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.
Jason Marshall, PhD
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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.>
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.
How to Annoy (and Impress) Your Math Teacher
The story goes that one day in young Gauss’ arithmetic class, his teacher wanted a little break (I have no idea whether or not that part is true—but we’ll go with it) and so assigned his students the rather laborious task of writing out and adding together all the integers between 1 and 100. While most students in the class toiled away by adding up 1 + 2 = 3, then 3 + 3 = 6, then 6 + 4 = 10, and so on (and on and on and on); one clever young lad—obviously I’m talking about Gauss—finished in mere seconds and handed in the correct answer! How did he do it so quickly?
For starters, there’s no way around the fact that solving this problem the long way—that is, by actually performing the 100 addition problems it requires—would take even a super-genius like Gauss a long time. Which means that he must have done something clever instead. What was it? Well, if you’d like a little challenge, I encourage you to stop for a few minutes and think about the problem to see if you can come up with a solution. Then, when you’re ready, read on to find out how Gauss was able to solve it so quickly…and annoy his teacher along the way.
How to Quickly Add the Integers From 1 to 100
The trick with this problem—as with so many problems in math—is to find a pattern that helps you solve it more easily. The easiest way to stumble upon the pattern that helps solve this problem is to use the fact that the associative property of addition says that we are free to add up the integers from 1 to 100 in any order we like. Of particular interest to us, this means that we are free to add them up in pairs—with one number coming from the head of the sequence and one from its tail.
In other words, instead of solving 1 + 2 + 3 + … + 100, let’s rearrange things and instead solve the identical problem (1 + 100) + (2 + 99) + (3 + 98) + … + (49 + 52) + (50 + 51). Does that help us out? It absolutely does! Why? Because each pair of numbers here—1 and 100, 2 and 99, 3 and 98, and so on—adds up to 101. And how many of these pairs that add up to 101 are there? A quick moment of thought reveals that there are 50 of them. Which means that the sum of all the integers from 1 to 100 must be equal to 50 x 101 = 5,050. It’s as easy as that!
So, what do you think of this old-school quick and dirty tip? Pretty cool, right? As far as I’m concerned, doing 1 multiplication problem is a lot better than 100 addition problems any day!
How to Quickly Add the Integers From 1 to n
But, you might be wondering: Does this technique only work when adding the numbers from 1 to 100? Or can we somehow generalize it to use to quickly calculate the sum of the first 50, 200, or maybe even 1,000,000 positive integers? In other words, can we use it to quickly add the integers from 1 to some number—any number—that we call “n”? And, you may remember me mentioning that all of this is somehow related to the triangular numbers—those are the numbers in the sequence 1, 3, 6, 10, 15, 21, and so on that we discovered when talking about Pascal’s triangle. How exactly is that?
Unfortunately, we’re all out of time for today. Which means that the answers to these questions—and my instructions for making a very cool sketch that will graphically prove these results and let you visualize how they’re related to the sequence of triangular numbers—will have to wait until next week.
Wrap Up
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Carl Friedrich Gauss image courtesy of Shutterstock
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.