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 Many scientific theories that worked have been superseded by theories that work better.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.