This is a really tough question, but I'll try my best!
First, let's understand the 1
1 (one, also called unit, and unity) is a number and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer.[1] It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. Â The fundamental mathematical property of 1 is to be a multiplicative identity,[2] meaning that any number multiplied by 1 returns that number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; although universal today, this was a matter of some controversy until the mid-20th century.
Now let's understand the mathematics & properties of 1
Mathematics:
Mathematically, 1 is:
in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.
Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.
In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e[2] (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.
By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.
By definition, 1 is the probability of an event that is absolutely or almost certain to occur.
In category theory, 1 is sometimes used to denote the terminal object of a category.
In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.
Properties:
Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.
Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).
There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.
In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.
Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.
It is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.
The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.
1 is the most common leading digit in many sets of data, a consequence of Benford's law.
1 is the only known Tamagawa number for a simply connected algebraic group over a number field.
Thus, I can conclude that 1 + 1 is indeed equal to 2
