Course Cheatsheet
Section 01: Mathematical Foundations & Algebra
Number Systems
The hierarchy of number systems:
\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{I} \subset \mathbb{R}\]
| Symbol | Name | Definition | Examples |
|---|---|---|---|
| \(\mathbb{N}\) | Natural numbers | \(\{1, 2, 3, ...\}\) | \(1, 7, 42\) |
| \(\mathbb{N}_0\) | Natural with zero | \(\{0, 1, 2, 3, ...\}\) | \(0, 1, 7\) |
| \(\mathbb{Z}\) | Integers | \(\{..., -2, -1, 0, 1, 2, ...\}\) | \(-5, 0, 3\) |
| \(\mathbb{Q}\) | Rationals | \(\{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\}\) | \(\frac{1}{3}, 0.5, -4\) |
| \(\mathbb{I}\) | Irrationals | Cannot be expressed as fractions | \(\pi, \sqrt{2}, e\) |
| \(\mathbb{R}\) | Reals | All points on the number line | All of the above |
Every natural number is also an integer, every integer is also rational, and every rational is also real. For example, \(3 \in \mathbb{N}\) means \(3 \in \mathbb{Z}\), \(3 \in \mathbb{Q}\), and \(3 \in \mathbb{R}\) as well. When classifying a number, list all sets it belongs to.
Repeating Decimals are Rational:
To convert \(0.\overline{36}\) to a fraction:
- Let \(x = 0.363636...\)
- Multiply by \(100\): \(100x = 36.363636...\)
- Subtract: \(99x = 36\), so \(x = \frac{36}{99} = \frac{4}{11}\)
General Rule: For \(0.\overline{abc}\) with \(n\) repeating digits, multiply by \(10^n\), subtract the original, and solve.
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:
| Operation | Symbol | Meaning | Example |
|---|---|---|---|
| Union | \(A \cup B\) | Elements in A or B | \(\{1,2\} \cup \{2,3\} = \{1,2,3\}\) |
| Intersection | \(A \cap B\) | Elements in A and B | \(\{1,2\} \cap \{2,3\} = \{2\}\) |
| Difference | \(A \setminus B\) | In A but not in B | \(\{1,2,3\} \setminus \{2,3\} = \{1\}\) |
| Complement | \(\bar{A}\) | All elements not in A | Requires universal set \(U\) |
Essential Symbols:
| Symbol | Meaning | Example |
|---|---|---|
| \(\in\) / \(\notin\) | Element of / Not element of | \(3 \in \mathbb{N}\), \(\pi \notin \mathbb{Q}\) |
| \(\subset\) / \(\subseteq\) | Proper subset / Subset or equal | \(\mathbb{N} \subset \mathbb{Z}\), \(A \subseteq A\) |
| \(\emptyset\) | Empty set | \(A \cap B = \emptyset\) (disjoint sets) |
| \(\forall\) | For all | \(\forall x \in \mathbb{R}: x^2 \geq 0\) |
| \(\exists\) | There exists | \(\exists x \in \mathbb{Z}: x < 0\) |
| \(\Rightarrow\) | Implies | \(x = 2 \Rightarrow x^2 = 4\) |
| \(\Leftrightarrow\) | If and only if | \(x^2 = 4 \Leftrightarrow x = \pm 2\) |
Interval Notation
| Notation | Type | Meaning |
|---|---|---|
| \([a, b]\) | Closed | \(a \leq x \leq b\) (both endpoints included) |
| \((a, b)\) | Open | \(a < x < b\) (both endpoints excluded) |
| \([a, b)\) | Half-open | \(a \leq x < b\) (left included, right excluded) |
| \((a, b]\) | Half-open | \(a < x \leq b\) (left excluded, right included) |
| \((-\infty, a]\) | Unbounded left | \(x \leq a\) |
| \((a, \infty)\) | Unbounded right | \(x > a\) |
Square bracket \([\) means the endpoint is included (the bracket “grabs” it). Round parenthesis \((\) means the endpoint is excluded. Infinity always uses a round parenthesis because you can never reach infinity.
Commutative, Associative, and Distributive Laws
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | \(a + b = b + a\) | \(a \times b = b \times a\) |
| Associative | \((a + b) + c = a + (b + c)\) | \((a \times b) \times c = a \times (b \times c)\) |
Distributive: \(a(b + c) = ab + ac\)
Non-commutative operations: Subtraction (\(5 - 3 \neq 3 - 5\)), division (\(6 \div 2 \neq 2 \div 6\)), and exponentiation (\(2^3 \neq 3^2\)) are not commutative.
