Overview
This lecture covers essential math concepts and formulas for ultrasound physics, focusing on manipulating equations, understanding relationships, unit conversions, and powers, all foundational for later study and board exams.
Formula Manipulation & Algebra
- Formulas typically have one variable isolated (e.g., A = B/C).
- To solve for a different variable, perform the same mathematical operations on both sides.
- Inverting operations (multiplying/dividing) helps isolate the target variable.
- Always maintain the relationships between variables when rearranging formulas.
- Check your manipulated formula by plugging in sample numbers.
Describing Relationships in Formulas
- Types of relationships: unrelated, related, directly related/proportional, inversely related/proportional, reciprocal.
- Direct relationship: both variables increase or decrease together.
- Inverse relationship: one variable increases while the other decreases.
- Reciprocal: two factors multiplied together equal one (e.g., a × 1/a = 1).
Rules for Recognizing Relationships in Equations
- For products (A × B = C): Each factor is directly related to the product.
- For quotients (A = B/C): Numerator is directly, denominator is inversely related to the quotient.
- Changing one variable affects the product/quotient according to these rules.
Units and Dimensional Analysis
- Always specify units for numerical answers (e.g., meters, seconds).
- Dimensional units include length (m, cm), area (cm²), volume (cm³), time (s, ms), velocity (m/s), frequency (Hz, MHz), and percent (%).
- Use complementary units for related measurements (e.g., MHz and μs).
Metric System & Unit Conversion
- Metric prefixes represent powers of ten (e.g., kilo = 10³, centi = 10⁻²).
- Use a metric staircase to move decimal points for conversions between units.
- Move decimal right for large to small units; left for small to large.
- Practice converting both up and down the metric scale.
Powers of Ten & Scientific Notation
- Positive exponent: move decimal right, number increases.
- Negative exponent: move decimal left, number decreases.
- Scientific notation writes numbers as a × 10ⁿ (n is the number of decimal moves).
- Ten to the zero power (10⁰) equals one.
Exponents & Powers
- Raising a base to a positive exponent multiplies it multiple times (e.g., 2³ = 2 × 2 × 2).
- Raising to a negative exponent means 1 divided by the base to the positive exponent (e.g., 2⁻² = 1/4).
- "Squared" means exponent 2; "cubed" means exponent 3.
Fractions & Decimals
- Convert fractions to decimals using long division (numerator ÷ denominator).
- Be prepared to interpret and convert repeating or terminating decimals.
Graph Axes
- X-axis: horizontal, Y-axis: vertical, Z-axis: depth (for 3D).
- Used for mapping data (e.g., time vs. velocity in ultrasound).
Key Terms & Definitions
- Numerator — The top value in a fraction/division.
- Denominator — The bottom value in a fraction/division.
- Directly related — Both variables move in the same direction.
- Inversely related — Variables move in opposite directions.
- Reciprocal — Two numbers whose product is 1.
- Metric Prefix — A symbol added to a unit to indicate a multiple/fraction of ten.
- Scientific notation — Expressing numbers as a × 10ⁿ.
Action Items / Next Steps
- Create and update a personal formula sheet with definitions and units.
- Complete unit conversion and scientific notation practice problems in your workbook.
- Review metric prefixes and their exponents for memorization.
- Answer nerd check questions to prepare for your quiz.