Why is 1+1 =2 ?

This might seem arbitrary to most people, even intuitive. But, This is one of the deepest questions to have been answered in the entirety of mathematics. 

In the thresholds of common day experience one might also find ‘things’ that do not seem to obey this proposition:

  • 1 drop of water + 1 drop of water = 1 drop of water.  1 + 1 = 1.
  • 1 cup of water + 1 cup of alcohol = about 1 ¾ cups of 100 proof alcohol.  So 1 + 1 = 1 ¾
  • 01 + 01 = 10 ( In binary )
  • 1+1 =0 ( Modular addition by 2 )
  • 1+1=11 ( concatenation of two strings )

But hey, 1 + 1 is 2. Conflicts arise due to improper usage of ‘1′, ‘+’ and ‘2′.

So, What does ‘1′, ‘+’ and ‘2′ mean?

Peanos axioms :

1) 1 is a natural number. *
2) Every natural number has a successor. *
3) No natural number has successor 1 (or 1 has no predecessor) *
4) Every natural number has a predecessor except for 1. *
5) You don’t need to know this one right now. * 

Defining Addition.

We define addition a+b, for a,b natural numbers, as follows:

If a is 1, then a + b is b’s successor. * 

Otherwise, a has a predecessor, denoted a’. Then, a+b equals the successor of a’ + b. * (Axiom 4) 

For example : 

3+2 = > successor ( 2+2 ) = > successor ( 1+2 ) = > successor( 2 ).

2 = > predecessor ( 3 ) = > predecessor ( 4 ) = > predecessor ( 5 ) .

Defining Equality.

If a is 1, and b is 1, then a = b is true.

If either a or b is 1, but the other is not, then a = b is false.

Otherwise, there exist a’ and b’, predecessors of a and b, respectively (Axiom 4). Then, if a’ = b’ is true, a = b is true, otherwise a = b is false.

The decimal number system and Arabic numerals are not part of this theory - they simply represent natural numbers. In any case, the successor of “1” is denoted “2”, the successor of “2” is denoted “3”, and so on. *

So given these definitions, it is clear that 1 + 1 = 2. 


The Russel and Whitehead proof.

Believe it or not, it took Mathematicians Russel and Whitehead hundreds of pages to get to this result in the Principia Mathematica. This is considered to be a more rigorous proof for the proposition.

Going down that road is something only a truly dedicated mathematician would dare to endure. 


Thanks for reading! Hope you enjoyed reading it as much as I enjoyed writing it.

What else is Pythagoras famous for?

Many of you may remember Mathematics teachers talking about Pythagoras and his theorem: a² + b² = c². But what else is he famous for? In Classical Philosophy: A History of Philosophy Without Any Gaps, Peter Adamson explores the man behind the triangle:

  1. There is no good evidence that Pythagoras himself discovered the Pythagorean Theorem but his followers did know the theory.
  2. However, Pythagoras can be credited with the belief that you shouldn’t eat beans or meat.
  3. Pythagoras and his students supposedly observed a code of silence to prevent a leakage of ideas to those outside their circle.
  4. Like Aristotle, Pythagoras never wrote any books on his ideas which may have been one of the reasons why he was so famous.
  5. Pythagoreans concluded that there must be an unseen heavenly being due to their belief in the importance of the number 10.
  6. According to (unverified) legend from his life time, Pythagoras was supposedly the first to fuse mathematics and philosophy, be able to see into the future, and he was the son of either Apollo or Hermes.
  7. He was also said to have had a thigh made of gold, and be able to talk to animals and geographical features.

If you want to make sure there are no other gaps in your philosophy knowledge, you can follow Peter Adamson on Twitter, listen to his podcasts, or check out the first instalment of the History of Philosophy Without Any Gaps.

Image: Pythagoras emerging from the Underworld, by Salvator Rosa. Public domain via Wikimedia Commons.

Last call to sign up for Discovery Precalculus before course opening!

This summer played host to the most unreal, thrilling events in my math career to date. I’m still in the process of calming the heck down. I started work on my first publication and covered a lot of ground. I met the professor who will very, very likely be my PhD advisor and learned about the technologies I’ll use in my work upcoming. Last but far, far from least, I got to participate in the course building and delivery for Discovery Precalculus. 

If you haven’t signed up and want to, you can do so here: https://www.edx.org/course/discovery-precalculus-creative-connected-utaustinx-ut-prec-10-01x 

The course opens tomorrow. It’s go time. It’s free, self-paced, and covers pre-calculus topics. It’s an awesome resource for calculus review, too. Unit 7 even has an induction section that really helps build understanding of proof writing. 

So, this is the final plug for this course, but I can’t even begin to thank the people who follow this blog for their interest, patience, questions, and enthusiasm. The class has jaw-dropping enrollment numbers, and that makes me so happy. Seriously. Of all the summer’s events, this is the one that encourages me the most that I’ve got my stuff together. Nothing I do and none of my work matters at all if it doesn’t help advance mathematics, and the best way I can do that is help make math more understandable for everyone who wants to learn it. That’s the most important part of the advancement of mathematics: Teaching math right so we can have more and more mathematicians. (Plus, who doesn’t want to live in a world full of mathematicians? We’re awesome! I’m wearing my Pi Day of the Century commemorative shirt right now, why would you not want more people who do that?) 

