By the end of this chapter, you will be able to:
Mastering these skills will help you tackle real-world engineering challenges with confidence and precision.
Every count, interval and reading a biomedical engineering technician handles is a number: the spare electrode pads on a shelf, the days between two equipment services, the volume of solution in a reservoir, the temperature of a cold room. This chapter develops the number systems behind that work, from whole numbers, factors, the greatest common divisor (GCD) and the least common multiple (LCM), through integers, fractions and decimals, to the four arithmetic operations, squares and square roots. With these skills the technician can total stock, plan service schedules, prepare solutions and check instrument readings accurately.
Numbers are grouped into types by the properties they share, and each type has a use at the biomedical bench. Whole numbers count items such as sample tubes and spare parts, odd and even numbers split those items into groups, prime numbers cannot be split into smaller equal groups, and ratios and percentages compare one quantity with another. Every counted or measured value is also reported to a sensible degree of accuracy, so this topic begins with rounding off and then covers each type in turn.
Two families of counting numbers are used throughout. Natural numbers are the counting numbers 1, 2, 3, 4 and so on. Whole numbers are the natural numbers together with zero, so they run 0, 1, 2, 3 and so on. Integers, fractions and decimals extend these types and are taught under their own headings later in the chapter.
Rounding off replaces a number with a nearby number that has zeros in its lower places, so that it is easier to read, compare and remember. A technician rounds a stock count of 12,480 spare pads to 12,000 when reporting to a supervisor, and rounds a scale reading of 995 g to 1,000 g when only the nearest ten grams matters. The rounded value is not exact, so it is used for reports, estimates and checks, while exact values are kept for calculations that need them.
Each digit of a number has a value that depends on its place. The table shows the places used for rounding, with the digit that sits in each place of the number 7,654,321,098.
| Place | Value | Digit in 7,654,321,098 |
|---|---|---|
| Units | 1 | 8 |
| Tens | 10 | 9 |
| Hundreds | 100 | 0 |
| Thousands | 1,000 | 1 |
| Millions | 1,000,000 | 4 |
| Billions | 1,000,000,000 | 7 |
The same steps are used for every place:
When the digit to be raised is 9, adding 1 gives 10: write 0 and carry 1 to the next place on the left, so 995 rounded to the nearest ten becomes 1,000.
Example 1: Round 3,476 to the nearest ten, the nearest hundred and the nearest thousand.
Nearest ten: the units digit is 6, so round up: 3,480.
Nearest hundred: the tens digit is 7, so round up: 3,500.
Nearest thousand: the hundreds digit is 4, so round down: 3,000.
Answer: 3,480 to the nearest ten, 3,500 to the nearest hundred, 3,000 to the nearest thousand
Example 2: A production line records 48,250 specimen labels in a month. Round the figure to the nearest hundred and to the nearest thousand.
Nearest hundred: the tens digit is 5, so round up: 48,300.
Nearest thousand: the hundreds digit is 2, so round down: 48,000.
Answer: 48,300 to the nearest hundred and 48,000 to the nearest thousand
Example 3: A weighing scale reads 995 g. Round the reading to the nearest 10 g.
The units digit is 5, so the tens digit 9 is raised by 1. Since 9 + 1 = 10, write 0 in the tens place and carry 1 to the hundreds place, which turns 9 hundreds into 10 hundreds, then 1 thousand.
995 → 1,000
Answer: 1,000 g
Example 4: The planned cost of a test bench is Ksh 2,364,500. Round it to the nearest thousand and to the nearest million.
Nearest thousand: the hundreds digit is 5, so round up: 2,364,500 → 2,365,000.
Nearest million: the hundred thousands digit is 3, so round down: 2,364,500 → 2,000,000.
Answer: Ksh 2,365,000 to the nearest thousand and Ksh 2,000,000 to the nearest million
Example 5: A data logger has stored 7,654,321,098 bytes. Round the number to the nearest billion and to the nearest million.
Nearest billion: the hundred millions digit is 6, so round up: 8,000,000,000.
