Course Cheatsheet

Section 01: Mathematical Foundations & Algebra

Number Systems

The hierarchy of number systems: \[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{I} \subset \mathbb{R}\]

  • \(\mathbb{N} = \{1, 2, 3, ...\}\) - Natural numbers
  • \(\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}\) - Integers
  • \(\mathbb{Q} = \{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\}\) - Rational numbers
  • \(\mathbb{I}\) - Irrational numbers (cannot be expressed as fractions)
  • \(\mathbb{R}\) - Real numbers (all points on the number line)

Key Facts:

  • Repeating decimals are rational: \(0.\overline{36} = \frac{36}{99} = \frac{4}{11}\)
  • \(\pi\), \(\sqrt{2}\), \(\sqrt{7}\) are irrational

Set Theory Essentials

Set Notation:

  • Roster: \(A = \{1, 2, 3, 4, 5\}\)
  • Set-builder: \(B = \{x \in \mathbb{N} : x < 6\}\)
  • Interval: \([0, 1] = \{x \in \mathbb{R} : 0 \leq x \leq 1\}\)

Set Operations:

  • Union: \(A \cup B\) (elements in A or B)
  • Intersection: \(A \cap B\) (elements in A and B)
  • Difference: \(A \setminus B\) (elements in A but not in B)

Essential Symbols:

  • \(\in\) (element of), \(\notin\) (not element of)
  • \(\subset\) (subset), \(\subseteq\) (subset or equal)
  • \(\forall\) (for all), \(\exists\) (there exists)
  • \(\Rightarrow\) (implies), \(\Leftrightarrow\) (if and only if)

Commutative, Associative, and Distributive Laws

Commutative: \(a + b = b + a\), \(a \times b = b \times a\)

Associative: \((a + b) + c = a + (b + c)\)

Distributive: \(a(b + c) = ab + ac\)

Laws of Exponents

Rule Formula Example
Product \(a^m \cdot a^n = a^{m+n}\) \(x^3 \cdot x^4 = x^7\)
Quotient \(\frac{a^m}{a^n} = a^{m-n}\) \(\frac{x^5}{x^2} = x^3\)
Power \((a^m)^n = a^{mn}\) \((x^3)^2 = x^6\)
Product Power \((ab)^n = a^n b^n\) \((2x)^3 = 8x^3\)
Quotient Power \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) \(\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}\)

Special Values:

  • \(a^0 = 1\) (for \(a \neq 0\))
  • \(a^1 = a\)
  • \(a^{-n} = \frac{1}{a^n}\)
  • \(a^{1/n} = \sqrt[n]{a}\)

Scientific Notation

Format: \(a \times 10^n\) where \(1 \leq |a| < 10\)

Examples:

  • \(56,700,000 = 5.67 \times 10^7\)
  • \(0.00000423 = 4.23 \times 10^{-6}\)

Operations:

  • Multiply: \((3 \times 10^5) \times (2 \times 10^3) = 6 \times 10^8\)
  • Divide: \(\frac{8.4 \times 10^7}{2.1 \times 10^4} = 4 \times 10^3\)

Absolute Value

Definition: \(|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\)

Solving Equations: \(|ax + b| = c\) has solutions:

  • \(ax + b = c\) and \(ax + b = -c\)

Inequalities:

  • \(|x| < a\) means \(-a < x < a\)
  • \(|x| > a\) means \(x < -a\) or \(x > a\)

Basic Factorization

Common Factor: Always check first!

  • \(12x^3 - 18x^2 + 6x = 6x(2x^2 - 3x + 1)\)

Difference of Squares:

  • \(a^2 - b^2 = (a + b)(a - b)\)
  • \(x^2 - 9 = (x + 3)(x - 3)\)

Perfect Square Trinomials:

  • \((a + b)^2 = a^2 + 2ab + b^2\)
  • \((a - b)^2 = a^2 - 2ab + b^2\)

Advanced Factorization

AC Method: For \(ax^2 + bx + c\) when \(a \neq 1\)

  1. Find \(ac\)
  2. Find factors of \(ac\) that sum to \(b\)
  3. Rewrite middle term and group
  4. Factor by grouping

Example: \(6x^2 + 13x + 5\)

