Pure Mathematics D1: Numbers

48 flashcards · Mathematics · by lakesare

1. [Decimal representation] Write 0.85 as the arithmetic series.

2. [Decimal representation] What do terminating decimals have to do with 2 and 5?

3. [Decimal representation] Decimal representation and: rational numbers irrational numbers

   rational number ≡ represented by the recurring decimal irrational number ≡ represented by the non-recurring decimal

4. [Ordering] Roughly speaking, what's Archimedean Property?

   No infinitely large or infinitely small elements. Roughly speaking, it is the property of having no infinitely large or infinitely small elements.It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.

5. [Ordering] Roughly speaking, what's Density Property?

   Between 2 distinct real numbers there are: ∞ many rational numbers ∞ many irrational numbers.

6. What's the strangest definition of real numbers ℝ?

   ℝ = ℚ ∪ ~ℚ

7. What is a succinct way to describe (ℝ, +, ×) field?

   (ℝ, +) is an abelian group (ℝ*, ×) is an abelian group Distributivity connects + and ×

8. [Inequalities] Name 2 operations that switch the sign of the inequality?

   a × negative number = b × negative number reciprocal of a = reciprocal of b (ONLY when a and b have the same sign)

9. [Inequalities] Name an interesting way to write down the solution set.

10. [Inequalities] How should you think of |a| on a number line?

    |a| is the distance from 0 to a. The modulus of 3 and –3

11. [Inequalities] Name 2 ways to think of |a – b| on a number line?

    |a – b| is the distance from 0 to a – b. |a – b| is the distance from a to b. The modulus of a – b

12. [Inequalities] How to expand |a| < b? Rules for rearranging the inequalities. For a, b ∈ R.

13. [Inequalities] What rule should you be thinking of when you're solving this inequality?

14. [Inequalities] How would you solve it?

    By taking squares of each side. Because both are positive (Rule 5), and because |a|2 = a2 (Property 3).

15. [Inequalities] Interpret this inequality geometrically. Thinking about what an inequality means geometrically can often give you an idea of its solution set.

16. [Inequalities] Interpret this inequality geometrically. Thinking about what an inequality means geometrically can often give you an idea of its solution set.

    The solution set of |x − 2| ≤ |x + 1|

17. [Inequalities] What's the difference between: Rules for rearranging inequalities Sum rule, product rule, transitive rule

    Rules for rearranging inequalities - ⟺, ≡ rules Sum rule, product rule, transitive rule - → rules

18. [Inequalities] Describe this in your own words.

    x is always squeezed between its magnitude and negative its magnitude.

19. [Inequalities] What's the name of these formulas?

    Triangle Inequality Backwards form of the Triangle Inequality

20. [Inequalities] Why is the Triangle Inequality also called Donkey Inequality?

    Because it's so simple even a donkey gets it: the donkey will walk a straight path to get to water, not walk in a jagged path. My linear algebra professor used to call it the donkey inequality because it's so simple, even a donkey gets it! (The donkey of course will walk a straight line to get to water, not go in a jagged path.) In Euclidean geometry, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths.

21. [Inequalities] What does this "for" even mean?

22. [Inequalities] What geometric interpretation may this inequality have?

    For the same perimeter, area of the rectangle is always ≤ than area of the square. Comparison between the area of a rectangle and the area of a square with the same perimeter.

23. [Binomial Theorem] What would be the coefficient of xy2 in the expansion of the binomial? n choose k

24. [Binomial Theorem] What's to the left of the Pascal's Triangle?

25. [Upper/Lower Bounds] Visually imagine each of: the upper bound of the set the maximum of the set (max E) the supremum of the set (sup E).

    M is upper bound ≡ all numbers M where x ≤ M M is maximum ≡ M is upper bound + M ∈ E M is supremum ≡ M is least upper bound

26. [Upper/Lower Bounds] In "least upper bound" what does least imply?

    That among all upper bounds, it's the smallest.

27. [Upper/Lower Bounds] Give an example of the upper bounds of [0, 2).

    2, 800, etc.

