Pure Mathematics A2: Number Systems

65 flashcards · Mathematics · by lakesare

1. List the first few natural numbers (ℕ).

   { 1, 2, 3, ~ }

2. For addition, what's the name for: 0 –a

   0 - additive identity –a - additive inverse

3. For multiplication, what's the name for: 1 a-1

   1 - multiplicative identity a-1 - multiplicative inverse

4. What's a field?

   A set of numbers that satisfies these 12 properies: 12th property: the additive and multiplicative identities are different numbers. The twelfth property is included for technical reasons to ensure that the set { 0 } with addition and multiplication is not a field.

5. Polynomial VS polynomial equation?

   polynomial - p(x) polynomial equation - p(x) = 0

6. What does p(x) stand for?

   For the polynomial anxn + ~ + a0.

7. What's meant by the roots of p(x)?

   Solutions to p(x) = 0. So finding the roots of a polynomial p(x) means finding the solutions of the polynomial equation p(x) = 0.

8. What's another name for roots of p(x)?

   Zeros of p(x).

9. What's a real polynomial?

   p(x) with coefficients ∈ ℝ.

10. What are the factors of the polynomial?

    s(x), t(x) are polynomials with degrees < degree of p(x)

11. What's the Factor Theorem?

    Factor Theorem: p(x) = (x – a)(...) iff p(a) = 0 (meaning a is root)

12. How would you go about finding p(x) = (x – 2)(...)?

    Write p(x) = (x – 2) (ax2 + bx + c). Since p(x) is a cubic polynomial, it has the linear and quadratic polynomials as factors.Equate the coefficients.

13. If p(x) has roots 1, 2, 3, and –2, what's p(2)?

    p(2) = 0

14. What will be our coefficients b and c if we multiply out these brackets?

15. What's the logic behind this formula?

16. How would you approach finding the roots of p(x) (knowing they must be integers)?

    By our formula, a0 = (–1)n (α1 • ... • αn). Since we know a0 = 14, we need to think of what factors of a0 there are: [±1, ±2, ±7, ±14]. Check each of these!

17. What's Argand diagram?

    Complex plane, a cartesian plane with the real axis and imaginary axis. Argand diagram refers to a geometric plot of complex numbers as points z=x+iy using the x-axis as the real axis and y-axis as the imaginary axis. Such plots are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian–Danish land surveyor and mathematician Caspar Wessel (1745–1818).Also called complex plane.

18. Definition of the modulus of: the real number VS the complex number.

    Both are about distance to origin. real number complex number

19. How to find the distance between: 2 real numbers 2 complex numbers?

    2 real numbers: |x1 - x2| |–3–4| = 72 complex numbers: |z1 - z2|

20. What does it mean that complex numbers (ℂ) satisfy these properties?

    That the set of complex numbers ℂ is a field.

21. arg z VS Arg z?

    arg z - argument for a complex number in polar form. Arg z - principal argument (−π, π] for a complex number in polar form.

22. What's the principal argument of z (Arg z) if it's either real or imaginary?

    [0, π/2, π, or −π/2] Arg z - principal argument (−π, π] for a complex number in polar form.

23. What's the difference between: roots of p(z) VS nth roots of a?

    roots of p(z) - such z that p(z) = 0. nth roots of a - such z that zn - a = 0. A very particular polynomial!

24. [Finding nth roots of a] How would you find θ from this equation? Solutions.

    3θ = π + 2kπ

25. [Finding nth roots of a] What's nth roots of unity?

    Such z that zn - 1 = 0. roots of p(z) - such z that p(z) = 0.nth roots of a - such z that zn - a = 0.A very particular polynomial!

26. [Finding nth roots of a] Give the polynomial equation meant by "6th roots of unity".

    z6 - 1 = 0

27. [Finding nth roots of a] How do the solutions to nth roots of a usually look like on the Argand diagram?

    Equally spaced around a circle (centered in the origin). Always a sun! The nth roots of any complex number are always equally spaced around a circle with center in the origin, but the circle may not have radius 1 and there may not be a root on the real axis, as the following exercises illustrate.

28. When is a number system algebraically closed?

    When every polynomial equation with coefficients ∈ this number system has at least 1 solution ∈ this number system. Closed: CNot closed: R, Q

29. What's the Fundamental Theorem of Algebra VS Fundamental Theorem of Arithmetic?

    Fundamental Theorem of Algebra Moreover, if 'repeated' solutions are counted separately, then a polynomial equation of degree n with coefficients in C has exactly n solutions (all in C).Fundamental Theorem of Arithmetic

30. How are these equations connected?

    Every p(z) with coefficients in C has this factorization. Moreover, if 'repeated' solutions are counted separately, then a polynomial equation of degree n with coefficients in C has exactly n solutions (all in C).

