Here’s an interesting story about Carl Gauss, a mathematician born in the 18th century. This is from when he was young - when his math skills were not known even to himself!
When Gauss was in elementary school, he was punished by his strict math teacher, who asked him to find the sum of all natural numbers from 1 to 100.
1 + 2 + 3 + ... + 98 + 99 + 100 = ?
And remember, this was way back in the 18th century when people didn’t have mobile phones or even calculators. The only way this sum could be calculated was by adding the numbers one after the other. That is:
- First, add 1 and 2 to get 3.
- Then add this 3 to the next number (3) to get 6.
- Then, add this 6 to the next number (4) to get 10, and so on.
A painstaking calculation indeed.
However, Gauss was no ordinary student. He was able to quickly answer correctly, leaving the teacher in shock and awe.
How could a young child find this sum so quickly, something that would take most people a lot of time?
You’re probably wondering that too. Well, here’s how Gauss did it. He noticed a pattern in the numbers. Here's the sum:
S = 1 + 2 + 3 + ... + 98 + 99 + 100
What Gauss noticed was that the first and the last numbers added to 101 (i.e. 1 + 100). Similarly, the second and second last numbers added to 101 (i.e. 2 + 99), and so on.
So, what Gauss did was, he reversed the order of the numbers. Tap the play button in the simulation to see that in action.
Notice that this reversing doesn’t change the sum.
Gauss then added the two equations. Tap the play button in the simulation to do that.
Now, here’s the cool thing about the numbers on the right. In pairs, they all add up to 101.
2S = 101 + 101 + … + 101 + 101
Since 101 appears 100 times (because we had 100 pairs in the previous step), Gauss simplified this sum to get:
2S = 101 x 100 = 10100
And now, for the final step. Since twice the sum (or 2S) equals 10100, the sum (S) will be half of 10100. That is:
S = 10100 ÷ 2 = 5050
And this is how he found the sum at warp speed! Instead of doing the ‘busy work’ as a punishment, Gauss solved the problem like a boss and had what we call a mic drop moment.
Want to relive the moment?
Try finding the sum of the first 20 natural numbers using Gauss's method. How long did it take?