Adding sequential numbers (I)
Let us begin with a very small sequence of numbers. Suppose we want to find out total of numbers from 1 to 6.
One way to add these numbers would be to write down these numbers below one another and then use any method to add. If you want to know these methods go through article Speedy and accurate addition (Part III)
However, if you try to look at this exercise differently you can solve it quickly without making mistakes. How to look at this sequence is shown below:
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What we have done above is we have written the sequence of numbers from 1 to 6 twice horizontally. We have written it once in usual way. But the same sequence is written again beginning from 6 and completed by writing it in the backward direction.
This has made the addition easier. As you can see each pair of numbers one below the other, add to 7. We then have 6 such pairs. If we add all these pairs, we will get a total of seven times 6, that is 42. However, in simplifying the addition of numbers from 1 to 6 in this fashion we have actually added 1 to 6 numbers twice. So if we simply want the total of numbers from 1 to 6 we have to take half of 42 or we must divide 42 by 2. This simple calculation results into the answer 21.
Let us check our answer adding numbers from 1 to 6 mentally. 1 and 2 is 3, plus 3 is 6, plus 4 is 10, plus 5 is 15, plus 6 is 21. Thus our answer tallies with the answer we have obtained earlier.
You must have noticed that we have found out the addition by using three basic operations viz.: addition, multiplication and division.
This method of finding out total of sequential numbers was found out by a great mathematician Carl Friedrich Gauss, just when he was about 10 yrs old. It so happened that Gauss at the age of ten years was allowed to attend an arithmetic class taught by a man, Buttner. Mr. Buttner had a reputation for being cynical and having little respect for the peasant children he was teaching.
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