| General > General Technical Chat |
| Rational, Irrational, and Integer. The correct definition. |
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| TimFox:
Take it up with the Encyclopedia Britannica https://www.britannica.com/science/irrational-number |
| magic:
--- Quote from: TimFox on December 21, 2022, 06:11:49 pm ---Britannica --- End quote --- Neat, let's see if we need to further restrict the Privileged Boys Club to continental Europe only :box: --- Quote ---irrational number, any real number that cannot be expressed as the quotient of two integers --- End quote --- OK, this definition is not an irrational number itself, but let's see the reals... --- Quote ---real number, in mathematics, a quantity that can be expressed as an infinite decimal expansion --- End quote --- So say I have a sequence of infinitely many zeros, followed by seven. Like, 0.(0)7. What sort of real number is that? >:D |
| Nominal Animal:
In my opinion, complex numbers are between scalars and vectors. (There are others there too, like quaternions.) The \$i^2 = -1\$ thing is just a "mathematical" way of expressing the idea of rotation by 90°. If you think of the one-dimensional number line concept, then such a rotation by itself doesn't make any sense, but if applied twice, you mirror the number line. And that's exactly what happens. The math boffins examined the concept for a bit, and realized that it also naturally aligned with the exponential function, so that \$e^{a + i b}\$ corresponds to a scaling (multiplication) by \$e^a\$, and rotation by \$e^{i b}\$. Everything else stems from there. So, while the "imaginary number" concept doesn't make much sense for real numbers (integers, rationals, irrationals), it does work as a stepping stone for extensions to more complex concepts. In certain ways, complex numbers remind me of homogenous coordinates. Cartesian coordinates \$(x_1, \dots, x_N)\$ are represented by \$(X x_1, \dots, X x_N, X)\$ in homogenous coordinates, and somewhat similarly to complex numbers, homogenous coordinates can be used to express both translation and rotation (full transform) in a single operation. (In fact, the vast majority of computer 3D graphics libraries use homogenous coordinates at least internally for transformations.) A particularly interesting facet of homogenous coordinates is that they let one trivially distinguish between position vectors (\$(x_1, \dots, x_N, 1)\$) and direction vectors (\$(x_1, \dots, x_N, 0)\$). Mathematically, there is no reason to distinguish between the two, but in 3D visualization, the ability to easily do so while applying the exact same transformations (with translation only affecting position vectors and not direction vectors) simplifies things quite a bit. --- Quote from: magic on December 21, 2022, 06:06:40 pm ---So what the hell are the irrationals, then? --- End quote --- Any number that you cannot express as the ratio of two integers (or an extension of the same concept for e.g. complex numbers). A step further are transcendental numbers. They are numbers you cannot obtain as a root of a finite-degree nonzero polynomial with rational coefficients, like \$\pi\$ and \$e\$. --- Quote from: magic on December 21, 2022, 06:06:40 pm ---But you have heard about those "atoms" and "quantum" stuff, right? There is nothing real about real numbers. --- End quote --- The two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'. --- Quote from: magic on December 21, 2022, 06:06:40 pm ---The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off :P --- End quote --- Except they weren't initially European and never even "mostly" European, men, or even heterosexual. That's what makes "decolonizing math" so ridiculous, and extremely racist and bigoted concept. In particular, Indians and Arabs are quite, quite offended by your suggestion, and for good reason. Sthananga Sutra is about 2400 years old, and covers things like fractions and elementary number theory. In the 800s Arabs made huge leaps forward in maths, especially people like Muhammad ibn Musa al-Khwarizmi. |
| magic:
--- Quote from: Nominal Animal on December 21, 2022, 07:06:19 pm --- --- Quote from: magic on December 21, 2022, 06:06:40 pm ---So what the hell are the irrationals, then? --- End quote --- Any number that you cannot express as the ratio of two integers (or an extension of the same concept for e.g. complex numbers). --- End quote --- Can the number of all real numbers be expressed as a ratio of two integers, or is it an irrational number? ;) --- Quote from: Nominal Animal on December 21, 2022, 07:06:19 pm ---The two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'. --- End quote --- Yes, and hardly anyone knows what the latter actually are, that's my point which started this whole exchange. All I hear is "irrationals are not rational this", "reals are not imaginary that". So what they are, if they are not all those other things? --- Quote from: Nominal Animal on December 21, 2022, 07:06:19 pm --- --- Quote from: magic on December 21, 2022, 06:06:40 pm ---The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off :P --- End quote --- Except they weren't initially European and never even "mostly" European, men, or even heterosexual. That's what makes "decolonizing math" so ridiculous, and extremely racist and bigoted concept. --- End quote --- No, it's precisely true :D People of Color invented Mathematics. Heterosexual European Males came up with the formal concept of "Real Numbers". Heterosexual European Males teach that their "Real Numbers" are no more, no less, but all numbers that are real. Reality is a social construct now. This is Colonization. |
| SiliconWizard:
Good lord. :-DD |
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