Order of Operations (PEMDAS)
- Parentheses (brackets, braces)
- Exponents (powers, roots)
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Example: \(2 + 3 \times 4^2 - (5 - 3) \div 2\)
- Parentheses: \((5 - 3) = 2\)
- Exponents: \(4^2 = 16\)
- Multiply/Divide: \(3 \times 16 = 48\) and \(2 \div 2 = 1\)
- Add/Subtract: \(2 + 48 - 1 = 49\)
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 \(a \neq 0\))
- \(a^1 = a\)
- \(a^{-n} = \frac{1}{a^n}\)
- \(a^{1/n} = \sqrt[n]{a}\)
- \(a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\)
These are wrong – do not fall into these traps:
- \((x + y)^2 \neq x^2 + y^2\) — Correct: \((x + y)^2 = x^2 + 2xy + y^2\)
- \((x \cdot y)^n \neq x \cdot y^n\) — Correct: \((xy)^n = x^n y^n\)
- \(a^m \cdot b^m \neq (ab)^{2m}\) — Correct: \(a^m \cdot b^m = (ab)^m\)
- \(a^{-n} \neq -a^n\) — Correct: \(a^{-n} = \frac{1}{a^n}\)
Remember: Exponent rules only apply when bases are the same (for product/quotient rules) or when exponents are the same (for product power rule).
Percentage Calculations
Basic Percentage: \(\text{Result} = \frac{x}{100} \times \text{Base}\)
Percentage Change: \(\text{Change \%} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%\)
Compound Growth: \(\text{Final} = \text{Initial} \times (1 + r)^n\) where \(r\) = rate as decimal, \(n\) = periods
Scientific Notation
Format: \(a \times 10^n\) where \(1 \leq |a| < 10\)
| Direction | Rule | Example |
|---|---|---|
| Large numbers | Move decimal left, positive exponent | \(56{,}700{,}000 = 5.67 \times 10^7\) |
| Small numbers | Move decimal right, negative exponent | \(0.00000423 = 4.23 \times 10^{-6}\) |
Operations:
- Multiply: \((3 \times 10^5) \times (2 \times 10^3) = 6 \times 10^8\) (multiply coefficients, add exponents)
- Divide: \(\frac{8.4 \times 10^7}{2.1 \times 10^4} = 4 \times 10^3\) (divide coefficients, subtract exponents)
Absolute Value
Definition: \(|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\)
Think of it as the distance from zero on the number line.
Properties:
- \(|x| \geq 0\) always
- \(|xy| = |x| \cdot |y|\)
- \(|x|^2 = x^2\)
Solving Equations: \(|ax + b| = c\) splits into two cases:
- \(ax + b = c\) and \(ax + b = -c\)
Solving Inequalities:
| Type | Condition | Solution |
|---|---|---|
| Less than | \(|x - a| < d\) | \(a - d < x < a + d\) |
| Greater than | \(|x - a| > d\) | \(x < a - d\) or \(x > a + d\) |
Business Application: Quality tolerance \(|w - 100| \leq 2\) means acceptable range is \([98, 102]\).
Basic Factorization
Always check for common factors first!
- \(12x^3 - 18x^2 + 6x = 6x(2x^2 - 3x + 1)\)
Difference of Squares: \(a^2 - b^2 = (a + b)(a - b)\)
- \(x^2 - 9 = (x + 3)(x - 3)\)
- \(4x^2 - 25 = (2x + 5)(2x - 5)\)
Perfect Square Trinomials:
- \((a + b)^2 = a^2 + 2ab + b^2\)
- \((a - b)^2 = a^2 - 2ab + b^2\)
\(a^2 + b^2\) cannot be factored over the real numbers. Only the difference \(a^2 - b^2\) factors. Do not waste time trying to factor a sum of squares.
Advanced Factorization
Factoring Quadratics (\(a = 1\))
For \(x^2 + bx + c\): find two numbers that multiply to \(c\) and add to \(b\).
