Units, standard form and significant figures

Introduction to Mathematical Skills in Biology

In Biology, we don't just look at pictures of cells or dissect plants; we measure them! Whether you are calculating the size of a mitochondrion or the rate of an enzyme reaction, you need to be precise. This chapter covers the fundamental "rules of the road" for handling numbers: units, standard form, and significant figures. Mastering these basics will help you avoid losing "easy marks" in your OCR A Level exams.

1. Units of Measurement

In science, we use the SI system (International System of Units). This ensures that a biologist in London and a biologist in Tokyo can compare their results without confusion.

Base Units

Most biological measurements are based on these core units:

• Length: Metre (\(m\))
• Mass: Kilogram (\(kg\)) or Gram (\(g\))
• Time: Second (\(s\))
• Amount of substance: Mole (\(mol\))

Unit Prefixes

Biological structures vary massively in size, from a whole whale to a single DNA molecule. We use prefixes to make these numbers manageable. You must be able to convert between these fluently:

• Kilo (k): \(10^3\) (e.g., \(1 \text{ km} = 1000 \text{ m}\))
• Milli (m): \(10^{-3}\) (e.g., \(1 \text{ mm} = 0.001 \text{ m}\))
• Micro (\(\mu\)): \(10^{-6}\) (e.g., \(1 \text{ \)\mu\)m} = 0.000001 \text{ m}\))
• Nano (n): \(10^{-9}\) (e.g., \(1 \text{ nm} = 0.000000001 \text{ m}\))

Quick Tip: To go from a larger unit to a smaller unit (e.g., \(mm\) to \(\mu m\)), multiply by \(1000\). To go from a smaller unit to a larger unit (e.g., \(nm\) to \(\mu m\)), divide by \(1000\).

Derived Units

Some units are created by combining base units:

• Area: \(m^2\) or \(cm^2\)
• Volume: \(m^3\), \(cm^3\) (often used interchangeably with \(ml\)), or \(dm^3\) (equivalent to \(1\) litre).
• Concentration: \(mol \text{ } dm^{-3}\) or \(g \text{ } dm^{-3}\).

Note: For more on how to use these units in specific biological contexts, see the chapter on Magnification, scale and surface area to volume.

Key Takeaway: Always check the units requested in a question. If the data is in \(mm\) but the answer box asks for \(\mu m\), you must convert your final value.

2. Standard Form

Biologists deal with very large numbers (like the number of cells in the human body) and very small numbers (like the diameter of a cell membrane). Writing all those zeros is tedious and leads to mistakes. Standard form is our shorthand.

How to Write Standard Form

A number in standard form is written as: \(A \times 10^n\)

• \(A\) is a number between \(1\) and \(10\) (it can be \(1\), but must be less than \(10\)).
• \(n\) is the index or power. It tells you how many places to move the decimal point.

Step-by-Step Conversion

Example 1: Large Numbers
The number of red blood cells in a sample is \(5,400,000\).
1. Move the decimal point until you have a number between \(1\) and \(10\): \(5.4\)
2. Count how many places you moved the decimal: \(6\) places to the left.
3. The standard form is \(5.4 \times 10^6\).

Example 2: Small Numbers
The width of a plant cell is \(0.000035 \text{ m}\).
1. Move the decimal point until you have a number between \(1\) and \(10\): \(3.5\)
2. Count how many places you moved the decimal: \(5\) places to the right.
3. Because you moved to the right (a small number), the power is negative.
4. The standard form is \(3.5 \times 10^{-5} \text{ m}\).

Quick Review: Positive power = Large number. Negative power = Small number (less than 1).

3. Significant Figures (sf)

Significant figures tell us about the precision of a measurement. Using too many "random" digits in your answer suggests your equipment was more precise than it actually was.

The Rules for Counting Significant Figures

1. All non-zero digits are significant. (\(1.23\) has \(3 \text{ sf}\))
2. Zeros between non-zero digits are significant. (\(102\) has \(3 \text{ sf}\))
3. Leading zeros are NOT significant. They are just placeholders. (\(0.005\) has only \(1 \text{ sf}\))
4. Trailing zeros after a decimal point ARE significant. (\(2.50\) has \(3 \text{ sf}\); it shows the measurement was precise to that second decimal place).

How Many Significant Figures Should I Use?

This is a common "trap" in OCR exams! The general rule for calculations is:
Give your answer to the same number of significant figures as the least precise piece of data used in the calculation.

Example:
If you multiply \(2.3\) (\(2 \text{ sf}\)) by \(3.56\) (\(3 \text{ sf}\)), your calculator gives \(8.188\).
Since the least precise number (\(2.3\)) has only \(2 \text{ sf}\), you should round your answer to \(2 \text{ sf}\): \(8.2\).

Rounding Rules

• If the next digit is \(5\) or above, round up.
• If the next digit is \(4\) or below, keep it the same.

Don't worry if this seems tricky! Just remember: look at the numbers the examiner gave you in the question. If they all use \(2\) or \(3\) significant figures, your answer should too.

Key Takeaway: Never just copy every digit from your calculator screen. Rounding to an appropriate number of significant figures is a required skill (M1.1).

Summary Checklist

• Can you convert between \(mm\), \(\mu m\), and \(nm\) by multiplying or dividing by \(1000\)?
• Is your standard form "A" value between \(1\) and \(10\)?
• Did you remember that small numbers have a negative power in standard form?
• Have you checked your final answer to ensure it matches the significant figures of the data provided in the question?

Note: To understand how these measurements relate to the "realness" of your data, please see the chapter on Uncertainty, precision and accuracy.