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Apply number systems in biomedical engineering is fundamental for interpreting measurements, calculating dosages, and analyzing data accurately. Precision in rounding numbers affects medical device calibration and patient safety. Understanding types of numbers, ratios, and percentages aids in diagnostic imaging, lab result interpretation, and biomedical signal processing.
Biomedical engineers often work with various types of numbers to quantify biological phenomena, calibrate instruments, and analyze data. This section covers rounding numbers to different place values, classification of numbers, and applications of ratios and percentages in biomedical contexts.
Rounding numbers to the nearest tens simplifies data for quick estimation and communication, especially when dealing with large datasets from medical tests or measurements.
The rule for rounding to the nearest tens is:
$$ \text{Rounded number} = 10 \times \text{round}(\frac{\text{original number}}{10}) $$
Example 1: A biomedical sensor reading is 134 units. Round to the nearest tens.
Given: 134 units
$$ \text{Rounded} = 10 \times \text{round}(\frac{134}{10}) $$
$$ = 10 \times \text{round}(13.4) $$
$$ = 10 \times 13 = 130 $$
Answer: 130 units
Example 2: A lab measurement shows 167.8 mg. Round to the nearest tens.
Given: 167.8 mg
$$ \text{Rounded} = 10 \times \text{round}(\frac{167.8}{10}) $$
$$ = 10 \times \text{round}(16.78) $$
$$ = 10 \times 17 = 170 $$
Answer: 170 mg
Example 3: A hospital inventory count is 2453 devices. Round to the nearest tens.
Given: 2453 devices
$$ \text{Rounded} = 10 \times \text{round}(\frac{2453}{10}) $$
$$ = 10 \times \text{round}(245.3) $$
$$ = 10 \times 245 = 2450 $$
Answer: 2450 devices
Rounding to the nearest hundreds is useful when dealing with large quantities such as hospital budgets or patient data summaries.
Rule:
$$ \text{Rounded number} = 100 \times \text{round}(\frac{\text{original number}}{100}) $$
Example 1: The total cost of biomedical equipment is Ksh 4789. Round to the nearest hundreds.
Given: Ksh 4789
$$ \text{Rounded} = 100 \times \text{round}(\frac{4789}{100}) $$
$$ = 100 \times \text{round}(47.89) $$
$$ = 100 \times 48 = 4800 $$
Answer: Ksh 4800
Example 2: A research lab has 12,345 samples. Round to the nearest hundreds.
Given: 12,345 samples
$$ \text{Rounded} = 100 \times \text{round}(\frac{12345}{100}) $$
$$ = 100 \times \text{round}(123.45) $$
$$ = 100 \times 123 = 12300 $$
Answer: 12,300 samples
Example 3: A hospital processes 87,654 test results annually. Round to the nearest hundreds.
Given: 87,654 test results
$$ \text{Rounded} = 100 \times \text{round}(\frac{87654}{100}) $$
$$ = 100 \times \text{round}(876.54) $$
$$ = 100 \times 877 = 87700 $$
Answer: 87,700 test results
Large-scale biomedical data or budgets require rounding to thousands, millions, or billions for clarity.
Rules:
Nearest thousands:
$$ \text{Rounded} = 1000 \times \text{round}(\frac{\text{number}}{1000}) $$
Nearest millions:
$$ \text{Rounded} = 1,000,000 \times \text{round}(\frac{\text{number}}{1,000,000}) $$
Nearest billions:
$$ \text{Rounded} = 1,000,000,000 \times \text{round}(\frac{\text{number}}{1,000,000,000}) $$
Example 1: A county hospital budget is Ksh 3,456,789. Round to the nearest thousands.
Given: Ksh 3,456,789
$$ \text{Rounded} = 1000 \times \text{round}(\frac{3456789}{1000}) $$
$$ = 1000 \times \text{round}(3456.789) $$
$$ = 1000 \times 3457 = 3,457,000 $$
Answer: Ksh 3,457,000
Example 2: The national health research fund is Ksh 12,345,678. Round to the nearest millions.
