Sum Of Positive Integers Calculator

What this calculator does

There is a story that Gauss, still a schoolboy, was set the busywork of adding every number from 1 to 100. He had it in seconds: 5,050. He had noticed that pairing the numbers from the outside in, 1 with 100, 2 with 99, gives fifty pairs that each come to 101. No adding them one at a time. This tool uses that same shortcut.

Give it the two ends of the range, and it returns the total.

Using the calculator

1. Type the first number, the start of the range.
2. Type the second number, the larger end.
3. Press Calculate.

The result is the sum of every whole number from the first to the second.

The formula that skips the adding

Adding a long run of numbers one at a time is slow, but there is a shortcut. The sum of the whole numbers from 1 to n is:

n(n + 1) / 2

For 1 to 100 that is 100 × 101 / 2 = 5050, all in a single step. To add up a range that does not start at 1, you take the sum up to the top number and subtract the sum sitting below the start.

The Gauss story

The formula comes with a famous tale. As the story goes, the mathematician Carl Friedrich Gauss found it as a schoolboy, set the task of adding the numbers 1 to 100. Rather than grinding through them, he paired the first with the last, 1 + 100, the second with the second-last, 2 + 99, and so on, each pair making 101. With 50 such pairs, the answer fell out as 50 × 101 = 5050.

The triangular numbers

The sums that start from 1 have a name of their own, the triangular numbers: 1, 3, 6, 10, 15, and onward. They are called that because that many dots stack neatly into a triangle. Each one is the running total of the whole numbers up to that point, so the sequence grows by adding the next number each time.

A worked example | 1 to 100

Find the sum from 1 to 100.

1. Apply the formula: 100 × 101 / 2.
2. That is 10100 / 2 = 5050.

The same 5050 that Gauss reached by pairing the numbers up.

Questions people ask

What does this add up?

Every whole number in the range you give, from the first number to the second.

What is the sum from 1 to 100?

5050, from 100 × 101 / 2.

What is the formula?

The sum of the whole numbers from 1 to n is n(n + 1) / 2.

What are triangular numbers?

The sums starting from 1: 1, 3, 6, 10, 15, and so on, each stacking into a triangle of dots.

References

A note on the idea behind it. The sum of the whole numbers from 1 to n equals n(n + 1) / 2, and a range not starting at 1 is handled by subtracting the sum below the start. The sums from 1 are the triangular numbers. For further reading, see Triangular number.

1. The sum of the first n positive integers, equal to n(n + 1) / 2.
2. The triangular numbers, the running totals of the whole numbers from 1.