The observation of lengths that can not be represented by rational numbers was noticed if I recall correctly by some Pythagorean disciple over applying the Pythagorean theorem on a triangle with side units of $1$ and realizing that there is no choice for a rational number that when squared is $2$. This discovery led to the real number set.
May be it is a silly question, but I was wondering if this (real numbers) is a side-effect/consequence of how we have structured/invented our number system e.g. base $10$ or if not, how can it be explained in such basis?
2026-04-02 19:15:30.1775157330
What is the reason for the existence of real numbers? Is it an artifact/side effect of our thought process?
123 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REAL-NUMBERS
- How to prove $\frac 10 \notin \mathbb R $
- Possible Error in Dedekind Construction of Stillwell's Book
- Is the professor wrong? Simple ODE question
- Concept of bounded and well ordered sets
- Why do I need boundedness for a a closed subset of $\mathbb{R}$ to have a maximum?
- Prove using the completeness axiom?
- Does $\mathbb{R}$ have any axioms?
- slowest integrable sequence of function
- cluster points of sub-sequences of sequence $\frac{n}{e}-[\frac{n}{e}]$
- comparing sup and inf of two sets
Related Questions in MATH-HISTORY
- Are there negative prime numbers?
- University math curriculum focused on (or inclusive of) "great historical works" of math?
- Did Grothendieck acknowledge his collaborators' intellectual contributions?
- Translation of the work of Gauss where the fast Fourier transform algorithm first appeared
- What about the 'geometry' in 'geometric progression'?
- Discovery of the first Janko Group
- Has miscommunication ever benefited mathematics? Let's list examples.
- Neumann Theorem about finite unions of cosets
- What is Euler doing?
- A book that shows history of mathematics and how ideas were formed?
Related Questions in RATIONAL-NUMBERS
- What is the smallest integer $N>2$, such that $x^5+y^5 = N$ has a rational solution?
- I don't understand why my college algebra book is picking when to multiply factors
- Non-galois real extensions of $\mathbb Q$
- A variation of the argument to prove that $\{m/n:n \text{ is odd },n,m \in \mathbb{Z}\}$ is a PID
- Almost have a group law: $(x,y)*(a,b) = (xa + yb, xb + ya)$ with rational components.
- When are $\alpha$ and $\cos\alpha$ both rational?
- What is the decimal form of 1/299,792,458
- Proving that the sequence $\{\frac{3n+5}{2n+6}\}$ is Cauchy.
- Is this a valid proof? If $a$ and $b$ are rational, $a^b$ is rational.
- What is the identity element for the subgroup $H=\{a+b\sqrt{2}:a,b\in\mathbb{Q},\text{$a$ and $b$ are not both zero}\}$ of the group $\mathbb{R}^*$?
Related Questions in FOUNDATIONS
- Difference between provability and truth of Goodstein's theorem
- Can all unprovable statements in a given mathematical theory be determined with the addition of a finite number of new axioms?
- Map = Tuple? Advantages and disadvantages
- Why doesn't the independence of the continuum hypothesis immediately imply that ZFC is unsatisfactory?
- Formally what is an unlabeled graph? I have no problem defining labeled graphs with set theory, but can't do the same here.
- Defining first order logic quantifiers without sets
- How to generalize the mechanism of subtraction, from naturals to negatives?
- Mathematical ideas that took long to define rigorously
- What elementary theorems depend on the Axiom of Infinity?
- Proving in Quine's New Foundations
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Whether mathematics is invented or discovered and whether its constructs are "real" are interesting philosophical questions.
The Greeks discovered(?) that the rational numbers were not sufficient to model the idea of a continuous geometric line of points.
That kind of line turns out to be interesting mathematically and useful for modeling physical phenomena. So mathematicians have spent a lot of time figuring out how to describe such an object using the rational numbers and some kind of limiting process. Infinite decimals in base $10$ are one such way.