Given: Ksh 12,345,678
$$ \text{Rounded} = 1,000,000 \times \text{round}(\frac{12345678}{1,000,000}) $$
$$ = 1,000,000 \times \text{round}(12.345678) $$
$$ = 1,000,000 \times 12 = 12,000,000 $$
Answer: Ksh 12,000,000
Example 3: The government allocates Ksh 5,678,900,000 for medical infrastructure. Round to the nearest billions.
Given: Ksh 5,678,900,000
$$ \text{Rounded} = 1,000,000,000 \times \text{round}(\frac{5678900000}{1,000,000,000}) $$
$$ = 1,000,000,000 \times \text{round}(5.6789) $$
$$ = 1,000,000,000 \times 6 = 6,000,000,000 $$
Answer: Ksh 6,000,000,000
Whole numbers are non-negative integers used to count discrete biomedical items like blood samples, devices, or patients.
Definition:
Whole numbers = {0, 1, 2, 3, ...}
Example 1: Count the number of biomedical devices in a hospital: 150.
No calculation needed, 150 is a whole number.
Example 2: Number of patients admitted: 0 (no patients).
0 is a whole number.
Example 3: A lab records 234 test results. Confirm it is a whole number.
234 is a whole number.
Odd numbers are integers not divisible by 2, often used in sample labeling or sequencing in biomedical experiments.
Definition:
An integer \(n\) is odd if
$$ n = 2k + 1, \quad k \in \mathbb{Z} $$
Example 1: Is 157 an odd number?
Calculate \(157 \mod 2\)
$$ 157 \mod 2 = 1 $$
Since remainder is 1, 157 is odd.
Example 2: Identify the 5th odd number starting from 1.
The \(n\)th odd number is \(2n - 1\).
$$ 2 \times 5 - 1 = 9 $$
Answer: 9
Example 3: Count the odd numbers between 10 and 20.
Odd numbers: 11, 13, 15, 17, 19 (5 numbers).
Even numbers are integers divisible by 2, used in grouping samples or devices in pairs.
Definition:
An integer \(n\) is even if
$$ n = 2k, \quad k \in \mathbb{Z} $$
Example 1: Is 200 an even number?
Calculate \(200 \mod 2\)
$$ 200 \mod 2 = 0 $$
Since remainder is 0, 200 is even.
Example 2: Find the 7th even number starting from 2.
The \(n\)th even number is \(2n\).
$$ 2 \times 7 = 14 $$
Answer: 14
Example 3: Count even numbers between 1 and 10.
Even numbers: 2, 4, 6, 8, 10 (5 numbers).
Prime numbers have exactly two distinct positive divisors: 1 and themselves. They are important in cryptography for medical data security.
Definition:
A prime number \(p > 1\) satisfies:
No integer \(d\) where \(1 < d < p\) divides \(p\).
Example 1: Is 29 a prime number?
Check divisibility by primes less than \(\sqrt{29} \approx 5.38\): 2, 3, 5
No divisor found; 29 is prime.
Example 2: Is 51 prime?
Check divisibility:
51 is not prime.
Example 3: Find the next prime after 97.
Check 98 (even), 99 (divisible by 3), 100 (even), 101.
101 is prime.
Ratios compare quantities; proportions express equality of two ratios. Used to analyze biomedical data like drug concentrations or biomarker levels.
Definition:
Ratio of \(a\) to \(b\):
$$ \frac{a}{b} $$
Proportion:
$$ \frac{a}{b} = \frac{c}{d} $$
Example 1: In a sample, 40 cells are healthy and 60 are infected. Find the ratio healthy:infected.
Given: healthy = 40, infected = 60
$$ \text{Ratio} = \frac{40}{60} = \frac{2}{3} $$
Answer: 2:3
Example 2: If 5 ml of drug A is mixed with 15 ml of drug B, what is the proportion of A to B?
Given: A = 5 ml, B = 15 ml
$$ \frac{5}{15} = \frac{1}{3} $$
Answer: 1:3
Example 3: Solve for \(x\) if \(\frac{2}{5} = \frac{x}{20}\).
$$ \frac{2}{5} = \frac{x}{20} $$
Cross multiply:
$$ 2 \times 20 = 5 \times x $$
$$ 40 = 5x $$
$$ x = \frac{40}{5} = 8 $$
Answer: x = 8
Percentages express a fraction out of 100, useful for calculating patient recovery rates, test accuracy, or concentration levels.
