Mathematics • Year 10 • Unit 1 • Lesson 10
Tiny Numbers in the Real World
Use zero and negative indices in everyday contexts: scientific notation for very small things (the width of a hair, virus diameter, milliseconds), unit conversions, and the surprising fact that anything-to-the-zero equals 1. Then explain your method.
1. Word problems
Each problem uses the rules from Lesson 10: a⁰ = 1 (a ≠ 0) and a⁻ⁿ = 1/aⁿ. Show your working.
1.1 — Width of a human hair (scientific notation). A typical human hair is about 10⁻⁴ metres wide.
(a) Convert 10⁻⁴ to a fraction with a positive index.
(b) Convert it to a decimal in metres.
(c) Convert it to millimetres (1 m = 10³ mm, so multiply by 10³). Give your answer as a power of 10 and as a decimal. 3 marks
1.2 — Virus diameter. A SARS-CoV-2 virus particle is about 10⁻⁷ metres across.
(a) Convert 10⁻⁷ to a fraction with a positive index.
(b) How many virus diameters fit across the width of a 10⁻⁴ m human hair? Use 10⁻⁴ ÷ 10⁻⁷, applying Index Law 2. 3 marks
1.3 — Frame rate (milliseconds per frame). A 60 frames-per-second video shows 60 frames every second. The time per frame is 1/60 seconds.
(a) Write 1/60 in milliseconds (multiply by 10³).
(b) Express the time per frame as 60⁻¹ seconds — what does the negative index mean here?
(c) For a 120 fps video, write the time per frame using a negative index. 3 marks
1.4 — Empty product (a⁰ = 1). A photographer takes 0 photos at an event. The total file size of those 0 photos is 5⁰ MB.
(a) What is the value of 5⁰?
(b) Explain in plain English why "0 photos × any file size = 0 bytes" is true in real life, but in maths we still have 5⁰ = 1, not 0. (Hint: 5⁰ comes from the pattern 5³ → 5² → 5¹ → 5⁰, dividing by 5 each step.)
(c) State why we exclude a = 0 from the zero-index rule. 3 marks
1.5 — Compound interest with a "zero years" check. An investment formula gives A = P × (1 + R)ⁿ. A student claims "if you invest $1,000 at 5% for 0 years, you get nothing." Test their claim.
(a) Substitute P = 1,000, R = 0.05, n = 0 into the formula.
(b) What does (1.05)⁰ equal, by the zero-index rule?
(c) What does A equal, and what does that tell you about the student's claim? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "2⁻³ = −8 because a negative power gives a negative answer". In your own words, explain (i) why this is wrong, (ii) what a negative index actually means (use the word "reciprocal" somewhere), and (iii) what 2⁻³ actually equals as a fraction and as a decimal. Then describe the pattern 2³ → 2² → 2¹ → 2⁰ → 2⁻¹ → 2⁻² → 2⁻³ to make your case.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Human hair
(a) 10⁻⁴ = 1/10⁴ = 1/10,000.
(b) 1/10,000 m = 0.0001 m.
(c) In mm: 10⁻⁴ × 10³ = 10⁻⁴⁺³ = 10⁻¹ mm = 0.1 mm.
1.2 — Virus diameter
(a) 10⁻⁷ = 1/10⁷ = 1/10,000,000 m (one ten-millionth of a metre).
(b) Hair / virus = 10⁻⁴ ÷ 10⁻⁷ = 10⁻⁴⁻⁽⁻⁷⁾ = 10⁻⁴⁺⁷ = 10³ = 1,000. About 1,000 virus particles fit across the width of a single hair.
1.3 — Frame rate
(a) 1/60 s × 10³ ms/s = 1,000/60 ≈ 16.67 ms per frame.
(b) 60⁻¹ s = 1/60 s — the negative index means "the reciprocal of 60", which is the fraction of a second each frame takes.
(c) 120 fps: time per frame = 120⁻¹ s = 1/120 s ≈ 8.33 ms.
1.4 — Empty product
(a) 5⁰ = 1.
(b) The maths definition is chosen so the rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ extends nicely: 5³ ÷ 5³ = 5⁰, but 5³ ÷ 5³ = 1 by ordinary division — so 5⁰ must equal 1 to keep the pattern consistent. It is not about "zero photos" — it is about completing the pattern of dividing by 5 each step.
(c) For a = 0: 0³ ÷ 0³ would be 0/0, which is undefined. So 0⁰ is undefined, and we exclude a = 0 from the rule.
1.5 — Compound interest with n = 0
(a) A = 1,000 × (1.05)⁰.
(b) (1.05)⁰ = 1 (zero-index rule — 1.05 is non-zero, so it applies).
(c) A = 1,000 × 1 = $1,000. The student is partly right — you earn $0 of interest — but they are wrong that you "get nothing": you still have your original deposit of $1,000.
2.1 — Explain your thinking (sample response)
My classmate is wrong because a negative index does not make the answer negative — it creates a reciprocal. The rule a⁻ⁿ = 1/aⁿ means we flip the power into a fraction with the same positive index underneath. So 2⁻³ = 1/2³ = 1/8 = 0.125 — a small positive fraction, not −8.
The pattern that proves it: each time we drop the index by 1, we divide the value by 2. So 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8. The values get smaller and stay positive — they never become negative.
Marking: 1 mark for noting "not negative"; 1 for using "reciprocal"; 1 for 1/8 = 0.125; 1 for the descending pattern.