Print PDF
Favorite
Email
Step 7Runs
This is the most obscure of the techniques I am going to present. It's not as simple a method as the others and has limited application, but also has some incredibly powerful uses. The story goes that when he was a child, the mathematician Carl Friedrich Gauss was asked to mentally add all the integers between 1 and 100 to keep him busy for a period. It didn't work, as he summed all the number in seconds by realising that they form a simple pattern.
Gauss saw that he could break the numbers down into matched pairs, remove an amount from the high number and add it to the low number to make all the pairs the same. Taking the numbers 1 to 5 as a simpler example, you would take 2 from the 5 and add it to the 1, take 1 from the 4 and add it to the 2. This turns 1+2+3+4+5 into the rather more manageable 3+3+3+3+3 or 15.
To do this for the numbers 1 to 100 is a little trickier because there are an even number of them so the pairs end up summing to 50.5 (the mean average of all the numbers). The general rule is that to add a run of increasing integers, add the first and last together, halve the value and multiply by the number of numbers you are adding together. To go back to our smaller example, 1+5=6, 6/2=3 and 3*5=15. The diagram below shows in a more visual way why this works. The intricacies of this technique are outside the scope of this Instructable, but for now you can remember the formula (first+last)/2 * number of numbers
Again, this method isn't particularly general because it requires an addition of the form 1+4+7 (properly, an arithmetic series), but when it is applicable it is a very powerful method (as Gauss demonstrated). Runs of evenly-spaced additions are unlikely to come up much in "real life" (unless you go to restaurants with mathematicians) but pure maths features them more often.
Gauss saw that he could break the numbers down into matched pairs, remove an amount from the high number and add it to the low number to make all the pairs the same. Taking the numbers 1 to 5 as a simpler example, you would take 2 from the 5 and add it to the 1, take 1 from the 4 and add it to the 2. This turns 1+2+3+4+5 into the rather more manageable 3+3+3+3+3 or 15.
To do this for the numbers 1 to 100 is a little trickier because there are an even number of them so the pairs end up summing to 50.5 (the mean average of all the numbers). The general rule is that to add a run of increasing integers, add the first and last together, halve the value and multiply by the number of numbers you are adding together. To go back to our smaller example, 1+5=6, 6/2=3 and 3*5=15. The diagram below shows in a more visual way why this works. The intricacies of this technique are outside the scope of this Instructable, but for now you can remember the formula (first+last)/2 * number of numbers
Again, this method isn't particularly general because it requires an addition of the form 1+4+7 (properly, an arithmetic series), but when it is applicable it is a very powerful method (as Gauss demonstrated). Runs of evenly-spaced additions are unlikely to come up much in "real life" (unless you go to restaurants with mathematicians) but pure maths features them more often.
| « Previous Step | Download PDFView All Steps | Next Step » |
 |
Add Comment
|