- \(x^2 + 7x + 12 = (x + 3)(x + 4)\) because \(3 \times 4 = 12\) and \(3 + 4 = 7\)
The AC Method (\(a \neq 1\))
For \(ax^2 + bx + c\):
- Calculate \(ac\)
- Find factors of \(ac\) that sum to \(b\)
- Rewrite middle term using those factors
- Factor by grouping
Example: \(6x^2 + 13x + 5\)
- \(ac = 30\), factors \((3, 10)\) sum to \(13\)
- \(6x^2 + 3x + 10x + 5 = 3x(2x + 1) + 5(2x + 1) = (3x + 5)(2x + 1)\)
The Discriminant Test
Before attempting to factor \(ax^2 + bx + c\), compute \(\Delta = b^2 - 4ac\):
- If \(\Delta\) is a perfect square (\(0, 1, 4, 9, 16, 25, ...\)), the expression factors over the integers
- If \(\Delta\) is not a perfect square, do not waste time trying to factor
Follow this order when factoring any expression:
- Common factor – always check first
- Recognize special patterns – difference of squares, perfect square trinomials, sum/difference of cubes
- Quadratic with \(a = 1\) – find factors of \(c\) that add to \(b\)
- AC method – when \(a \neq 1\), check discriminant first
- Grouping – for four-term polynomials, group in pairs
- Substitution – when you see repeated expressions or higher powers
- Always check if factors can be factored further
Factoring by Grouping
For four-term polynomials, group terms with common factors:
- \(x^3 + 2x^2 - 3x - 6 = x^2(x + 2) - 3(x + 2) = (x + 2)(x^2 - 3)\)
Sum and Difference of Cubes
| Pattern | Formula |
|---|---|
| Sum of Cubes | \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) |
| Difference of Cubes | \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) |
Examples:
- \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\)
- \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\)
- \(8x^3 + 125 = (2x + 5)(4x^2 - 10x + 25)\)
Roots and Radicals
Properties of Radicals:
| Property | Formula | Example |
|---|---|---|
| Product | \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) | \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\) |
| Quotient | \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) | \(\sqrt{\frac{16}{4}} = \frac{4}{2} = 2\) |
| Power | \(\sqrt[n]{a^m} = a^{m/n}\) | \(\sqrt[3]{x^6} = x^2\) |
Simplifying Strategy: Extract perfect powers from under the radical.
- \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\)
- \(\sqrt{50x^5y^3} = 5x^2y\sqrt{2xy}\)
Combining Radicals: Simplify each term first, then combine like radicals.
- \(3\sqrt{12} + 2\sqrt{27} - \sqrt{48} = 6\sqrt{3} + 6\sqrt{3} - 4\sqrt{3} = 8\sqrt{3}\)
- Even roots (square, fourth, etc.) are always non-negative by convention: \(\sqrt{9} = 3\), not \(-3\)
- Odd roots (cube, fifth, etc.) keep the original sign: \(\sqrt[3]{-8} = -2\)
- You cannot take an even root of a negative number (within the reals): \(\sqrt{-4}\) is undefined in \(\mathbb{R}\)
Rationalizing Denominators
Simple radical: Multiply numerator and denominator by the radical.
\[\frac{3}{\sqrt{5}} = \frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}\]
Binomial with radical: Multiply by the conjugate.
\[\frac{2}{\sqrt{3} + 1} = \frac{2(\sqrt{3} - 1)}{(\sqrt{3})^2 - 1^2} = \frac{2(\sqrt{3} - 1)}{2} = \sqrt{3} - 1\]
The conjugate of \(a + b\sqrt{c}\) is \(a - b\sqrt{c}\). Their product eliminates the radical: \((a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2c\).
Substitution for Factorization
Strategy: Replace repeated or complex sub-expressions with a single variable.
Common Substitution Patterns
| Pattern | Substitution | Example |
|---|---|---|
| \(x^4 + bx^2 + c\) | Let \(u = x^2\) | \(x^4 - 13x^2 + 36 \to u^2 - 13u + 36\) |
| \(x + b\sqrt{x} + c\) | Let \(u = \sqrt{x}\) | \(x + 6\sqrt{x} + 8 \to u^2 + 6u + 8\) |
| \(x^6 + bx^3 + c\) | Let \(u = x^3\) | \(3x^6 - 11x^3 - 20 \to 3u^2 - 11u - 20\) |
| \(x^{2/3} + bx^{1/3} + c\) | Let \(u = x^{1/3}\) | \(x^{2/3} - 5x^{1/3} + 6 \to u^2 - 5u + 6\) |
| \(3^{2x} + b \cdot 3^x + c\) | Let \(u = 3^x\) | \(3^{2x} - 4 \cdot 3^x + 3 \to u^2 - 4u + 3\) |
| \((f(x))^2 + b \cdot f(x) + c\) | Let \(u = f(x)\) | \((2x+1)^2 - 3(2x+1) - 10 \to u^2 - 3u - 10\) |
Steps:
- Identify the repeated building block
- Substitute with a simple variable \(u\)
- Factor the resulting expression
- Substitute back to original variable
- Check if further factoring is possible
Full Example: Factor \(x^4 - 13x^2 + 36\)
- Let \(u = x^2\), so \(u^2 - 13u + 36 = (u - 4)(u - 9)\)
- Substitute back: \((x^2 - 4)(x^2 - 9)\)
- Factor further: \((x - 2)(x + 2)(x - 3)(x + 3)\)
Look for expressions where the same sub-expression appears multiple times, especially when the higher power is the square of the lower power. If you see \(x^4\) and \(x^2\), or \(x^6\) and \(x^3\), or \(\sqrt{x}\) and \(x\), substitution will likely simplify the problem.
