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How to Quickly Add the Integers From 1 to 100 - Part 1

What does 1 + 2 + 3 + … + 100 equal? Learn how to amaze your friends by quickly calculating the sum of the integers from 1 to 100.

By
Jason Marshall, PhD
October 12, 2012
Episode #125

Page 1 of 3

How to Quickly Add the Integers From 1 to 100 - Part 1

If you’re anything like me, you probably enjoy a good number trick every now and then. Which is exactly why I’m excited that today we’re going to take a look at one of the very best number tricks I know. And not only is it a great trick, it also comes with a fabulous story about how, once upon a time, a school-aged version of a man who would eventually become an extremely famous mathematician and scientist both annoyed and impressed his math teacher. As if that isn’t enough, today’s trick is also related to the very cool class of triangular numbers that we talked about last week. So what’s the trick? And how can you use it to amaze your friends? Stay tuned because that’s exactly what we’re going to find out today.

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Who Is Carl Friedrich Gauss?

Our story begins sometime around 1780, a few years after the birth of the German boy who would grow up to become the famous mathematician Carl Friedrich Gauss. As seems to so often be the case with young budding geniuses, Gauss’ childhood was filled with early indications of his talents—at least if you believe the stories (which I for the most part do). It’s said that the three year old Gauss started double-checking his father’s figures—in his head, of course—and saved him from making many-a-bookkeeping blunder. Within a few years, he was pretty much in charge of the bookkeeping.

Later in life, Gauss went on to do truly amazing things…lots and lots of them. Seriously, if you study math or physics in college and beyond as I did, you quickly realize that he’s one of those guys whose name is plastered all over the place. By the age of 15 he was discovering patterns in prime numbers, by 22 he was coming up with a proof of something known as the Fundamental Theorem of Algebra, by 24 he was doing a bunch of other awesome stuff (too much to detail right now), later in life he did even cooler stuff with probability and statistics (Ever heard of the bell curve? Yep, that’s also known as a Gaussian function), toss in some modular arithmetic, and (just for fun) he even went ahead and made some big-time contributions to physics and astronomy. And that’s kind of just the tip of the iceberg!

But those aren’t the things I want to talk about today. Instead, I want to talk about the math trick that the somewhere between 7 and 10 year old Gauss figured out and used to annoy and impress his dear old math teacher.

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