Gauss: Sum from 1 to 100 trick

Interesting math trick. I wanted to relearn it. What if you wanted to add all the integers from 1 to 100?

S being the sum of integers from 1 to 100

S = 1 + 2 + 3 + 4 + ... + 98 + 99 + 100

We can rewrite the sum like this:

S = (100+1)+(99+2)+(98+3)+...+(53+48)+(52+49)+(51+50)

Then, we realize that 100+1=99+2=98+3=...=(51+50) = 101

We know there will be 50 pairs of 101, so the sum of all integers from 1 to 100 is 50*101 or 5050.

In general what we are saying is that the sum integers from 1 to some greater integer, N, is the product of 1 more than N (e.g. 100 in the case above) and half N.

S = (N/2)*(N+1)

example from 1 to 10

S = (10/2)*(10+1) = 55

http://www.wolframalpha.com/input/?i=1%2B2%2B3%2B4%2B5%2B6%2B7%2B8%2B9%2B10

This math trick is interesting because it was supposedly invented/discovered by a great mathematician when he was a child, Carl Friedrich Gauss.

A Low Blow for Fictionalism

I am have been a proponent of fictionalism for a short while now, but MinutePhysics has come out with this short clip on how the universe boils down to the mathematical. I wonder if the speaker in this video has ever considered fictionalism, so it would be interesting to hear how he addresses that branch of thought directly. I don't yet have any good arguments as to why this video is wrong or misleading, but maybe I'll come up with something soon. Enjoy!

For an introduction to fictionalism and other schools of thought about math check out my post Do Numbers Exist?

Do numbers exist?

"People always think that math is just a resource, like someone goes into a mine with a pickaxe and discovers math for us, when in reality it's the accumulation of brilliant thoughts, thought by brilliant people."
 -slightly paraphrasing my MTH 255 professor.

I've talked to this professor about his philosophy of math a bit during an appointment. I think any discussion about someone's philosophy of math boils down to the question "Do numbers exist?" and I liked numberphile's description of some roads one may take to answer that question. 

 For now, I'm a fictionalist. I don't see much reason to say math is true. Fictionalism is also nice because it reminds us that just because something works and is useful does not necessarily mean it's true. Many scientific theories that worked have been superseded by theories that work better. The best part about being a fictionalist is that it annoys mathematicians that would rather not call their work fiction.

"Solving Math Word Problems" book

I'm reading another one of the Master Math series books. This one is "Solving Word Problems" by Brita Immergut. I'm usually pretty good at solving word problems but sometimes I get confused even with the simple things. Reading through this book is going to be a good review on how to speak math in English. I got a few other books took look at while I'm in the library for the rest of the day. I got "Biography of E=mc^2" by David Bodanis, "Field Guide to Snow Crystals" by Edward R. LaChapelle, Radioactivity by Lauren Redniss, and Calculus simplified and self-taught by Sanderson M Smith.

Mathless Blog

Sorry to everyone that reads this blog. I'm sure there are many that love to read my blog with little to do with math. Actually, I'm going to be taking Physics with Calculus in the near future as a college course and I hope to post some updates concerning that course. I will also be taking some Chemistry in the near future, but the math in that chemistry course will probably not be as advanced as the math I'll be using in Physics. I have been talking about math education with friends. I have basically been promoting Khan Academy's whole philosophy that one should go through the fundamental math courses with a 100% proficiency. Sal Khan compares going through math at only an 80% proficiency to learning how to ride a bike at only 80% proficiency.

My Math Lately

Lately I have been doing much Calculus. I have a Math 252 final coming up in a couple weeks and I just might get 4.0 in the class if I keep studying until the term is over and not slack off too much playing video games. I beat Portal 2 today, by the way.

Math as a Language

Any time I mention math in a room full of people, they are quick to say 'I'm not good at that' This view that math is something you are naturally good at or not is a popular misconception circulating schools today. Math should be treated like learning any other language. We don't just know French or Spanish or whatever language. We were taught the meaning of the words and how those words are used. In math, there are symbols and those symbols have properties and representations. Learning the definitions of a word in another language is very similar to learning the properties of a symbol or notation in mathematics.