Mathematics

Mathematical principles are not just relevant for examination questions; they are also essential for routine tasks in medicine, such as adjusting medication dosages based on a patient's weight or kidney function. This article provides a selection of mathematical concepts pertinent to health care professionals.

Physical units can be scaled by powers of 10 using standard SI prefixes.

• Powers of 10 with a positive exponent: The exponent indicates how many times 10 is multiplied by itself.
• Powers of 10 with a negative exponent: The exponent indicates division by 10 multiplied by itself that many times.

The rule of three

The rule of three is a mathematical method for solving simple proportional relationships.

• Initial question: If a/b = c/X, what is the value of X?
• Solution strategy 1: the formula
  • Rearrange a/b = c/X to solve for the unknown X → X = (c × b)/a
    • X = unknown value to be calculated, a = first value, b = second value (corresponds to a), c = third value (corresponds to X)
  • Example: If 300 g (a) of cheese costs $1.50 (b), how many dollars (X) does 500 g (c) of cheese cost?
    • Calculation
      • a/b = c/X
      • 300 g/$1.50 = 500 g/X
      • X = (500 g × $1.50)/300 g
      • X = $750/300 g
      • X = $2.50
• Solution strategy 2: the unit rate
  1. Determine the value for a single unit of X.
     • E.g., find the cost of just 1 g of cheese: $1.50/300 g = $0.005 per g
  2. Multiply this unit rate by the desired amount (c).
     • E.g., $0.005 per g × 500 g = $2.50

Calculation examples

• Example 1 (unit rate)
  • An athlete produces 200 W of mechanical power at an efficiency of 20%. What is the athlete's total metabolic power consumption?
    • Given: 20% of total power corresponds to 200 W
    • Question: How much power would be produced at 100% efficiency?
    • Solution
      1. Divide the amount of power at 20% by 20 to find the power at 1% (unit rate): 200 W/20 = 10 W.
      2. Multiply the unit rate by 100 to find the power at 100%: 10 W × 100 = 1000 W.
• Example 2 (formula): A patient needs a 30 mg dose of a painkiller. The drug is available in a solution with a concentration of 50 mg per 2 mL.
  • Given: a = 50 mg, b = 2 mL, c = 30 mg, X = ? mL
  • Question: How many mL should be administered?
  • Solution (direct formula)
    • a/b = c/X → X = (c × b)/a
    • X = (30 mg × 2 mL)/50 mg
    • X = 1.2 mL
  • Solution (cross multiplication)
    • a/b = c/X
    • 50 mg/2 mL = 30 mg/X mL
    • Cross-multiply: (50 mg) × (X mL) = (2 mL) × (30 mg)
    • Solve for X: X = (2 mL) × (30 mg)/50 mg = 60/50 mL = 1.2 mL

When describing a force, specifying its magnitude is not enough; its direction is equally important.

• Vectors: quantities that have both magnitude and direction (e.g., force, velocity, displacement, acceleration)
• Scalars: quantities that have magnitude but no direction (e.g., mass, distance, speed)

Vector components and resolution

• By placing a vector on a coordinate axis, it can be visualized as the hypotenuse of a right triangle.
  • Vector resolution: A vector can be broken down into two components (the legs of the right triangle).
    • The horizontal component (V_x) is the projection of the vector onto the x-axis.
    • The vertical component (V_y) is the projection of the vector onto the y-axis.
  • If the vector's magnitude (V) and its angle (θ) relative to the horizontal axis are known, trigonometry can be used to find the magnitude of the x and y components.
    • x-component (V_x) = V × cos(θ)
    • y-component (V_y) = V × sin(θ)
  • If the magnitudes of the x and y components are known, the Pythagorean theorem can be used to find the magnitude of the original vector: V² = V_x² + V_y².

Vector addition

When adding vectors, both their magnitudes and directions must be considered. The sum of two or more vectors is known as the resultant vector (R).

Graphical method (tip-to-tail)

The tail (start) of the second vector is placed at the tip (arrowhead) of the first vector. The resultant vector (R) is then drawn from the tail of the first vector to the tip of the second, forming the third side of a triangle.

