Math & Science

PEMDAS / BODMAS

The order of operations is the set of rules that tell you which part of an expression to evaluate first. Every calculator and programming language follows the same hierarchy:

1. P/B — Parentheses / Brackets — Evaluate innermost first
2. E/O — Exponents / Orders — Powers and roots (right-to-left)
3. M/D — Multiply & Divide — Left to right, equal priority
4. A/S — Add & Subtract — Left to right, equal priority

Common Mistakes

• Multiplication vs Implied
  2(3) means 2×3 = 6. The implied multiplication has the same priority as explicit ×.
• Unary Minus
  −3² = −9, not 9. The exponent applies to 3 first, then the negative sign.
• Left-to-Right Rule
  12 ÷ 4 × 3 = 3 × 3 = 9, NOT 12 ÷ 12 = 1. Division and multiplication are evaluated left to right.

SOH-CAH-TOA

For the unit circle, sin θ = y-coordinate, cos θ = x-coordinate. tan θ = sin θ / cos θ.

Key Angle Values

Radians vs. Degrees in Calculus

While degrees are standard in geometry, calculus and advanced math exclusively use radians. A radian is the angle subtended by an arc equal in length to the radius (1 rad ≈ 57.3°). Why? Because the beautiful simplicity of derivative formulas like d/dx(sin x) = cos x only holds true when x is in radians. If using degrees, d/dx(sin x) = (π/180)cos x, which makes the math needlessly messy.

Measures of Central Tendency

• Mean (Average) — Sum of all values ÷ count. Sensitive to outliers.
• Median — Middle value when sorted. Robust against outliers. For even n, average the two middle values.
• Mode — Most frequently occurring value. A dataset can have multiple modes (bimodal, multimodal) or none.
• Geometric Mean — ⁿ√(x₁×x₂×…×xₙ) — Used for growth rates and ratios.

Measures of Spread

Why n-1? Bessel's correction. Using n-1 in the denominator for sample SD gives an unbiased estimator of the population variance. Always use sample SD unless you have the entire population.

Key Operations

• Addition / Subtraction — Only same-dimension matrices. Add element by element.
• Multiplication (A×B) — Columns of A must equal rows of B. Result is m×p if A is m×n and B is n×p.
• Transpose (Aᵀ) — Rows become columns. If A is m×n, Aᵀ is n×m.
• Determinant — A scalar that encodes the scaling factor of the linear transformation. det = 0 means the matrix is singular (non-invertible).
• Inverse (A⁻¹) — A × A⁻¹ = I. Only exists if det(A) ≠ 0.

2×2 Determinant

2×2 Inverse

The trace of a matrix = sum of diagonal elements. It equals the sum of eigenvalues and is invariant under similarity transformations.

Mechanics

• F = ma — Newton's 2nd Law. Force = mass × acceleration (N)
• v = u + at — Final velocity = initial + (accel × time)
• s = ut + ½at² — Displacement under constant acceleration
• v² = u² + 2as — Velocity without time
• p = mv — Linear momentum (kg·m/s)
• F = mv²/r — Centripetal force (circular motion)

Energy & Electricity

• KE = ½mv² — Kinetic energy (Joules)
• PE = mgh — Gravitational potential energy
• W = Fd·cos� — Work done by a force
• P = W/t — Power = work per unit time (Watts)
• V = IR — Ohm's Law: Voltage = Current × Resistance
• P = IV — Electric power = current × voltage

Why Percent Error Matters

In scientific experiments, the Percent Error indicates how far an observed (experimental) value deviates from the expected (theoretical) value. It is calculated by dividing the absolute error by the exact value and multiplying by 100. High percent error often indicates systemic flaws in experimental design or equipment calibration.

Solving Proportional Ratios

A ratio expresses the proportional relationship between two or more quantities. By setting two ratios equal to each other (e.g., A : B = C : D), you create a proportion. If three of the values are known, you can solve for the missing fourth value using simple cross-multiplication.