  • \(ac = 30\), factors \((3,10)\) sum to \(13\)
  • \(6x^2 + 3x + 10x + 5 = (3x + 5)(2x + 1)\)

Sum and Difference of Cubes:

  • \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
  • \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Factoring by Grouping:

  • Group terms with common factors
  • Factor each group, then factor the common binomial

Roots and Radicals

Properties of Radicals:

Property Formula Example
Product \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) \(\sqrt{12} = 2\sqrt{3}\)
Quotient \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) \(\sqrt{\frac{16}{4}} = 2\)
Power \(\sqrt[n]{a^m} = a^{m/n}\) \(\sqrt[3]{x^6} = x^2\)

Sign Rules:

  • Even roots: Always positive (\(\sqrt{9} = 3\), not \(-3\))
  • Odd roots: Keep original sign (\(\sqrt[3]{-8} = -2\))

Rationalizing Denominators:

  • Simple: \(\frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}\)
  • Binomials: Use conjugates \((a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2c\)

Substitution for Factorization

Strategy: Replace repeated expressions with a simpler variable to reveal hidden patterns.

Common Patterns:

  • Quadratic in form: \(x^4 - 13x^2 + 36\)
    • Let \(u = x^2\): \(u^2 - 13u + 36 = (u-4)(u-9) = (x^2-4)(x^2-9)\)
  • Repeated expressions: \((2x+1)^2 - 3(2x+1) - 10\)
    • Let \(u = 2x+1\): \(u^2 - 3u - 10 = (u-5)(u+2) = (2x-4)(2x+3)\)
  • Radical expressions: \(x + 6\sqrt{x} + 8\)
    • Let \(u = \sqrt{x}\): \(u^2 + 6u + 8 = (u+2)(u+4) = (\sqrt{x}+2)(\sqrt{x}+4)\)

Steps:

  1. Identify the repeated pattern
  2. Substitute with a simple variable
  3. Factor the resulting expression
  4. Substitute back
  5. Check if further factoring is possible

Logarithms

Definition: If \(a^x = b\), then \(\log_a(b) = x\)

Basic Properties:

  • \(\log_a(1) = 0\) (since \(a^0 = 1\))
  • \(\log_a(a) = 1\) (since \(a^1 = a\))
  • \(\log_a(a^x) = x\) (inverse operations)
  • \(a^{\log_a(x)} = x\) (inverse operations)

Laws of Logarithms:

Rule Formula Example
Product \(\log_a(xy) = \log_a(x) + \log_a(y)\) \(\log(20) = \log(4) + \log(5)\)
Quotient \(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\) \(\log(\frac{100}{4}) = \log(100) - \log(4)\)
Power \(\log_a(x^n) = n\log_a(x)\) \(\log(8^3) = 3\log(8)\)

Change of Base Formula: \[\log_a(x) = \frac{\log_b(x)}{\log_b(a)} = \frac{\ln(x)}{\ln(a)} = \frac{\log(x)}{\log(a)}\]

Common Notations:

  • \(\log(x)\) means \(\log_{10}(x)\) (common logarithm)
  • \(\ln(x)\) means \(\log_e(x)\) (natural logarithm, \(e \approx 2.718\))

Solving Exponential Equations:

  • If bases are the same: \(a^x = a^y \Rightarrow x = y\)
  • If bases differ: Take logarithms of both sides

Pascal’s Triangle and Binomial Expansion

Pascal’s Triangle:

Row 0:              1
Row 1:            1   1
Row 2:          1   2   1
Row 3:        1   3   3   1
Row 4:      1   4   6   4   1
Row 5:    1   5   10  10  5   1

Pattern: Each number = sum of the two numbers above it

Common Expansions:

  • \((a + b)^2 = a^2 + 2ab + b^2\)
  • \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Binomial Theorem: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) (later important!)

Compound Growth & Interest

Core Forms (from Sessions 01-03, 01-05, 01-06):

  • (Discrete once per period) \(A = P(1 + r)^t\)
    • \(P\) principal, \(r\) rate per period, \(t\) number of periods
  • (Compounded \(n\) times per year) \(A = P\left(1 + \frac{r}{n}\right)^{nt}\)
    • \(r\) nominal annual rate, \(n\) compounding frequency (12 monthly, 4 quarterly, etc.)
  • (Continuous compounding) \(A = Pe^{rt}\)