Definition:
Percentage of \(a\) in \(b\):
$$ \text{Percentage} = \frac{a}{b} \times 100\% $$
Example 1: Out of 500 patients, 125 recovered. Find the recovery percentage.
Given: recovered = 125, total = 500
$$ \text{Percentage} = \frac{125}{500} \times 100\% $$
$$ = 0.25 \times 100\% = 25\% $$
Answer: 25%
Example 2: A diagnostic test has 95 positive results out of 120 samples. Find the positive rate percentage.
Given: positive = 95, total = 120
$$ \text{Percentage} = \frac{95}{120} \times 100\% $$
$$ = 0.7917 \times 100\% = 79.17\% $$
Answer: 79.17%
Example 3: A drug solution has 0.8 grams of active ingredient in 5 grams total. Find the percentage concentration.
Given: active = 0.8 g, total = 5 g
$$ \text{Percentage} = \frac{0.8}{5} \times 100\% $$
$$ = 0.16 \times 100\% = 16\% $$
Answer: 16%
Natural numbers are positive integers used for counting discrete biomedical objects or events.
Example 1: A hospital receives 150 new patients every day. How many patients after 7 days?
Given: daily patients = 150, days = 7
$$ \text{Total} = 150 \times 7 = 1050 $$
Answer: 1050 patients
Example 2: A lab tests 25 samples each hour. How many samples in 12 hours?
Given: samples/hour = 25, hours = 12
$$ \text{Total} = 25 \times 12 = 300 $$
Answer: 300 samples
Example 3: A biomedical engineer needs 120 screws for one machine. How many screws for 15 machines?
Given: screws/machine = 120, machines = 15
$$ \text{Total} = 120 \times 15 = 1800 $$
Answer: 1800 screws
These problems include operations on numbers relevant to biomedical engineering such as addition, subtraction, multiplication, and division.
Example 1: A hospital has 300 beds; 120 are occupied. How many are free?
Given: total beds = 300, occupied = 120
$$ \text{Free} = 300 - 120 = 180 $$
Answer: 180 beds free
Example 2: A biomedical device costs Ksh 15,000. If 4 devices are bought, what is the total cost?
Given: cost/device = 15,000, quantity = 4
$$ \text{Total cost} = 15,000 \times 4 = 60,000 $$
Answer: Ksh 60,000
Example 3: A technician divides 480 test strips equally into 8 boxes. How many strips per box?
Given: total strips = 480, boxes = 8
$$ \text{Strips per box} = \frac{480}{8} = 60 $$
Answer: 60 strips per box
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Create a free accountThis chapter introduced various types of numbers including whole, odd, even, and prime numbers, and covered rounding off numbers to the nearest tens, hundreds, thousands, millions, and billions. It explained ratios, proportions, and percentages, and included word problems involving natural numbers. The concept of factors was explored through composite numbers, prime factors, and factors expressed in power form. Divisibility tests were presented alongside the calculation and application of the greatest common divisor (GCD) or highest common factor (HCF) in practical scenarios. The chapter also discussed multiples and the least common multiple (LCM) with real-life applications. Integers were examined using the number line, including operations and order of operations, with examples related to everyday contexts. Fractions were covered extensively, including proper, improper, and mixed numbers, conversions between these forms, comparisons, arithmetic operations, and word problems. Decimals were linked to fractions, with focus on recurring decimals, decimal places, standard form, operations, and order of operations applied to real-life problems. Finally, arithmetic operations of addition, subtraction, multiplication, and division were reviewed, followed by methods of calculating squares and square roots both manually and using calculators.
Round off the number 47,856 to the nearest thousand as it might represent the number of cells counted in a biomedical assay. (2 marks)
A biomedical device costs Ksh 1,235,400. Round this cost to the nearest million for budgeting purposes. (2 marks)
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