This is a moderated community.
The goal is to encourage people to approach mathematics in a fashion that makes it accessible. In order to do this, we're going to need a few basic rules.
- Try to be on topic.
- Be civil.
- No drama.
Pretty simple, huh? Yeah... I reduced them to the lowest common denominator. If you have any questions, feel free to ask.
Off-topic is tolerated to some extent. Submissions should all be about mathematics, or clearly relate to mathematics. However, once you get into the comments section, it's a much looser standard. Discourse is encouraged, so long as civility is maintained. Good conversations tend to meander around topics and that's perfectly acceptable.
If you do vote, concentrate on promoting content and not downvoting content. Encourage and comment, if you see something that could use improvement. Downvotes don't convey very much information and aren't really helpful. If you must, you must... But try to assume the person making the post was just communicating poorly. Try to assume the best possible interpretation of their post.
Some terms:
- WDT Weekly Discussion Thread
Greek Alphabet:
Α α, Β β, Γ γ, Δ δ, Ε ε, Ζ ζ, Η η, Θ θ, Ι ι, Κ κ, Λ λ, Μ μ, Ν ν, Ξ ξ, Ο ο, Π π, Ρ ρ, Σ σ/ς, Τ τ, Υ υ, Φ φ, Χ χ, Ψ ψ, Ω ω
Hebrew Alphabet:
א ב ג ד ה ו ז ח ט י כ/ך ל מ/ם נ/ן ס ע פ/ף צ/ץ ק ר ש ת
Other common symbols:
¬ → ⇒ ⇔ ∀ ∂ ∃ ∅ ∇ ∈ ∉ √ ∞ ∧ ∨ ∩ ∪ ⊕ ∫ ≈ ≠ ≡ ≤ ≥ ⊆ ⊂ ⊃ ⊄ ° ± · × ⟌ ÷ ⌊ ⌋ ⌈ ⌉ ➀ ℤ ℕ ℙ ℚ ℝ ℂ ℍ ℭ
If you're interested in academia, you can feel free to also join us at v/PrincipiaAcademia where we discuss all things academic.
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[–] VIP740 0 points 2 points 2 points (+2|-0) ago
Can zero become an obstacle? Odd as it may sound there have been times when zero stood in the way of mathematicians, and they had to develop some fairly sophisticated tools to get around it.
To the Greeks math was foundation of philosophy. Upon Plato's academy was said to be the sign "Let no-one without knowledge of geometry enter". Philosophers such as Plato and Zeno saw our world as an illusion, and math as the truth. By studying mathematics they would free themselves from the falsehoods which bind us to this world of imperfections.
Can you draw a perfect circle? Imperfections can be found even in the best painter's work if enough scrutiny is applied. But the mathematician should be able to create one through pure logic. Now we are taught the formulas C = 2πr, and A=πr2 in grade school, but most aren't taught where they came from. Before these could be taught, someone had to figure them out, history names this person as Archimedes. The visualizations @plankO shared with us will cast some light on the mathematics involved.
It's easy to see the area of a rectangle is given by the base by the hight. Cut it into triangles and you'll see all triangles have half the area of their base times height. This lets you measure the area of any polygon, as they can all be divided into triangles.
But a circle isn't a polygon. We can approximate circles from polygons, but the Greeks demanded perfection. While we could think of a circle as a polygon with infinite sides, the Greeks saw infinity as representing incompletion. Besides, the area of each triangle would get smaller and smaller, if you never stop cutting your figure into smaller triangles, can there be anything left when you get done?
The Greeks would view this as complete nonsense. We could use this argument to make anything we want to represent 0 * ∞, it would be worthless as a formula, giving us no details distinguishing one thing from another. Archimedes found a way around this problem. He said no matter how many sides you give the regular polygon you inscribe inside the circle, it will always have less area than the circle enclosing it.
This doesn't give us a perfect measure, but you could also place a larger polygon "hugging" the circle (so that the middle of each side rather than the corners touched the circle). So the area of the circle is greater than any regular polygon which could be inscribed inside of it, and less than any regular polygon that could be built around it. Archimedes used this idea to come up with his formula for a perfect circle without having to worry about zero or infinity. His method reveals how closely the mathematics of circles and triangles are tied.
Newton and Leibniz came up with a more general way to get around the problems zero could cause. While calculus served as a great tool for scientists, philosophers were skeptical of Newton's vanishing quantities and the infinitesimals of Leibniz. So to solidify the reputation of calculus, mathematicians such as Weierstrass set out to banish unclear ideas such as infinities from the science. They went through calculus, and when they found statements like this:
They replaced them with statements like this:
Now we don't have to worry about infinities or division by zero. Even if the details are hard to follow, we do at least have the details. No longer will zero and the ambiguity surrounding it stand in our way. Many thought infinity had been laid to rest at this point, but a student of Weierstrass brought it back into modern mathematics, rekindling some ancient debates.
[–] TheBuddha [S] 0 points 1 point 1 point (+1|-0) ago
pondering
I'm so going to cop out and day that any mathematical concept can be a an obstacle. For instance, we have those who think linear math allows division by zero, concepts such as infinity breaking basic maths, and things like Zeno's paradoxes (such as Achilles and the tortuous) that we work around.
Yeah, I'm so taking the low road with this answer. ;-)
[–] VIP740 ago
Did the spell-checker fix deny for you? Obstacles or not, there's definitely some confusion and frustration in the study of mathematics.