Listening to music while studying is proven to stimulate your senses and keep you focused. 

here is my updated study playlist

Its back to school season again… yay? I’m excited, yet nervous for my life. This year I’m graduating 8th grade into high school… AHH.  But to start off this awesome year, here are some of my favorite songs to listen to while I study.

-Classical-

https://play.spotify.com/track/3l8pZPbcBebYWkYcQJtIm3

https://play.spotify.com/track/6N7JzrteJv8lsr1GWYyu0b

https://play.spotify.com/track/4PCNrrTgcW5lBnUSGGCkhe

https://play.spotify.com/track/7JvzXZDVLUWAQAy43TmYKF

https://play.spotify.com/track/53KU8BrYZuxZOb3jqDm4ix

https://play.spotify.com/track/5rSR9a0inBSyYlNTLa8BLi

https://play.spotify.com/track/2jjVsWQEUeZ06qSqF8xiBr

https://play.spotify.com/track/6wXbdX2b6Q87kJ8e1G3SmT

https://play.spotify.com/track/0L0WblNH2uSs6lvskY73OQ

https://play.spotify.com/track/7uTyDwZOpODAIIWpkyX678

-Not Classical…-

https://play.spotify.com/track/0AXym3uMYhCOngivsnwjkz

https://play.spotify.com/track/0PywzrV955BnXDC8FHAf0n

https://play.spotify.com/track/0efT4YKQLQx2YHbp6vgRX8

https://play.spotify.com/track/2XRCxFtPqSsnY94jZCFuYu

https://play.spotify.com/track/7fvgzmfXxrdTcrL3xh2cBD

https://play.spotify.com/track/7fvgzmfXxrdTcrL3xh2cBD

https://play.spotify.com/track/0k73nWaD6RPx2sHFEkGPcn

https://play.spotify.com/track/3CRDbSIZ4r5MsZ0YwxuEkn

https://play.spotify.com/track/4Sfa7hdVkqlM8UW5LsSY3F

https://play.spotify.com/track/76EeScTnI2sCjDY0SfEoSb

https://play.spotify.com/track/0VxocBntP1XTZRsR9ZURPS

https://play.spotify.com/track/1pHP4JeQV9wDx87D6qH9hD



Have a great schoolyear to all you students!

2

Any shuffled deck of playing cards, believe it or not, has never before, in the history of our planet, been in that order. 

How can we know that? It’s a simple mathematical fact. The order of cards is a gigantic number. It’s a number known by mathematicians as ‘52 factorial’, otherwise known as 52! or 52 shriek (52 x 51 x 50 x 49 and so on). And that number is big. It’s this big:  

80,658,175,170,943,878,571,660,636,856,403,766, 975,289,505,440,883,277,824,000,000,000,­000.

This number is so big that were you to imagine if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people had a trillion packs of cards, and somehow they managed to shuffle them all one thousand times a second, and they had been doing that since the Big Bang, they would only just now be starting to repeat shuffles. 

So when you shuffle a deck of cards, you can say with all the mathematical certainty that is possible that the deck you shuffled has never been in that order before. It’s an absolute world first.

[see full video]

3

Free Calc I & II Book Available for Download

A handful of professors from Grand Valley State University have a running project called Active Calculus, a free calculus book for students to download. Since it always helps to have more than one way to look at things, I would seriously suggest downloading this if you’re taking Calc I or Calc II this coming semester.

There’s also a beta version of a Calc III version of the text [pdf] if you’re taking multivariable calculus next semester.

Each disk is ten times larger than the previous one. If the first disk you see is the size of the palm of your hand, then the second is the size of a coffee table, the third the size of a room. The seventh is the size of Belgium. The ninth is the size of the Earth. The 23rd is around the size of the galaxy, and less than 2 minutes into watching, the 28th is the size of the Universe.   [code]

Theano circa 500s BCE

Art by collapsiblechair (tumblr)

A woman named Theano has long been counted among the early Pythagorean philosophers.  Various sources list her as the student, daughter or wife of Pythagoras.  She may have run Pythagoras’s school after his death. Some modern scholars have suggested there might have been two female philosophers named Theano while others have suggested that Theano was a pseudonym used by multiple writers.  First century Roman authors such as Plutarch and Clement referenced Theano as a well known model of female scholarship, modesty, and morality.  The philosophical ideas ascribed to Theano are not always progressive by modern standards, but the story of Theano as a great thinker gave ancient women an intellectual role model.

Studying for Math

  • Memorize all necessary formulas. Some tests give you reference sheets, some don’t, some leave out necessary shit anyways, but it’s always nice to know your formulas. You could point blank recite them with flash cards, but often times you’ll remember them simply with the repetition of using them in problems.
  • Take out your textbook, and do a couple practice problems from each unit involved in the test. Get variation; don’t do thirty multiple choice questions and two word problems. If you don’t have your textbook, go online. There are tons of websites with math resources and practice questions, and I’ll probably post a few of them in the math tag later.
  • Once you’ve done your problems, see if you’ve understood everything. Check how many questions you got right, and understand the mistakes you made to get some of them wrong.
  • Clear up anything you don’t understand with a friend, google, your teacher, or me :)
  • Math is one of those subjects where it’s easy to fall behind and get frustrated. Don’t worry.  A grade is just a number, and if you don’t intend to pursue math/science/tech, even less of a reason to stress.

Science // English // Social Studies