Analytical method (using components)

1. Formula: R = A + B
2. Resolve each vector into its horizontal (x) and vertical (y) components using trigonometry.
   • For vector A at its angle, θ_A:
     • x-component: A_x = A × cos(θ_A); also written as A_x = ∣A∣cos(θ_A)
     • y-component: A_y = A × sin(θ_A); also written as A_y = ∣A∣sin(θ_A)
   • For vector B at its own angle, θ_B:
     • x-component: B_x = ∣B∣cos(θ_B)
     • y-component: B_y = ∣B∣sin(θ_B)
3. Sum: Add the components in each direction separately to find the components of the resultant vector (R).
   • Resultant x-component: R_x = A_x + B_x +…
   • Resultant y-component: R_y = A_y + B_y +…
4. Recompose: Use the resultant's components (R_x and R_y) to find its final magnitude and direction.
   • Magnitude (R)
     • Use the Pythagorean theorem: R² = R_x² + R_y² → R = √(R_x² + R_y²).
   • Direction (θ)
     • Use the inverse tangent: θ = tan^-1(R_y/R_x).

Special case: When two vectors of equal magnitude are 120° apart, their resultant vector has the same magnitude because their tip-to-tail arrangement forms an equilateral triangle.

Example calculation: vector addition

• Given: vector A = (3, 4) and vector B = (1, 2)
• Identify the components.
  • Vector A: A_x = 3 and A_y = 4
  • Vector B: B_x = 1 and B_y = 2
• Apply the component method.
  • Substitute the components into the formula: R = A + B = (A_x+ B_x, A_y + B_y) = (3 + 1, 4 + 2)
  • Perform the calculations.
    • R_x = 3 + 1 = 4
    • R_y = 4 + 2 = 6
  • R = (4, 6)
• Magnitude of the resultant vector: ∣R∣ = √(R_x²+ R_y²) = √(4² + 6²) = √(16 + 36) = √52 ≈ 7.21
• Direction (angle θ) of the resultant vector: tan(θ) = R_y/R_x → θ = tan^-1(6/4) ≈ tan^-1 (1.5) ≈ 56.31°

Vector subtraction

• Vector subtraction is the addition of a negative vector. The vector -B has the same magnitude as B but points in the opposite direction (180° away).
  • Tip-to-tail method
    • Draw vector A.
    • Reverse vector B to get -B.
    • Place the tail of -B at the tip of A.
    • Draw the resultant vector R from the tail of A to the tip of -B.
  • Algebraic calculation using the component method: if A = (A_x, A_y) and B = (B_x, B_y), then A - B = (A_x - B_x, A_y - B_y)

Example calculation: vector subtraction

• Given: vector A = (3, 4) and vector B = (1, 2)
• Identify the components.
  • Vector A: A_x = 3 and A_y = 4
  • Vector B: B_x = 1 and B_y = 2
• Apply the component method.
  • Substitute the components into the formula: R = A - B = (A_x - B_x, A_y - B_y) = (3 - 1, 4 - 2)
  • Perform the calculations.
    • R_x = 3 - 1 = 2
    • R_y = 4 - 2 = 2
  • R = (2, 2)

Vector multiplication

Vector multiplication can be performed in two ways: the dot product and the cross product.

Dot product

• Results in a scalar quantity and measures the extent to which two vectors point in the same direction
• Definition: A ⋅ B = |A||B|cos(θ)
  • ∣A∣ = magnitude of vector A, ∣B∣ = magnitude of vector B, cos(θ) = ratio of the adjacent side to the hypotenuse in a right triangle (unitless), θ = angle between the two vectors
• Formula: If A = (A_x, A_y) and B = (B_x, B_y), A ⋅ B = A_xB_x + A_yB_y.
• Use case: calculating work done by a force (W = F⋅d = ∣F∣∣d∣cos(θ))
  • W = work (J), F = force (N), d = displacement (m), θ = angle between the force vector (F) and the displacement vector (d)

Example calculation: dot product

• Given: A = (3, 4) and B = (1, 2)
• Identify the components.
  • Vector A: A_x = 3 and A_y = 4
  • Vector B: B_x = 1 and B_y = 2
• Perform the calculation: A ⋅ B = (3)(1) + (4)(2) = 3 + 8 = 11

