plot any irrational number on number line.

Choosing a unit length, you can use the spiral of Theodorus to find any length of the form $\sqrt{n}$ in which n is an integer:

enter image description here

As a matter of fact, as shown in the picture below, using a similarity argument we can deduce that $CH^2=AH\times BH$, which means by choosing AH to be the unit lenght, we can construct $\sqrt{BH}$. You just have to find a way to construct such a triangel (Hint: Use a semicircle!).

enter image description here

For other irrational numbers, it depends on the number AND the tools you are allowed to use. For example, it has been proved that these numbers are not constructible using just a straight edge and compass: $2^{\frac{1}{3}}$, $\pi$, $e$. But, if you're allowd to use a tool to construct a hyperbola and another for a parabola, then you can construct $2^{\frac{1}{3}}$.


By definition, you cannot plot most of the numbers on the real line. Only a countably infinite subset of irrational numbers can be plotted (in a sense that there exists a procedure (finite or infinite) that leads you to the point you like).

Using a ruler and a compass, you can construct any constructible number (no kidding), which includes all solutions of quadratic polynomials with rational coefficients (and iterated application of solutions of quadratic polynomials).

Then there are larger sets - solutions of higher-order polynomials (angle trisector helps you with 3rd order, then you need more sophisticated plotting devices), then you have computable transcendental numbers, where you usually need an infinite summation or iteration or something like that. However, that can still be argued to produce digits that tell you which point you mean, even if you cannot get the exact result in a finite time.

Then there are incomputable numbers. These are the ones that are incountably numerous (which means most of the reals). Incomputable? That means, there is no way of even saying which number you mean, but it is there. By "saying which number you mean", I'm referring to giving an algorithm - a way of defining it, computing it,... most of the real numbers are such, that you cannot "name" them - because there are only countably infinite "names" (formulas/computer programs) to tell which number you are referring to. That's quite obvious if you think about it right. If computer programs (which are a generalization of mathematical formulas - more powerful in the sense of being Turing machines) are just lists of 0s and 1s, that's just an integer number in binary form (long, but still...). So... countably infinite number of programs/formulas/algorithms/recipes, and a continuum of incountably many real numbers. That's what doesn't add up. So in short... most of the real numbers, you cannot even point out or refer to in any computable way. They are just there, nameless and hidden forever.

But... you can plot any square root with a ruler and a compass, and you can plot any computable real number to a reasonable finite precision (truncation just means you get a rational approximation). So it's not that bad really.


Hints:

Draw a line of length $\,1\,$ , from one of its end points draw another line perpendicular to the first one (how? Can you use a compass...?), and now draw the diagonal between the other end of the original line and the farthest end of the perpendicular one. This perpendicular has length $\,\sqrt 2\,$ (why?) , so with a compass measure it and then place the compass on the origin *the "zero: of your line) and "plot" $\,\sqrt 2\,$ ...you can do something similar with "any" non-rationa square root.

Tags:

Irrational Numbers

Related

Prove that the zeros of an analytic function are finite and isolated Prove if a power series is zero on an interval then all coefficients are zero How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$? Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$ Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$ How to represent the floor function using mathematical notation? Dominant morphism between affine varieties induces injection on coordinate rings? If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well 10 to the power of 3.5: $10^{3.5}$ Regular open set whose boundary has nonzero volume. Proving that if $n$ is odd and $\gcd(m, n) = 1$, then $\gcd(2m + n, 2n) = 1$ Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.

Recent Posts