Mathematics • Year 9 • Unit 1 • Lesson 15
Scientific Notation in the Real World
Read scientific notation in real measurements: planets, atoms, computers and biology. Compare orders of magnitude, decide which is bigger, and explain why standard form makes science usable.
1. Word problems
Each problem reads or compares numbers given in scientific notation. Show your working — final answers alone earn half marks.
1.1 — Earth–Sun distance. The distance from Earth to the Sun is about $1.5 \times 10^{11}$ m.
(a) Write this distance as an ordinary number.
(b) What is the "order of magnitude" of this distance (i.e. the power of $10$)? 3 marks
1.2 — Hydrogen atom. A hydrogen atom is about $1 \times 10^{-10}$ m across.
(a) Write this width as an ordinary number.
(b) How many orders of magnitude apart are the Earth–Sun distance ($1.5 \times 10^{11}$) and a hydrogen atom ($1 \times 10^{-10}$)? 3 marks
1.3 — Phone storage. A new phone has $2.56 \times 10^{11}$ bytes of storage. An older phone has $6.4 \times 10^{10}$ bytes.
(a) Which phone has more storage? Justify using the powers, NOT the coefficients alone.
(b) Roughly how many times more storage does the bigger phone have? 3 marks
1.4 — Population of Australia. The population of Australia in $2026$ is about $2.7 \times 10^7$ people.
(a) Write this population as an ordinary number.
(b) Is "$27 \times 10^6$" the same number? Why is that form NOT in correct scientific notation? 3 marks
1.5 — Bacterium length. A common bacterium is about $2 \times 10^{-6}$ m long.
(a) Write this length as an ordinary number, in metres.
(b) Convert your length to millimetres. (Hint: $1$ m $= 1000$ mm, so multiply by $10^3$.) Write the answer in scientific notation. 3 marks
2. Explain your thinking
This question is about communication. Use full sentences. 4 marks
2.1 Your friend says "scientific notation is just a fancy way of writing the same number — why do scientists bother?". In your own words, give a clear reply that includes (i) why $1.5 \times 10^{11}$ is easier than $150\,000\,000\,000$ in practice, (ii) why $1 \times 10^{-10}$ is easier than $0.000\,000\,000\,1$, (iii) what "order of magnitude" lets you compare instantly, and (iv) one real-life setting (science, technology or everyday life) where standard form matters.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Earth–Sun distance
(a) $1.5 \times 10^{11} = \mathbf{150\,000\,000\,000}$ m.
(b) Order of magnitude: $\mathbf{10^{11}}$ — about a hundred billion metres.
1.2 — Hydrogen atom
(a) $1 \times 10^{-10} = \mathbf{0.000\,000\,000\,1}$ m.
(b) Difference $= 11 - (-10) = \mathbf{21}$ orders of magnitude apart. The Earth–Sun distance is about $10^{21}$ (one sextillion) times bigger than an atom.
1.3 — Phone storage
(a) The newer phone wins. $10^{11} = 10 \times 10^{10}$, so $6.4 \times 10^{10}$ is the same as $0.64 \times 10^{11}$, which is much less than $2.56 \times 10^{11}$. The bigger power swamps the difference in coefficients.
(b) Ratio $= \dfrac{2.56 \times 10^{11}}{6.4 \times 10^{10}} = \dfrac{2.56}{6.4} \times 10^{11-10} = 0.4 \times 10 = \mathbf{4}$ times as much storage.
1.4 — Population of Australia
(a) $2.7 \times 10^7 = \mathbf{27\,000\,000}$ people.
(b) Yes, $27 \times 10^6 = 27\,000\,000$ — the SAME number. But $27 \times 10^6$ is NOT in correct scientific notation because the coefficient $27$ is bigger than $10$. The rule requires $1 \leq a < 10$. Correct form: $2.7 \times 10^7$.
1.5 — Bacterium length
(a) $2 \times 10^{-6} = \mathbf{0.000\,002}$ m.
(b) Convert to mm: $2 \times 10^{-6}$ m $\times 10^3$ mm/m $= 2 \times 10^{-6+3} = \mathbf{2 \times 10^{-3}}$ mm (which is $0.002$ mm).
2.1 — Explain your thinking (sample response)
Scientific notation isn't just shorter — it's safer. (i) $1.5 \times 10^{11}$ tells you the size INSTANTLY (a hundred billion or so), while $150\,000\,000\,000$ has eleven zeros that you can easily mis-count. (ii) For tiny numbers like $1 \times 10^{-10}$, the negative power tells you exactly how small without you needing to count nine or ten zeros after the decimal point — and counting zeros is where most slips happen. (iii) "Order of magnitude" is just the power of $10$, so you can compare two huge numbers at a glance: $10^{11}$ and $10^{8}$ differ by a factor of $10^3 = 1000$, no calculator needed. (iv) Real-life examples are everywhere — astronomers writing the distance to a galaxy as $4 \times 10^{22}$ m, biologists writing a cell size as $3 \times 10^{-5}$ m, or computer engineers describing a phone's $2.56 \times 10^{11}$ bytes of storage. Anywhere the numbers get extreme, scientists use standard form to stay accurate.
Marking: 1 mark each for points (i), (ii), (iii), and (iv).