Chapter 1: Apply algebra

Biomedical Engineering professionals in Kenya frequently encounter complex systems that require precise quantitative analysis. Algebra forms the foundation for modelling biological processes, analysing medical device performance, and solving engineering problems related to healthcare technology. Mastery of algebraic concepts enables biomedical engineers to interpret data, optimise device parameters, and ensure safety and efficacy in clinical settings.

1.1 Importance of algebra in engineering

Algebra is the language in which biomedical engineers write down how a quantity depends on others. A letter stands for a number that is unknown or can change, and an equation states a relationship between such letters. With it, an engineer can describe a physiological process, a medical device or a prosthetic part in one line, rearrange that line to find the quantity needed, and check that the answer is reasonable. It supports drug dosage calculations, signal processing in imaging equipment and the mechanical analysis of prosthetic devices, and it is the base for every later part of this unit.

From words to equations

A technician usually meets a problem in words: how long will a bag of fluid last, what current flows through a heating element, how heavy is a moulded socket. Algebra turns each sentence into a statement with symbols that can be handled by fixed rules. Once the relationship is written, the same rules solve it for any set of numbers.

Example: an infusion pump delivers fluid at a flow rate Q (mL/h). The volume delivered V (mL) over a time t (h) is related by one formula, and the same formula answers three different questions.

Formula:           Q = V ÷ t

Flow rate:         V = 600 mL, t = 5 h
                   Q = 600 ÷ 5 = 120 mL/h

Time to deliver:   Q = 80 mL/h, V = 240 mL
                   t = V ÷ Q = 240 ÷ 80 = 3 h

Volume delivered:  Q = 60 mL/h, t = 2.5 h
                   V = Q × t = 60 × 2.5 = 150 mL

Answer: the flow rate is 120 mL/h, the time is 3 h and the volume is 150 mL. One formula, rearranged three ways, replaced three separate calculations.

Where biomedical engineers use algebra

• Formulas and rearranging: Equipment manuals often give a formula for one quantity, and the technician needs another. Rearranging V = I × R to find R, or Q = V ÷ t to find t, is a routine algebra skill in the workshop.
• Calibration and measurement: A sensor reading is often a straight-line function of the input. Two known points fix the line, and the line is then used to convert every later reading into a true value. Linear equations do this work.
• Dosage and delivery: Delivery devices work from relationships between dose, body mass, concentration, volume and time. The clinician prescribes, and the engineer checks that the device applies the stated equation correctly.
• Growth and decay: Cell cultures grow, tracer activity falls and drug levels decline in ways described by powers and logarithms. Indices and logarithms give the tools to calculate how many cells there are after a time, or how long a level takes to halve.
• Mechanical design of prostheses and supports: Mass equals density times volume, stress equals force divided by area, and areas and volumes involve squares and cubes of dimensions. Equations of the second degree appear whenever a required area or volume is known and a dimension must be found.
• Signals and decibels: Signal strength comparisons use logarithms, and filter and amplifier behaviour is written as functions of frequency or time.
• Graphs and data: A table of readings becomes useful when it is shown as a graph of a function, so that values can be read between measurements and trends predicted.

A short illustration from device design

Example: the grip of a hand-held scanner is moulded from a plastic of density 1050 kg/m³. The moulded part has a volume of 400 cm³ (0.0004 m³). The designer wants its mass.

Formula:       m = ρ × V
Substitution:  m = 1050 × 0.0004
Working:       m = 0.42

Answer: the mass is 0.42 kg. If the designer instead needs the mass to stay below 0.36 kg, the same formula rearranged as V = m ÷ ρ gives the largest allowed volume: 0.36 ÷ 1050 = 0.000343 m³, or about 343 cm³.

Habits of working with algebra

Careful habits matter in biomedical work, because a wrong figure can reach a patient. Write the formula before substituting numbers. Keep the units with every number and check that both sides of an equation have the same unit. Carry full precision through the working and round only the final answer. Finally, test the result by substituting it back into the original equation, and ask whether it is a sensible size for the device or the body being described.

This chapter builds the tools in order: indices and logarithms for growth, decay and decibels, linear and quadratic equations for solving unknowns, and algebraic functions and their graphs for showing relationships and reading answers from them.

Chapter Summary

This chapter showed how a biomedical engineering technician uses algebra to write a relationship once and then find any quantity in it. It began with the place of algebra in the work: letters stand for quantities that are unknown or change, a formula is rearranged so that the wanted quantity stands alone, units are kept with every number and each answer is checked by substituting it back. It then set out indices and logarithms. The laws of indices add the indices when powers of one base are multiplied, subtract them when the powers are divided and multiply them for a power of a power, and an indicial equation is solved by writing both sides as powers of the same base. A logarithm is an index, so its laws turn products into sums and quotients into differences, a logarithmic equation is solved by moving to index form and rejecting any root that makes a logarithm of a number that is not positive, and the change of base law gives a logarithm in any base from base 10 or base e values on the scientific calculator. Linear equations in one unknown are solved by keeping both sides balanced, and a pair of linear equations is solved by the graphical, substitution or elimination method, each checked in both equations. Quadratic equations in standard form ax² + bx + c = 0 are solved by factoring, by completing the square or by the quadratic formula, the discriminant b² − 4ac shows how many real roots there are, the sum and product of the roots give a check, and a root that cannot occur physically, such as a negative length, is rejected. Finally, the chapter classified functions as linear, quadratic, polynomial, rational and exponential, described their intercepts, turning points and asymptotes, and showed how to set out axes, plot a table of values and read roots, turning points and the answers to engineering questions from a graph.