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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.
Algebra is essential in biomedical engineering for formulating equations that describe physiological phenomena and engineering systems. It supports problem-solving in areas such as drug dosage calculations, signal processing in medical imaging, and mechanical analysis of prosthetic devices. This section introduces fundamental algebraic tools necessary for effective engineering analysis and design.
Indices (or exponents) simplify expression of repeated multiplication, which is common in modelling growth rates, decay processes, and signal amplification in biomedical systems.
The general rule for indices is:
$$a^{m} \times a^{n} = a^{m + n}$$
Example 1: Calculate \(3^{4} \times 3^{2}\).
Given: Base \(a = 3\), powers \(m=4\), \(n=2\).
$$3^{4} \times 3^{2} = 3^{4+2}$$
$$= 3^{6}$$
$$= 729$$
Answer: 729
Example 2: Simplify \( (2^{5})^{3} \).
Given: Base \(a=2\), powers \(m=5\), \(n=3\).
$$(2^{5})^{3} = 2^{5 \times 3}$$
$$= 2^{15}$$
$$= 32768$$
Answer: 32768
Example 3: Calculate \( \frac{5^{7}}{5^{3}} \).
Given: Base \(a=5\), powers \(m=7\), \(n=3\).
$$\frac{5^{7}}{5^{3}} = 5^{7-3}$$
$$= 5^{4}$$
$$= 625$$
Answer: 625
Example 4: Evaluate \( (4^{2} \times 4^{-3}) \).
Given: Base \(a=4\), powers \(m=2\), \(n=-3\).
$$4^{2} \times 4^{-3} = 4^{2 + (-3)}$$
$$= 4^{-1}$$
$$= \frac{1}{4^{1}} = \frac{1}{4}$$
Answer: \(\frac{1}{4}\)
Example 5: Calculate \( 9^{0} \).
Given: Base \(a=9\).
$$9^{0} = 1$$
Answer: 1
Kenyan Context Example:
At Nairobi Hospital, biomedical engineers calculate the mass of prosthetic limbs using the formula \( m = \rho \times V \). For example, if a prosthetic socket is manufactured by Revital Healthcare (Kenya) Ltd with a density of 1100 kg/m\(^3\), the mass calculation ensures the device meets safety standards set by the Kenya Bureau of Standards (KEBS).
Logarithms are the inverse operations of indices, useful in biomedical engineering for analysing exponential growth or decay, such as radioactive tracer decay or population growth of bacteria.
The logarithm formula is:
$$\log_{a} b = c \quad \text{if and only if} \quad a^{c} = b$$
Example 1: Find \(\log_{10} 1000\).
Given: Base \(a=10\), number \(b=1000\).
$$\log_{10} 1000 = c$$
Since \(10^{c} = 1000\),
$$10^{3} = 1000$$
So,
$$c = 3$$
Answer: 3
Example 2: Calculate \(\log_{2} 32\).
Given: Base \(a=2\), number \(b=32\).
$$\log_{2} 32 = c$$
Since \(2^{c} = 32\),
$$2^{5} = 32$$
So,
$$c = 5$$
Answer: 5
Example 3: Evaluate \(\log_{5} 1\).
Given: Base \(a=5\), number \(b=1\).
Since any number raised to 0 equals 1,
$$\log_{5} 1 = 0$$
Answer: 0
Example 4: Calculate \(\log_{10} 0.01\).
Given: Base \(a=10\), number \(b=0.01\).
$$\log_{10} 0.01 = c$$
Since \(10^{c} = 0.01\),
$$10^{-2} = 0.01$$
So,
$$c = -2$$
Answer: -2
Example 5: Find \(\log_{3} 81\).
Given: Base \(a=3\), number \(b=81\).
$$\log_{3} 81 = c$$
Since \(3^{c} = 81\),
$$3^{4} = 81$$
So,
$$c = 4$$
Answer: 4
Biomedical engineers use calculators to efficiently perform complex algebraic and numerical calculations, ensuring accuracy in device calibration and data analysis.
Example 1: Use a calculator to compute \( (1.25 \times 10^{3}) \times (3.6 \times 10^{2}) \).
Given: Numbers \(1.25 \times 10^{3}\) and \(3.6 \times 10^{2}\).
Calculate:
$$1.25 \times 10^{3} \times 3.6 \times 10^{2} = (1.25 \times 3.6) \times 10^{3+2}$$
$$= 4.5 \times 10^{5}$$
Answer: 450000
Example 2: Calculate \( \sqrt{625} \).