28. [Upper/Lower Bounds] When do we say that the set is bounded VS unbounded?

    bounded - has upper bound AND lower bound unbounded - ¬ bounded

29. [Upper/Lower Bounds] What do max(E) VS sup(E) stand for?

    max(E) - maximum of E sup(E) - supremum of E The least upper bound of a set is also called the supremum of a set. This comes from the Latin word supremus meaning 'highest'.

30. [Upper/Lower Bounds] What do min(E) VS inf(E) stand for?

    min(E) - minimum of E inf(E) - infimum of E The greatest lower bound or infimum of a set is defined in a similar way to the least upper bound. The word infimum comes from the Latin word infimus meaning 'least'.

31. [Upper/Lower Bounds] What's another name for supremum?

    The least upper bound.

32. [Upper/Lower Bounds] What's another name for infimum?

    The greatest lower bound.

33. [Upper/Lower Bounds] Describe "the youngest PhD" strategy for proving M is the least upper bound.

    I'm is the least upper bound iff: I'm the PhD (all schooling is behind me) Blue lines are schooling.If someone is younger than me, then some schooling is ahead of them! If M' is younger than me, then there is some x ∈ E ahead of M'.

34. [Mathematics] What's the existence theorem in mathematics?

    It's a theorem that: asserts a mathematical object exists, but doesn't tell you what it is. In mathematics, an existence theorem is purely theoretical if the proof given for it does not indicate a construction of the object whose existence is asserted. Such a proof is non-constructive, since the whole approach may not lend itself to construction.Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics. For example, John Nash's original proof…

35. [Upper/Lower Bounds] How would you know whether this set E has the least upper bound?

    By the Least Upper Bound Property of ℝ. We know all elements of E are smaller than 800, therefore there exists an upper bound, therefore there exists a least upper bound.

36. [Upper/Lower Bounds] Name one example of a set that: has the Least-Upper-Bound property, doesn't have the Least-Upper-Bound property.

    has "Least-Upper-Bound property" – ℝ no "Least-Upper-Bound property" – ℚ In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers.Not every (partially) ordered set has the least upper bound property. For example, the set Q of all rational numbers with its natural order does not have the least upper bound property.

37. [Upper/Lower Bounds] Visually imagine Least-Upper-Bound Property.

    If our set has upper bounds → it has the least upper bound.

38. [Upper/Lower Bounds] How do we define the addition of non-recurring decimals?

    We define √2 + π as the least upper bound (of the set of truncations). A natural way to obtain a sequence of approximations to the sum √2 + π is to truncate each of the above decimals and then form the sums of these truncated decimals. If each of the decimals is truncated at the same decimal place, then we obtain the following sequences of approximations, which are increasing.

39. What numbers does n imply?

    n means an integer. So n = 3, 4, 5, ~.

40. [Binomial Theorem] What's the strangest way to prove this inequality?

    Notice it looks like a Binomial Theorem.

41. [Inequalities] What's the name of this inequality?

    Bernoulli's Inequality.

42. [Inequalities] What are the names of these inequalities?

    Bernoulli's Inequality Triangle Inequality

43. [Inequalities] What does Bernoulli's Inequality remarkably look like?

    Bernoulli's Inequality look like the Binomial Theorem in (1 + x)n case. In the book Bernoulli's theorem is proved via induction, however you can also prove it via the Binomial Theorem (this is done in one of the exercises for n ≥ 1).

44. [Mathematics] Why do we say that the Bernoulli's Inequality: holds under a weaker assumption?

    Because the assumption is less restrictive.

45. [Mathematics] Why do we say that the Bernoulli's Inequality: is a stronger result?

    Because it applies more widely.

46. [Inequalities] What is this graph about? The graphs of y = (1 + x)n and y = 1 + nx.

    About Bernoulli's Inequality.

47. [History] Name 2 discoveries by the Bernoulli's family.

    Bernoulli's Equation for water pipes Bernoulli's Inequality - the first few terms of the Binomial Theorem

48. [Inequalities] Visually imagine the Bernoulli's Inequality on a Cartesian plane.

    The graphs of y = (1 + x)n and y = 1 + nx.