- Techniques for Adding the Numbers 1 to 100
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- The Story of Gauss
- Young Gauss and the sum of the first n positive integers
Techniques for Adding the Numbers 1 to 100
In elementary school in the late 's, Gauss was asked to find the sum of the numbers from 1 to The question was assigned as “busy work” by the teacher .with with metabolism is best described as
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Question: what is the sum of the first whole numbers?? The question you asked relates back to a famous mathematician, Gauss. His observation was as follows:. Gauss noticed that if he was to split the numbers into two groups 1 to 50 and 51 to , he could add them together vertically to get a sum of The sequence of numbers 1, 2, 3, … , is arithmetic and when we are looking for the sum of a sequence, we call it a series. Thanks to Gauss, there is a special formula we can use to find the sum of a series:.
Again, the number of x's in the pyramid = 1 + 2 + 3 + 4 + 5, or the sum from 1 to n. sum. If we have numbers (1 ), then we clearly have items. . Adding the first number to the last number is how we “pair” each number together .
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Find How to Quickly Add the Integers From 1 to n out how to impress your friends at parties by quickly calculating the sum of all the integers from 1 up to any number they choose. In the last episode, we learned an amazing trick that you can use to quickly add up all the integers from 1 to Which might lead you to wonder: Instead of just adding up the first positive integers, is there a way to quickly calculate the sum of the first 50, , or maybe even 1, positive integers? That would be a rather impressive trick, right? The key to this is our friend the associative property of addition which says that you are free to add together a group of numbers in any order you like. Why does that help? Because each of these pairs of numbers adds up to
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After the teacher asked this question, the class, which was full of young kids, fell into complete silence. A few students were stunned by the seemingly-impossible challenge and readily gave up; most students began scribbling on the paper, trying to add all the numbers one by one, from the very beginning. What a difficult question!
The Story of Gauss
Gauss displayed his genius at an early age. According to anecdotes, when he was in primary school, he was punished by his teacher due to misbehavior. He was told to add the numbers from 1 to He was able to compute its sum, which is , in a matter of seconds. Since it is very hard to add numbers at once, we start with smaller numbers and see if we can see a pattern. First, we observe that when the largest number is even, we can pair the numbers with the same color; that is, the last term and the first, the second and the second to the last terms, the third and the third to the last terms and so on, without a term left unpaired. Second, we observe that if we add the terms mentioned above, the sum of each pair is always equal to the same number.
The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to Gauss approached with his answer: So soon? The teacher suspected a cheat, but no. Manual addition was for suckers, and Gauss found a formula to sidestep the problem:. Pairing numbers is a common approach to this problem. An interesting pattern emerges: the sum of each column is
By Jane M. Wilburne , Posted October 10, —. I love the story of Carl Friedrich Gauss—who, as an elementary student in the late s, amazed his teacher with how quickly he found the sum of the integers from 1 to to be 5, Gauss recognized he had fifty pairs of numbers when he added the first and last number in the series, the second and second-last number in the series, and so on. Another way to represent the problem could be to list the integers from 1 to and write the same list in reverse order below the first list.
Young Gauss and the sum of the first n positive integers