis and y-axis together, a two-dimension can be described. To help them visualize, Dr. Hassan narrated them Anand’s Immortal, the famous chess game from 2013. He showed them how 2D representation in chess is like that in a typical matrix. Rows and columns are used together to locate the numbers in the matrix, just like how chess players memorize names of locations on the chess board.

The star of the program was Dr. Mohy Ud Din’s board game. The audience was divided into groups and each group was divided into two subgroups. Amongst the group, one of the subgroups was given a 4-by-4 matrix type board, and a stack of cards that had numbers 0-13. That subgroup randomly picked 4 cards and made a 2-by-2 matrix. The other subgroup guessed what those 4 numbers are. But they didn’t have to the guess casually! They were told the sums of each column and sums of each row. Then they must simultaneously solve the equations to get their answers. They did the same task with a 3-by-3 matrix, but it could not be solved! Students realized that to solve a system, the number of equations should be equal to the number of unknows. In a 3-by-3 system, the number of equations (3 for the rows and 3 for the columns) were lesser than the number of unknowns (nine). Amazingly, students suggested that they could use diagonal sums. Dr. Hassan realized that diagonals were a primary grade math concept that the students were able to connect to themselves. This made it easy for the professor to explain an upper triangular matrix, which led to students themselves wondering about a lower triangular matrix. Then, he explained sparse and dense matrices that he related to sparsely and densely populated areas.

Dr. Mohy Ud Din believes that such seminars can complement traditional education to keep the motivation and sharpen a student’s observation. They will be visualizing and using their memory by making relations to their experience to learn math!