Dalhousie Math Circles - Exploding Dots

On Wednesday, March 31^st, Dalhousie University hosted its latest virtual Math Circles evening.  Dr. Asmita Sodhi (Dalhousie Mathematics Department) led the seminar, providing interactive opportunities and the mathematical motivation to make the event a great success.  The topic itself, Exploding Dots, had our curiosity piqued, and the ensuing mathematical connection did not disappoint. The hour-long presentation was an uplifting mathematical experience.  Time itself seemed to “explode”.  First, participants were introduced to a ”2 to 1” exploding dot machine, where they introduced numbers into the furthermost right block of a series of attached boxes, and watched as pairs of dots exploded into one dot in the adjacent box to the left.  This pattern continued in adjacent boxes until participants discovered that they were actually expressing numbers in base 2, or binary form.  

The box at the far right represented 1 (2⁰), while the next box to the left represented 2 (2¹), then 4 (2²), and so on.  So, the number 13 as we know it in base 10 would fill the following boxes in the base 2 machine from left to right:  1x2³+ 1x2² + 0x2¹+ 1x2⁰ or 1101 (base 2). The number 20 would become: 10100 in base 2:  (1x2⁴+0x2³+ 1x2² + 0x2¹+ 0x2⁰=16+0+4+0+0=20); did you try it?  Students quickly transitioned to base 3 boxes (ternary form), then base 4 (quaternary form), base 5 (quinary form), etc.  We even revisited base 10 (decimal form).  How interesting to put 10 dots in the leftmost box and watch them explode into 1 dot in the next box to the left = 1x10¹+0x10⁰=10).  It all made sense of course; these are just numbers as we know them everyday in base 10.  Our culture has agreed that base 10 is best; but, wait a minute: what about time?  Time is an example of base 60 arithmetic; 3785 seconds can be written as 1x60²+3x60¹+5x60⁰ or 01:03:05 (1 hour: 3 minutes: 5 seconds).  The exploding dot box idea exists everywhere. Participants also learned how to add or subtract numbers in different bases. I commend our Junior School participants Lillian Blois, William Larder, Brin Lloyd, Anthony Wheeler and Owen Xu for their participation and insightful contributions.  For more information on this numeracy concept, visit Exploding Dots Experience or use the interactive website Polypad Dots 1.