Using calculator:
$$\sqrt{625} = 25$$
Answer: 25
Example 3: Compute \( \log_{10} 500 \).
Using calculator:
$$\log_{10} 500 = 2.69897$$
Answer: 2.69897
Example 4: Calculate \( \sin 30^{\circ} \).
Using calculator:
$$\sin 30^{\circ} = 0.5$$
Answer: 0.5
Example 5: Find \( \frac{7.2}{0.03} \) using calculator.
Calculate:
$$\frac{7.2}{0.03} = 240$$
Answer: 240
Whole numbers include zero and all positive integers, used in biomedical engineering for counting discrete items like cells, devices, or samples.
Example 1: Add whole numbers 125 and 375.
Calculate:
$$125 + 375 = 500$$
Answer: 500
Example 2: Subtract 250 from 600.
Calculate:
$$600 - 250 = 350$$
Answer: 350
Example 3: Multiply 15 by 20.
Calculate:
$$15 \times 20 = 300$$
Answer: 300
Example 4: Divide 144 by 12.
Calculate:
$$\frac{144}{12} = 12$$
Answer: 12
Example 5: Calculate \( (7 \times 8) + (9 - 4) \).
Calculate:
$$7 \times 8 = 56$$
$$9 - 4 = 5$$
$$56 + 5 = 61$$
Answer: 61
Odd numbers are integers not divisible by 2, relevant in biomedical engineering when designing alternating patterns or sequences.
Example 1: Identify if 57 is odd.
Since \(57 \div 2 = 28.5\) (not an integer), 57 is odd.
Answer: 57 is odd
Example 2: Sum of first five odd numbers: 1, 3, 5, 7, 9.
Calculate:
$$1 + 3 + 5 + 7 + 9 = 25$$
Answer: 25
Example 3: Multiply two odd numbers: 7 and 9.
Calculate:
$$7 \times 9 = 63$$
Answer: 63
Example 4: Find the next odd number after 101.
Add 2:
$$101 + 2 = 103$$
Answer: 103
Example 5: Check if 1001 is odd.
Since \(1001 \div 2 = 500.5\), 1001 is odd.
Answer: 1001 is odd
Even numbers are integers divisible by 2, often used in biomedical engineering for symmetric designs or sample grouping.
Example 1: Check if 84 is even.
Since \(84 \div 2 = 42\) (integer), 84 is even.
Answer: 84 is even
Example 2: Sum of first five even numbers: 2, 4, 6, 8, 10.
Calculate:
$$2 + 4 + 6 + 8 + 10 = 30$$
Answer: 30
Example 3: Multiply two even numbers: 12 and 16.
Calculate:
$$12 \times 16 = 192$$
Answer: 192
Example 4: Find the next even number after 44.
Add 2:
$$44 + 2 = 46$$
Answer: 46
Example 5: Check if 1000 is even.
Since \(1000 \div 2 = 500\), 1000 is even.
Answer: 1000 is even
Prime numbers have only two distinct positive divisors: 1 and itself. They are important in cryptography and secure data transmission in biomedical informatics.
Example 1: Check if 29 is prime.
Divisors to check: 2, 3, 5.
\(29 \div 2 = 14.5\) no
\(29 \div 3 = 9.67\) no
\(29 \div 5 = 5.8\) no
No divisors found; 29 is prime.
Answer: 29 is prime
Example 2: Is 51 prime?
Check divisors 2,3,5,7.
Not prime.
Answer: 51 is not prime
Example 3: Is 97 prime?
Check divisors 2,3,5,7.
No exact division.
97 is prime.
Answer: 97 is prime
Example 4: Is 1 prime?
No, 1 has only one divisor.
Answer: 1 is not prime
Example 5: Is 53 prime?
Check divisors 2,3,5,7.
No exact division.
53 is prime.
Answer: 53 is prime
Ratio and proportion are fundamental in biomedical calculations such as drug formulation, mixture preparation, and scaling device dimensions.
Ratio expresses the relative size of two quantities, proportion states equality of two ratios.
Ratio formula:
$$\text{Ratio} = \frac{a}{b}$$
Proportion formula:
$$\frac{a}{b} = \frac{c}{d}$$
Example 1: Find the ratio of 50 ml to 200 ml of a solution.
