You can add the integers from 1 to 10 by hand, but what if you need to add from 1 to 1000? The shortcut for adding from one to ten will work for numbers other than ten too. So let’s solve the problem for 10 first, and then generalize.
Numbers can represent something in the real world. 10 rocks or 10 dollars for instance. The beauty, is you can add 5 dollars to 4 dollars as numbers, and predict what happens when you add them as stacks of bills. Numbers model the world. You can add, subtract, multiply, and divide numbers and things. The model lets us manipulate the world on paper, and predict what will happen in real life.
But today, I am going to flip this on its head. Something can be obvious when you are playing with objects in y0ur hands. By manipulating our piles of things, we can then understand something about numbers.
The area of colored pieces of paper will be our things. A 1 by 1 square with area 1, will correspond to our number 1. A 10 by 1 rectangle with area 10 thus corresponds to our number 10.
So if you want to add the numbers form 1 to 10, picture ten pieces of paper as rectangles with the areas from 1 to 10. If you look at the drawing above, you see the 10 shapes next to each other in orange with blue corners. The area of those ten rectangles together will answer our original question. If we can find that, we also found the sum from 1 to 10.
The blue rectangle with a 5 over it represents the sum of the blue corner triangles. All the little areas pieced together fit exactly in that rectangle. The orange rectangles without the blue corners fit together and make a big triangle. That triangle is half a square with side lengths of 10.
So adding the area of the orange triangle to the area of the blue rectangle is the total area we are looking for.
- The area of the 10 by 10 square is 10 times 10.
- The area of the orange triangle then is half that.
- Put ten tall equal rectangles of area 10 next to each other, and you made a square.
- Cut that square on the diagonal, and we made the triangle.
- If we had one more area 10 rectangle to cut in half, we can create the area 5 blue rectangle.
- So eleven area 10 rectangles pieced together, and then divided in two creates the area that corresponds to the sum from 1 to 10.
- That would be 11 x 10 = 110 then divided by 2 = 55
- The answer for the sum of 1 to 10 is 55
The general answer, is the sum of all the integers from 1 to any number N, is N + 1 together times N and then divided by 2. In math:
Sum_1_to_N = ( N + 1 ) * N / 2
Try it for numbers under 10 using the graphic at the top of the page, and you can see that it works. It also works for numbers greater than 10.
- 1 = 1, 1 x 2 / 2 = 1
- 1 + 2 = 3, 2 x 3 / 2 = 3
- 1 + 2 + 3 = 6, 3 x 4 / 2 = 6
- 1 + 2 + 3 + 4 = 10, 4 x 5 / 2 = 10
- 1 + 2 + 3 + […] + 999 + 1000 = 1000 x 1001 / 2 = 500500