Nearest million: the hundred thousands digit is 3, so round down: 7,654,000,000.
Answer: 8,000,000,000 bytes to the nearest billion and 7,654,000,000 bytes to the nearest million
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 × 20 = 300
Answer: 300
Example 4: Divide 144 by 12.
Calculate:
144 ÷ 12 = 12
Answer: 12
Example 5: Calculate (7 × 8) + (9 − 4).
Calculate:
7 × 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 ÷ 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 × 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 ÷ 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 ÷ 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 × 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 ÷ 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 ÷ 2 = 14.5 no
29 ÷ 3 = 9.67 no
29 ÷ 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:
Ratio = a/b
Proportion formula:
a/b = c/d
Example 1: Find the ratio of 50 ml to 200 ml of a solution.
Calculate:
50/200 = 1/4
Answer: 1:4
Example 2: Solve for x if 3/4 = x/16.
3/4 = x/16
Cross multiply:
3 × 16 = 4 × x
48 = 4x
x = 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.
5/3 = 40/x
Cross multiply:
5x = 3 × 40
5x = 120
x = 120/5 = 24
Answer: 24 ml
Example 4: If 2/x = 5/20, find x.
Cross multiply:
2 × 20 = 5 × x
40 = 5x
x = 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?
12/1 = 120/x
Cross multiply:
12x = 120
x = 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:
Percentage = (Part ÷ Whole) × 100%
Example 1: Find 25% of 200 ml fluid.
Calculate:
(25 ÷ 100) × 200 = 50 ml
Answer: 50 ml
Example 2: A device has 95% efficiency. If input power is 150 W, find output power.
Calculate:
(95 ÷ 100) × 150 = 142.5 W
Answer: 142.5 W
Example 3: Increase 120 mg drug dose by 15%.
Calculate increase:
(15 ÷ 100) × 120 = 18 mg
New dose:
120 + 18 = 138 mg
Answer: 138 mg
Example 4: Decrease 500 ml solution by 10%.
Calculate decrease:
(10 ÷ 100) × 500 = 50 ml
New volume:
500 − 50 = 450 ml
Answer: 450 ml
Example 5: What percentage of 250 mg is 50 mg?
Calculate:
(50 ÷ 250) × 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:
45/120 = 3/8
Answer: 3/8
Example 2: A lab processes 15 samples per day. How many in 6 days?
Calculate:
15 × 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 80 heartbeats per minute. How many in 5 minutes?
Calculate:
80 × 5 = 400
Answer: 400 heartbeats
Example 5: A clinic has 8 doctors and 40 nurses. What is the ratio of nurses to doctors?
Calculate:
40/8 = 5
Answer: 5:1
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Create a free accountThis chapter set out the number systems a biomedical engineering technician uses at the workbench. It began with types of numbers: rounding off, whole, odd, even and prime numbers, ratio and proportion, percentages and word problems with natural numbers. It then showed how to break a number into its factors, write its prime factors in power form and count its factors, followed by the divisibility tests and the greatest common divisor (GCD) with its use in sharing items into equal packs. Multiples and the least common multiple (LCM) were applied to service schedules that repeat at different intervals. The chapter moved on to integers, with the number line, the four operations on signed numbers and the order of operations, and to fractions: proper and improper fractions, mixed numbers, conversion, comparison, operations and word problems. Decimals followed, covering conversion to and from fractions, recurring decimals, decimal places, standard form, operations and the order of operations. The written methods of addition, subtraction, multiplication and division were then set out with the laws that let each answer be checked. Finally, squares and square roots were found by multiplication, by factorisation and with a calculator.
A parts depot receives three deliveries of electrode pads, of 12,478, 9,865 and 6,094 pads. (a) Use the column method to find the total number of pads received. (b) Round the total to the nearest thousand. (3 marks)
The labels on four infusion pumps end in the numbers 119, 127, 144 and 2. For each number, state whether it is odd or even and whether it is prime or composite, and give a reason. (4 marks)
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