31. How many solutions does this complex polynomial have?

    6 Moreover, if 'repeated' solutions are counted separately, then a polynomial equation of degree n with coefficients in C has exactly n solutions (all in C).

32. Why might this be happening (if coeffs are real then "a is root → ā is root)?

33. What's ez defined as? z = x + iy

34. Where does the definition of ecomplex number come from?

    The definition comes from needing to make ez1ez2 = ez1+z2, and other basic properties work. The definition needs to satisfy basic properties of real powers of e.

35. How can we get Euler's formula from ez definition?

    You derive Euler's formula by simply inputting z = 0 + iy into ez defintion. Euler's formula is simply ez where z is imaginary.

36. What are the names for these equations?

37. If a = reiθ, then what is ā?

    ā = re –iθ

38. Why do some texts regard exponential form of z as just another version of polar form?

    Because both in polar form and exponential form, r and θ are used. Rather than using the term exponential form, some texts regard reiθ as another version of polar form, since it involves the modulus and angle of the complex number.

39. What's the pipeline from the Euler's formula and polar form to the exponential form of the complex number?

40. If we divide by n - what's divisor quotient remainder

41. What are the possible remainders after dividing by 4?

    { 0, 1, 2, 3 }

42. In what sense are the integers that are congruent to each other the same?

    a is congruent to b (modulo n) when they both have the same remainder on division by n.

43. Name 2 meanings of the word "modulus".

    |z| - distance of ℝ/ℂ number to the origin. if a ≡ b (mod n), n is the modulus of the congruence.

44. What's the modulus of this congruence?

    n.

45. What does ℤ7 contain?

    ℤ7 = { 0, 1, 2, 3, 4, 5, 6 }

46. How is this operation read?

    a plus b, mod n.

47. What's the simple meaning behind this operation? 3 times 6, mod 7.

    3 + 6 – 7 – 7 ... – 7 = 2

48. What's arithmetic modulo n?

49. mod VS modulo VS modulus?

    All the same, except used in different situations. mod a plus b, mod nmodulus modulus of this congruencemodulo arithmetic modulo n

50. How to write down the solution to this?

    Solve the congruence 29 × 18 (mod 31), but remember the answer should be in ℤ31

51. How to write down the solution to this?

    Solve the congruence 2 – 5 (mod 12), but remember the answer should be in ℤ12

52. Name 2 words for when two integers' only common factor is 1.

    coprime relatively prime

53. In this syntax, how would you express "multiplicative inverse of a"?

    a ×n b = 1 b is the multiplicative inverse of a.

54. How would you show that: if element in ℤn has a multiplicative inverse, then it's the only one multiplicative inverse.

55. Which numbers have a multiplicative inverse in ℤ4? Multiplication table for ℤ4

    1 ×4 1 = 1 3 ×4 3 = 1 2 ×4 ??? = 1 2 doesn't have a multiplicative inverse in ℤ4.

56. Why does it happen so that each element of ℤ7 has a multiplicative inverse. Multiplication table for ℤ7

    7 is a prime number, making it coprime with every element in ℤ7! An element a of ℤn has a multiplicative inverse iff a and n are coprime.

57. Which ℤn are fields, and which ℤn are not fields?

    ℤprime are fields ℤnot-prime are not fields

58. Why are only ℤprime fields?

    Because only in ℤprime-s, EACH element has a multiplicative inverse.

59. Why is it necessary that a and n are coprime for b to exist? Multiplicative inverse of a (mod n) exists only if a and n are coprime.

    ab - nk = 1 By Bezout's identity, we can find b (and k) now (because gcd(a, n) = 1).

60. [Euclid's Algorithm] Which of these (divisor, quotient, remainder) do we have to ignore in the Euclid's algorithm pipeline?

    Ignore the quotient. Euclid's algorithm is a pipeline from remainder to divisor. Each remainder shifts from position [3, 2, 1].

61. [Euclid's Algorithm] Which pairs of numbers have the same HCF in Euclid's algorithm? Finding the Highest Common Factor of 32 and 9.

    Divisor and remainder.

62. [Euclid's Algorithm] Why is it considered that these pairs have the same hcf?

63. How would you solve this equation knowing the multiplicative inverse of 10 in Z27 is 19?

    Multiply both sides by this multiplicative inverse.

64. Provided solutions exist - how many solutions does this equation have?

    hcf(a, n)

65. What's the form of linear equations in Zn? Solutions to the linear equation in Zn.