Calculate:
$$\frac{50}{200} = \frac{1}{4}$$
Answer: 1:4
Example 2: Solve for \(x\) if \(\frac{3}{4} = \frac{x}{16}\).
$$\frac{3}{4} = \frac{x}{16}$$
Cross multiply:
$$3 \times 16 = 4 \times x$$
$$48 = 4x$$
$$x = \frac{48}{4} = 12$$
Answer: 12
Example 3: A mixture requires a ratio of 5:3 of chemical A to B. If 40 ml of A is used, find volume of B.
$$\frac{5}{3} = \frac{40}{x}$$
Cross multiply:
$$5x = 3 \times 40$$
$$5x = 120$$
$$x = \frac{120}{5} = 24$$
Answer: 24 ml
Example 4: If \( \frac{2}{x} = \frac{5}{20} \), find \(x\).
Cross multiply:
$$2 \times 20 = 5 \times x$$
$$40 = 5x$$
$$x = \frac{40}{5} = 8$$
Answer: 8
Example 5: The ratio of patients to doctors in a hospital is 12:1. If there are 120 patients, how many doctors?
$$\frac{12}{1} = \frac{120}{x}$$
Cross multiply:
$$12x = 120$$
$$x = \frac{120}{12} = 10$$
Answer: 10 doctors
Percentages express quantities as parts per hundred, critical in biomedical engineering for calculating concentration, efficiency, and error margins.
Percentage formula:
$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$$
Example 1: Find 25% of 200 ml fluid.
Calculate:
$$\frac{25}{100} \times 200 = 50 \text{ ml}$$
Answer: 50 ml
Example 2: A device has 95% efficiency. If input power is 150 W, find output power.
Calculate:
$$\frac{95}{100} \times 150 = 142.5 \text{ W}$$
Answer: 142.5 W
Example 3: Increase 120 mg drug dose by 15%.
Calculate increase:
$$\frac{15}{100} \times 120 = 18 \text{ mg}$$
New dose:
$$120 + 18 = 138 \text{ mg}$$
Answer: 138 mg
Example 4: Decrease 500 ml solution by 10%.
Calculate decrease:
$$\frac{10}{100} \times 500 = 50 \text{ ml}$$
New volume:
$$500 - 50 = 450 \text{ ml}$$
Answer: 450 ml
Example 5: What percentage of 250 mg is 50 mg?
Calculate:
$$\frac{50}{250} \times 100 = 20\%$$
Answer: 20%
Natural numbers are positive integers used in biomedical engineering for counting cells, patients, or devices.
Example 1: A hospital has 120 patients. If 45 patients recover, what fraction recovered?
Calculate:
$$\frac{45}{120} = \frac{3}{8}$$
Answer: \(\frac{3}{8}\)
Example 2: A lab processes 15 samples per day. How many in 6 days?
Calculate:
$$15 \times 6 = 90$$
Answer: 90 samples
Example 3: A prosthetic limb costs Ksh 25,000. If a patient pays Ksh 10,000 upfront, how much remains?
Calculate:
$$25,000 - 10,000 = 15,000$$
Answer: Ksh 15,000
Example 4: A sensor records 200 heartbeats per minute. How many in 5 minutes?
Calculate:
$$200 \times 5 = 1000$$
Answer: 1000 heartbeats
Example 5: A clinic has 8 doctors and 40 nurses. What is the ratio of nurses to doctors?
Calculate:
$$\frac{40}{8} = 5$$
Answer: 5:1
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Create a free accountThis chapter begins by highlighting the importance of algebra in engineering, emphasizing its role in problem-solving and analysis. It then covers indices and logarithms, explaining bases, indices, their laws, and solving related equations, including the use of calculators for efficient computation. Linear equations are introduced with their definitions, standard forms, and various solution methods such as graphical, substitution, and elimination, followed by their practical applications in engineering contexts. Quadratic equations are explored next, detailing different solving techniques like factoring, completing the square, and the quadratic formula, along with real-world engineering examples. The chapter proceeds to discuss different types of algebraic functions, their properties, and how to plot and interpret their graphs using the Cartesian coordinate system to address engineering problems. It also reviews geometric concepts involving areas and perimeters of basic and composite shapes, extending to surface area and volume calculations for various solids, including composite solids with engineering applications. Further topics include the calculation of areas of sectors and segments of circles, operations with whole numbers, fractions, decimals, and arithmetic operations, concluding with methods for finding squares and square roots, all framed within practical engineering problem-solving scenarios.
Calculate \(3^{4} \times 3^{2}\) and express the answer as a single power of 3. (1 mark)
Solve for \(x\) in the equation \(2^{x} = 32\), where \(x\) is an integer. (1 mark)
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