Why 1+1 is 2

Theo's explanation of why 1+1 is 2.

So, we all say 1+1 is equal to two because its axionically correct right? Well, that's just an assumption. You see, the equation "1+1 = 2" has more, including the numbers 1 and 2, the plus sign and the equal sign.

The Numbers

The numbers 1 and 2 are adapted since we were born, for example, Indigenous people from the Amazon had learned numbers, but didn't quite have number words like "one" or "two" for quantity, they couldn't really tell the difference between quantity beyond three.

The numbers 1 and 2 etc, are known as Arabic Numerals. Funny, because they were actually invented within India. Within the Indian subcontinent, there are pillars called the "Pillars of Ashoka". These monuments were yet erected by the 3rd Mauryan Emperor, Ashoka the Great. Within these pillars, they have the numerals 1, 4, 6 etc. Around a century later, we had the Nana Ghat inscriptions, which are written in Sanskrit in the Brahmi script. These included the next numbers, 2, 4, 6, 7 and 9.

Even fun fact, the terms "Number 1" and "Number 2" as used for toilets, was analogy of Number 1, being where you have your business in a bush, and number 2 being the outhouse.

The Plus Symbol

Within Latin, the plus symbol (+) meant et, or "and". Within the 15th century, mathematicians had used this logic for a shorthand way to say "and". This unfortunately, wasn't adopted quickly back then. Since the plus sign didn't have a fixed conventional use, some individuals used the Maltese cross, or the one that resembles a fan.

It eventually, had gain popularity and of course, has adapted to mathematical notation. This was quite revolutionary, it had changed the way people had seen mathematics. Oh wow, thanks 16th century.

Addition

Conventionally, the plus sign as stated above usually is used in a mathematical operation called 'addition'. For example, 3 + 5 = 8. The word origin of 'addition' comes from the word addere, ad meaning "to" and dare, meaning "to give". Wow, love kindergarten, and elementary school and all that stuff from basic math, addition being one of the four main ones. Wow, four function calculator! My favorite. What the hell is a square root? -Me.

Addition doesn't give a crap about your order of the operands, a + b = b + a, so stupid. This is called commutativity, and associativity, which the order doesn't really matter like i just. AND WOW, ADDING 0 TO ANYTHING DOESNT CHANG E IT.

Set Theory

zzz boring ain't it? Anyways, set theory is a branch of math that describes collections of objects. A set is a group of objects, like a football team is a set of the players.

Richard Dedekind and Georg Cantor were idiots who invented this dumb "branch" of math. But anyways, let's use stupidity for this one, how can we prove 1+1 is 2 with set theory?

Ok like some dumb mathematician, let's say N is our set of NaTuRaL nUmBeRs. From 1, we can say it is the successor of 0. The successor function basically will move to the next natural number. In this simple logic, 1+1 is 2, because 2 is the successor of one. (applause)

Spongebob: BREAK TIME!

Sleeping zzzz catchuing zzzs I dotn know lol

Uniqueness quantification

The axiom simply states, for every natural number (n), there exists a unique natural number (m). The function is defined in the form S(m) = n.

Uniqueness quantification is an axiom (WHAT) in math that tries to assign things unique from others. The symbol denotation for uniqueness quantification is ∃! or ∃=1.

In our context, we want to define the uniqueness quantification function, being S(n) is the next natural number after n. Going with S(0) = 1, S(1) = 2 and so on. Now this will state that S(m) = 1, there has to be a unique number (m) that proves S(m) = 1. So, S(0) = 1. Using our rule, S(1), or the successor of one should be 2.

This however, has a flaw, which is because we don't know if addition is associative, meaning (a+b) + c = a+(b+c).

Is addition REALLY associative?

For something to be associative in here, we need to prove (a+b) + c = a+(b+c).

Using mathematical induction (BORING!!). Let's begin with the base case, let's say c is equal to 0. We can take addition rules and prove it, let's plug in c into both. (a+b) + 0 = a + b, and a + (b+0) = a + b, so those are similar using the base case of c = 0.

Now, let's bring in the induction hypothesis, stating ((a + b) + c = a + (b + c)). And let's try using c + 1:

(a + b) + (c + 1) = a + (b + (c + 1))
And using the induction hypothesis, it should be true (a + b) + (c + 1) = a + ((b + c) + 1).
And lastly, associativity of addition. a + ((b + c) + 1) = a + (b + (c + 1)).

The induction step holds, addition is associative for any natural number, including 1 and 2....but is a sum of two natural numbers really a natural number?

Closure under Addition

Closure under addition states that if numbers closed under addition, their sum will be the same set of numbers. For example, let's begin with Whole Numbers (ℤ+).

2 + 3 = 5, 2 and 3 are whole numbers.

8 + 12 = 20, the sum of 8 and 12 which are whole numbers create another whole number (20).

Natural numbers (the numbers used to count) are also closed under addition. (ℕ).

100 + 500 = 600. The sum of two natural numbers is another natural number.

As for integers, which are positive and negative whole numbers, they are also closed under addition. (ℤ)

-4 + 12 = 8, the sum of two integers, -4 and 12 is another integer, 8.

And lastly, rational numbers, or numbers expressed as a ratio of two integers are as well closed under addition. (ℚ).

For example, 1/2 + 3/4 = 5/4, all of the addends are rational, the same applies for 1/2 + 2/4, equaling 1. 1 is also a rational number. This is called the closure property as seen below.

Closure Property

Closure property states that if a set of numbers (real numbers, integers etc.) are closed under an operation, like addition (see above), subtraction and multiplication. Though, division has a special rule for this.

Conversely, if even a single result falls outside the set, the set is not closed under that operation.

Adding two real numbers will always yield a real number, no stupid i symbol.
For subtraction, it can depend, for natural numbers, it cannot be closed under subtraction, for example, 1 - 2 equals -1, which -1 is NOT a natural number, since -1 is less than 1, all natural numbers go from 1 to whatever infinity can be. So, natural numbers cannot be closed under subtraction, though it applies to all the other sub-categories of real numbers.
For multiplication, it's easy, the product of two real numbers is a real number.
And for division, like I said, all real numbers are closed under division EXCEPT for dividing by 0, which results in undefined.

So, real numbers are closed under addition, subtraction, multiplication and NON-ZERO DIVISION!!!!

Integers are only closed under addition, subtraction and multiplication.

Natural numbers and Whole numbers are only closed in addition and multiplication.

Why zero, Why???

Break time YEAH!

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