MAT1313 - Revision history

Eduardo.duenez: /* Topics List */

2022-08-23T22:33:58Z

Topics List

2022-07-25T18:57:27Z

Week 14

2022-07-25T18:47:53Z

2022-07-25T18:45:36Z

Week 13

2022-07-25T18:35:10Z

Week 12 amends.

2022-07-25T18:32:38Z

Week 12

2022-07-25T18:22:41Z

Expanded Week 11.

2022-07-25T18:16:31Z

Week 11 improvements.

2022-07-25T18:14:09Z

Week 11

2022-07-25T18:00:40Z

Gave up on Wikimedia TeX.

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 * Represent complex numbers in polar form.
 * Algebraically relate the Cartesian and polar forms of a complex number.
-* Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)^n = cis(n𝜃) (De Moivre's formula) for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 to evaluate products and powers both algebraically and geometrically.
+* Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)<sup>n</sup> = cis(n𝜃) (De Moivre's formula) for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 to evaluate products and powers both algebraically and geometrically.
 * Evaluate all n-th roots of a given complex number both in trigonometric and (when possible) in algebraic closed form, and represent them geometrically.
 |-  <!-- START ROW -->

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 ||  <!-- SLOs -->
 * Represent complex numbers in polar form.
-* Algebraically relate the Cartesian and polar forms of a complex numbers.
+* Algebraically relate the Cartesian and polar forms of a complex number.
 * Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)^n = cis(n𝜃) (De Moivre's formula) for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 to evaluate products and powers both algebraically and geometrically.
 * Evaluate all n-th roots of a given complex number both in trigonometric and (when possible) in algebraic closed form, and represent them geometrically.
 |-
 |}

@@ Line 242: / Line 242: @@
 * Represent complex numbers in polar form.
 * Algebraically relate the Cartesian and polar forms of a complex numbers.
-* Interpret the identity cis(𝜃+ɸ) = cis𝜃∙cisɸ for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 as an additive/multiplicative homomorphism property.
+* Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)^n = cis(n𝜃) (De Moivre's formula) for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 to evaluate products and powers both algebraically and geometrically.
 * Evaluate all n-th roots of a given complex number both in trigonometric and (when possible) in algebraic closed form, and represent them geometrically.
 |-
 |}

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 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
 ||  <!-- Topics -->
 ||  <!-- Prereqs -->
 ||  <!-- SLOs -->
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 |}

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 * Carry out arithmetic operations with complex numbers.
 * Interpret the geometric meaning of addition, subtraction and complex conjugation.
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 |}

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 * The rational number system 𝐐.
 * The real number system 𝐑.
 * Rational and irrational numbers. Existence of irrationals.
 ||  <!-- Prereqs -->
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 * Informally interpret the convergence of decimal expansions as the completeness of 𝐑.
 * Informally recognize that the universal existence of roots ⁿ√x and fractional powers x<sup>m/n</sup> of real numbers x>0 relies on the completeness of 𝐑.
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 |}

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 * Identify the entries in Pascal's Triangle as Binomial Coefficients.
 * State and apply the Binomial Expansion Formula.
-* Compute individual binomial coefficients using the quotient-of-falling powers formula (n|k) = n(n−1)…(n−k+1)/k!
+* Compute individual binomial coefficients using the quotient-of-falling powers formula (n𝑪k) = n(n−1)…(n−k+1)/k!
 |-  <!-- START ROW -->
 | <!-- Week# -->
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 * Divisibility of integers.
 * Unique factorization and the Fundamental Theorem of Arithmetic.
 * Roots and fractional powers of real numbers.
 ||  <!-- SLOs -->
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 * Prove the irrationality of √2 and, more generally, of √p for p prime.
 * Prove that fractional powers x<sup>m/n</sup> of real x>0 are well defined and unique.
-* Recognize that the universal existence of roots ⁿ√x and fractional powers x<sup>m/n</sup> of real numbers x>0 cannot be proved by algebraic methods alone.
+* Informally interpret the convergence of decimal expansions as the completeness of 𝐑.
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 |}

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 ==Topics List==
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 ||  <!-- Topics -->
 * Primes.
 * Euclid's Lemma: for p prime, p|ab implies p∣a or p∣b.
 * Unique factorization and the Fundamental Theorem of Arithmetic.
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 ||  <!-- SLOs -->
 * Define prime numbers and state their basic properties.
 * Prove Euclid's Lemma using Bèzout's identity.
 * Prove uniqueness of prime factorization using Euclid's Lemma.
@@ Line 179: / Line 185: @@
 * Identify the entries in Pascal's Triangle as Binomial Coefficients.
 * State and apply the Binomial Expansion Formula.
-* Compute individual binomial coefficients using the quotient-of-falling powers formula (n|k) = n(n-1)…(n-k+1)/k!
+* Compute individual binomial coefficients using the quotient-of-falling powers formula (n|k) = n(n−1)…(n−k+1)/k!
 |-
 |}