Cross product

• Results in a vector that is perpendicular to both of the original vectors and measures the area of the parallelogram formed by two vectors
• Definition: A × B = |A||B|sin(θ)
  • ∣A∣ = magnitude of vector A, ∣B∣ = magnitude of vector B, sin(θ) = ratio of the length of the opposite side to the hypotenuse in a right triangle (unitless), θ = angle between the two vectors
  • The direction of the resulting vector is found using the right-hand rule.
• Use cases
  • Calculate magnetic force on a charged particle moving in a magnetic field: ∣FB∣ = ∣qv × B∣ = ∣q∣∣v∣∣B∣sin(θ)
    • ∣FB∣ = magnitude of the magnetic force (N), ∣q∣ = magnitude of the charge (C), ∣v∣ = magnitude of the velocity of the charged particle (m/s), ∣B∣ = magnitude of the magnetic field (T), θ = angle between the velocity vector (v) and the magnetic field vector (B; degrees or radians)
  • Calculate torque: τ = rFsin(θ)
    • τ = torque (N·m), r = distance from the pivot point to the point of force application (lever arm; m), F = magnitude of the applied force (N), θ = angle between the force vector and the lever arm (degrees or radians)

Triangles

In any triangle, the sum of the interior angles is 180°. The geometric principles described also apply to vector calculations. Here, the focus is on right triangles.

Pythagorean theorem

In a right triangle (90° angle), the lengths of the sides are related by the Pythagorean theorem.

• Pythagorean theorem: In a triangle with a right angle between sides a and b, the length of the third side, c, is given by a² + b² = c².
• Calculation example: Two velocity vectors, perpendicular to each other, have magnitudes of 0.8 m/s and 0.6 m/s. What is the magnitude of the resultant velocity?
  • Solution: c = √(0.8² + 0.6²) = √(0.64 + 0.36) = √1.00 m/s = 1.00 m/s

Angle calculation

• In a right triangle, there is a fixed relationship between an angle and the ratio of the lengths of any two sides. To describe these relationships, the following terms are used:
  • Hypotenuse: the longest side of the triangle, located opposite the right angle
  • Adjacent side: the side next to angle α that is not the hypotenuse
  • Opposite side: the side across from angle α
• The trigonometric relationships are as follows:
  • Sine: sin(α) = opposite / hypotenuse
  • Cosine: cos(α) = adjacent / hypotenuse
  • Tangent: tan(α) = opposite / adjacent
• This table shows the exact sine, cosine, and tangent values for the most frequently used angles in trigonometry.

• Calculation example: A ladder rests against a wall, making a 60° angle with the ground. The base of the ladder is 1 m from the wall. How long is the ladder?
  • Solution: The wall and the ground form a right angle. The ladder is the hypotenuse. Using the cosine function (cos(α) = adjacent/hypotenuse, which can be rearranged to hypotenuse = adjacent/cos(α)), the ladder's length can be calculated: hypotenuse = 1 m/cos(60°) = 1 m/½ = 2 m.

When a measurement is performed multiple times, the results often vary. This variability can be described mathematically using concepts such as measurement error, mean, and standard deviation, which help assess how well a measurement reflects the true value.

Absolute and relative error

If a measured value differs from a known true value, the difference is called the absolute error. The relative error shows the absolute error in proportion to the true value.

• Absolute error: the difference between the true value and the measured value
  • Δx = x_true - x_measured
• Relative error: the ratio of the absolute error to the true value
  • δx = Δx/x_true

Example

• Absolute measurement error of 100 g
  • If a person weighs 70 kg (i.e., 70,000 g):
    • Relative error = absolute error/true value = 100 g/70,000 g ≈ 0.0014 or 0.14% (negligible difference)
  • If a medication dose is 200 mg (i.e., 0.2 g):
    • Relative error = 100 g/0.2 g = 500 or 50,000% (extreme overdose)

A measurement error of 100 g is often insignificant when weighing a person but could be life-threatening when measuring a medication dose. The absolute error is the same in both cases, but the relative errors differ significantly.

Measures of central tendency and standard deviation

Due to measurement errors, it is often impossible to determine the true value with a single measurement. Instead, the true value is estimated (e.g., using the mean) and the uncertainty of that estimate is quantified (e.g., using the standard deviation).

