Saturday, September 28, 2013

Carl Friedrich Gauss: The Child Prodigy

One of the most famous mathematicians of all time is the German mathematician Carl Friedrich Gauss (1777 - 1855). Gauss was one of the leading number theorists of all time, as well as a contributor to algebra, statistics, geometry, analysis, and applied mathematics.

There is a piece of mathematical folklore (which may or may not be 100% accurate) that involved a child Gauss. It is a great story, highlights a great point, and shows the intelligence of a great mathematician.

A fifth grade teacher is teaching a class, and started to get frustrated with the students. So, in an attempt to punish them, she demanded that they add up all of the numbers from 1 to 100. This is a daunting task for the average person. She expected to have the students start working on the problem, and she could leave and take a break.

As she was about to walk out the door, the young Gauss raised his hand and declared that the answer is 5050. The teacher was stunned. After checking his work, they found that 5050 was the correct answer.

How did he do it? Well, he visualized a horizontal line with all 100 numbers:

1   2   3   4   5   ...   96   97   98   99   100

And then he took the second half of that line (51 - 100) and flipped it around underneath to look like so:

    1     2     3     4     5   ...   46   47   48   49   50
100   99   98   97   96   ...   55   54   53   52   51

Each of these vertical columns is its own addition problem. And in all fifty columns, the sum is 101. So, the sum of the numbers from one to one-hundred is the same as fifty 101's, or 50 x 101. Since 50 ends in a zero, it is a pretty quick computation: 50 x 101 = 5050. And there is the answer.

I think this is a great story when it comes to historical mathematicians, regardless of how true it is. Gauss did go on to study triangular numbers, which are the sums of consecutive integers up to a point. And since triangular numbers are absolutely fascinating, this story is a great way to begin an endeavor in that topic.

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