Logarithms
Definition: If \(a^x = b\), then \(\log_a(b) = x\)
“What power do I raise \(a\) to in order to get \(b\)?”
Common Notations:
- \(\log(x)\) means \(\log_{10}(x)\) (common logarithm)
- \(\ln(x)\) means \(\log_e(x)\) (natural logarithm, \(e \approx 2.718\))
Basic Properties:
| Property | Formula | Reason |
|---|---|---|
| Log of 1 | \(\log_a(1) = 0\) | Because \(a^0 = 1\) |
| Log of base | \(\log_a(a) = 1\) | Because \(a^1 = a\) |
| Inverse | \(\log_a(a^x) = x\) | Logs undo exponents |
| Inverse | \(a^{\log_a(x)} = x\) | Exponents undo logs |
Laws of Logarithms:
| Rule | Formula | Example |
|---|---|---|
| Product | \(\log_a(xy) = \log_a(x) + \log_a(y)\) | \(\log(20) = \log(4) + \log(5)\) |
| Quotient | \(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\) | \(\log(\frac{100}{4}) = 2 - \log(4)\) |
| Power | \(\log_a(x^n) = n\log_a(x)\) | \(\log(8^3) = 3\log(8)\) |
\(\log(x + y) \neq \log(x) + \log(y)\) — There is no simplification rule for the log of a sum! The product rule says \(\log(x \cdot y) = \log(x) + \log(y)\), which applies to multiplication, not addition.
Change of Base Formula:
\[\log_a(x) = \frac{\log_b(x)}{\log_b(a)} = \frac{\ln(x)}{\ln(a)} = \frac{\log(x)}{\log(a)}\]
Solving Exponential Equations with Logs:
- If bases are the same: \(a^{f(x)} = a^{g(x)} \Rightarrow f(x) = g(x)\)
- If bases differ: take \(\log\) of both sides and use the power rule
Example: Solve \(5^x = 30\)
- \(\log(5^x) = \log(30)\)
- \(x \cdot \log(5) = \log(30)\)
- \(x = \frac{\log(30)}{\log(5)} \approx 2.113\)
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 1Pattern: Each number = sum of the two numbers above it.
Common Expansions:
- \((a + b)^2 = a^2 + 2ab + b^2\)
- \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
- \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)
Binomial Theorem: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
Compound Growth and Interest
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
Continuous compounding: \(A = Pe^{rt}\)
Rule of 72: At \(r\%\) interest, doubling time \(\approx \frac{72}{r}\) years.
Time to Double: \(t = \frac{\ln(2)}{\ln(1 + r)}\)
Quick Reference
| Topic | Key Formula |
|---|---|
| Exponent product | \(a^m \cdot a^n = a^{m+n}\) |
| Exponent quotient | \(\frac{a^m}{a^n} = a^{m-n}\) |
| Power of power | \((a^m)^n = a^{mn}\) |
| Negative exponent | \(a^{-n} = \frac{1}{a^n}\) |
| Fractional exponent | \(a^{m/n} = \sqrt[n]{a^m}\) |
| Difference of squares | \(a^2 - b^2 = (a+b)(a-b)\) |
| Sum of cubes | \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) |
| Difference of cubes | \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) |
| Quadratic formula | \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) |
| Log product | \(\log_a(xy) = \log_a(x) + \log_a(y)\) |
| Log power | \(\log_a(x^n) = n\log_a(x)\) |
| Change of base | \(\log_a(x) = \frac{\ln(x)}{\ln(a)}\) |
| Compound interest | \(A = P(1+r)^t\) |
Problem-Solving Strategies
- Read carefully and identify what type of expression or problem you have
- Simplify first – combine like terms, reduce fractions, simplify radicals
- Look for patterns – difference of squares, perfect squares, cubes, substitution opportunities
- Check your work – expand factored results, verify simplified expressions, substitute back
- Use the discriminant before attempting to factor quadratics
- Convert between forms – radicals to fractional exponents, logs to exponentials, and vice versa
- Exponents: \((x + y)^2 \neq x^2 + y^2\) and \(a^{-n} \neq -a^n\)
- Radicals: \(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\) and \(\sqrt{x^2} = |x|\), not \(x\)
- Logarithms: \(\log(x + y) \neq \log(x) + \log(y)\)
- Factoring: \(a^2 + b^2\) cannot be factored over the reals
- Sign errors: Forgetting to distribute the negative sign when factoring out \(-1\)
- PEMDAS: Multiplication and division have equal precedence (left to right), as do addition and subtraction