• Mean: calculated by summing all individual values and dividing by the number of values
  • Example: calculating the mean hair length of 5 medical students
    • 2 cm + 4 cm + 5 cm + 6 cm + 33 cm = 50 cm
    • 50 cm/5 = 10 cm
    • The mean hair length of the 5 students is 10 cm.
• Median: the middle value in a dataset when it is sorted in numerical order; if there is an even number of values, the median is the mean of the two middle values.
  • In the hair length example above (2 cm, 4 cm, 5 cm, 6 cm, 33 cm), the median is 5 cm.
• Mode: The value that appears most frequently in a dataset. A dataset can have one mode, more than one mode, or no mode.
  • In the hair length example above (2 cm, 4 cm, 5 cm, 6 cm, 33 cm), there is no mode, as each value appears only once.
• Variance (statistical): a measure of how spread out the data points are from the mean, calculated as the mean of the squared differences from the mean
  • Example: variance of the hair lengths of 5 medical students
    • Mean: 10 cm
    • Values: 2 cm, 4 cm, 5 cm, 6 cm, 33 cm
    • Number of values: 5
    • Variance = [(10 - 2)² + (10 - 4)² + (10 - 5)² + (10 - 6)² + (10 - 33)²]/5 = (64 + 36 + 25 + 16 + 529) cm²/5 = 670/5 = 134 cm²
  • The differences are squared for two main reasons: to prevent positive and negative deviations from canceling each other out and to give more weight to larger deviations. However, this means that the unit of variance is also squared (e.g., cm²), which can be unintuitive; therefore, variance is often converted to standard deviation.
• Standard deviation (SD): the square root of the variance, measuring the average distance of individual data points from the mean; it also restores the original unit of measurement
  • SD = √variance
  • For the hair length data, the variance is 134 cm², so the standard deviation is √134 cm² ≈ 11.6 cm.
    • This means the hair lengths deviate, on average, by about 11.6 cm from the mean.

For skewed data with outliers (such as the hair length example), the median is often a more representative measure of the “typical” value than the mean because it is less influenced by extreme values.

The Gaussian distribution, also known as the normal distribution, is a probability distribution that is symmetric around the mean, with data near the mean occurring more frequently than data farther away. Many biological variables, such as hemoglobin (Hb) levels, approximate a Gaussian distribution.

Key properties of the Gaussian distribution

• Graphical representation: a Gaussian curve, which is bell-shaped and symmetric around the mean
• Distribution of measured values
  • Due to its symmetry, half of the values lie below the mean, and the other half lie above it.
  • This distribution follows the 68-95-99.7 rule, a key concept for statistical interpretation.
    • Approximately 68% of values fall within one standard deviation of the mean.
    • Approximately 95% of values fall within two standard deviations of the mean; this leaves about 2.5% of values in each tail of the distribution.
    • Approximately 99.7% of values fall within three standard deviations of the mean.
• Application example
  • The mean Hb concentration in a population is 9.5 mmol/L with a standard deviation of 0.6 mmol/L. What is the probability that a randomly selected person has an Hb value below 8.3 mmol/L?
  • Solution
    • The range of ± two standard deviations is 9.5 mmol/L ± (2 × 0.6 mmol/L), which is 8.3–10.7 mmol/L. About 95% of people have an Hb value in this range.
    • Because the distribution is symmetric, the remaining 5% of people are split evenly between the two tails. Therefore, 2.5% of people have an Hb value below 8.3 mmol/L and 2.5% have a value above 10.7 mmol/L.

In a perfect Gaussian distribution, the mean, median, and mode are all identical.

Exponential functions are functions in which the independent variable appears in the exponent. They are used to model processes such as population growth, drug clearance, and radioactive decay. Logarithmic functions are the inverse of exponential functions; they “undo” the exponentiation.

Exponential function

Exponential functions have a constant base raised to a variable exponent.

