Johann Carl Friedrich Gauss

What is the sum of the first 100 counting numbers?

If you answered 5,050 − you are correct! But, how long did it take you to calculate for this sum? Did you use a calculator?

More than two centuries and two decades ago, a ten-year old pupil in Professor Herr Buttner's class instantly gave the correct answer to the question above. This student goes by the name of Johann Carl Friedrich Gauss (1777-1855) who was eventually known as the greatest German mathematician of the 19th century.

Carl Gauss began attending elementary school at the age of 7, his prowess was exhibited and noticed immediately. During these days, Carl had experienced attending classes with a teacher who was so heartless. This teacher is used to giving difficult problems which will keep the boys in his class busy for hours, without even discussing or explaining any.

It was during these days in basic school when Professor Buttner and his assistant, Martin Bartels, were astonished when Little Carl summed the integers from 1 to 100 instantly. How did he do it?

That evening on his way home, Little Carl stopped by his Uncle Friedrich's tapestry shop. He couldn't resist but tell his uncle what had happened at school. Before Little Carl left that night, his uncle asked him, "How did you solve that problem?"

"It was easy," Carl explained. "I realized that 1 + 100 = 101, 2 + 99 = 101, and 3 + 98 = 101. I could tell that if you add up all the integers from 1 to 100, you can find 50 pairs of numbers that each sums to 101. So my answer would be 50 times 101!"

Let us learn how Carl might have represented his mathematical representations in his mind that moment:
  
  
1 2 3... ... ...... 48 49 50
+ + + + + + + + + +
1009998... .........321
101 101 101... ... ... ... 101 101 101


Thus, 50 x 101 = 5,050.

Little Carl's teacher actually gave his class the entire period to perform the task and give a correct answer. But Carl was able to finish and submit his work right before Professor Buttner could turn around and sit down on his stool. Amazing? Indeed!

But, do you know that there is another technique which will also lead us to getting the same correct answer in much shorter time? It is proven by mathematical induction. We will discuss Mathematical Induction in detail on our succeeding topics. For the meantime, let us study closely how this type of reasoning helps us get the sum when all the integers from 1 to 100 are added up.

For that particular given, we let our number be 100. We represent that number as N.Thus N = 100. Next, we consider the next larger integer, which is 101. This, logically, will be represented as N + 1. We then get the product of these two numbers and divide the product by 2. We have these statements in symbols as follows:


StepIn WordsIn SymbolsActual Computations
Step 1Represent N and (N + 1)N = 100 and N + 1 = 101
Step 2Multiply N by (N + 1)N(N + 1)100(101) = 10,100
Step 3Divide the product you
got from Step 2 by 2. This will be our final answer.		[N(N + 1)] / 210,100 / 2 = 5,050


Take note that whatever integer we chose as the endpoint, we will always have an even factor for Step 2. For instance, if our N is an odd number, then the next number is even. Thus,N(N + 1) will always be even. It is also suggested that since exactly one of the two factors is even, i.e. either N or (N + 1), then we can directly divide that factor by 2 and multiply the result by the remaining number.

So, the next time somebody asks you to add up all the integers from 1 to 1000, you very well know that the sum is (1000 x 1001)/ 2 = 500 x 1001 = 500,500. And that the sum of the first 30 counting numbers is (30 x 31) / 2 = 465. It would really be impressive that you can instantly give the accurate answer without going through the tedious 1 plus 2 plus 3 plus 4, and so on.... Thanks to a great mathematician, Carl Gauss!