• Definition: a function of the form y = a^x (here, a is the base and x is the exponent)
• Special case: When the base is Euler's number, e (≈ 2.718), the function is called the natural exponential function (e-function); it has unique properties, such as being its own derivative.
• Rules of calculation
  • a^x × a^y = a^x+y
  • a^x × b^x = (ab)^x
  • (a^x)^y = a^xy
  • a^-x = 1/a^x
  • a^x/a^y = a^x-y
• Calculation examples
  • 10⁵ × 10⁶ = 10¹¹
  • 10⁵ × 2⁵ = 20⁵
  • (10⁵)⁶= 10³⁰
  • 10^-1= 1/10 = 0.1
  • (9 × 10^-4)/(3 × 10^-7) = 3 × 10^{-4 - (-7)} = 3 × 10³

Logarithm

The logarithm is the inverse of an exponential function. The logarithm of a number y to a given base a is the exponent x to which the base must be raised to produce that number y. Logarithms are used to calculate quantities such as pH from proton concentration and the half-life in radioactive decay.

• Definition: a function of the form x = log_a y; it is the inverse of the exponential function y = a^x
• Common types
  • Common logarithm: logarithm to the base 10, abbreviated as “log”
  • Natural logarithm: logarithm to the base e, abbreviated as “ln”
• Rules of calculation: Since a logarithm is an exponent, the rules for logarithms are derived from the rules for exponents.
  • log_a(xy) = log_a x + log_a y
  • log_a(x^y) = y × log_a x
  • log_a a = 1
• Calculation examples
  • log 25 + log 4 = log(25 × 4) = log 100 = 2
  • log₂ 4³ = 3 × log₂ 4 = 3 × 2 = 6
  • log 10 = 1
• Interpretation aid: If the common logarithm of a ratio is 2, then log(a/b) = 2. This means (a/b) = 10², or (a/b) = 100, which implies a = 100 × b.
• Application example
  • The pH scale is defined as pH = -log[H+], where [H+] is the concentration of hydrogen ions in moles per liter (mol/L).
  • Since the pH scale is logarithmic (base 10), each whole-number change on the scale represents a tenfold change in hydrogen ion concentration. E.g.:
    • A solution with a pH of 7 has a certain concentration of hydrogen ions.
    • A solution with a pH of 6 has ten times more hydrogen ions than the solution with a pH of 7.
    • If the concentration of hydrogen ions at pH 7 is 1 × 10^-7 mol/L, the concentration at pH 6 would be 1 × 10^-6 mol/L.

Because pH is a base-10 logarithmic scale (pH = -log[H+]), a change of one pH unit (e.g., from pH 7 to pH 6) represents a tenfold increase in hydrogen ion concentration.

In physics and mathematics, angles are often described using radians instead of degrees. The radian measure relates an angle to the arc length on a unit circle (a circle with a radius of 1). The circumference of the unit circle is 2π, so a full 360° rotation corresponds to 2π radians. Consequently, 180° corresponds to π radians, 90° to π/2 radians, and so on.

• Radian measure: the length of the arc on a unit circle that is subtended by an angle
  • Conversion from degrees to radians: angle in rad = (π/180°) × angle in degrees
  • Conversion from radians to degrees: angle in degrees = (180°/π) × angle in rad
• Example: On a watch, the minute hand moves 15 minutes (α = 90°). The hand is 1 cm long. What distance did the tip of the hand travel?
  • First, convert the angle to radians: angle in rad = (π/180°) × 90° = π/2 ≈ 1.57
  • The arc length, which is the distance traveled by the tip of the hand, is calculated by multiplying the radius (length of the hand) by the angle in radians. Here, the radius is 1 cm. Therefore, the distance traveled by the tip is: 1 cm × 1.57 = 1.57 cm.

The mathematical properties of circles and spheres are important for calculations in medicine, such as finding the cross-sectional area of a blood vessel or the volume of a tumor. Both shapes are defined by their radius (r), which is half of the diameter.

• The constant π
  • Required for many calculations involving circles and spheres
  • π = 3.14159... ≈ 3.14
• Circle
  • Area: A = πr²
  • Circumference: C = 2πr
• Sphere
  • Surface area: A = 4πr²
  • Volume: V = (4/3)πr³

The surface area to volume ratio is essential in cell biology. Many cells are spherical, with the surface area (4πr²) representing the cell membrane, which is critical for nutrient uptake and waste removal. In contrast, the volume (4/3πr³) indicates the cell's metabolic needs. As a cell grows, its volume increases more rapidly than its surface area, which limits transport efficiency. This relationship explains why most cells remain microscopic.