Secondary 3 · Mathematics · Number
Scientific notation and significant figures
Write very large and very small numbers in scientific notation, convert back to ordinary form, and round to a given number of significant figures.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Write very large and very small numbers in scientific notation.
- Convert numbers in scientific notation back to ordinary (decimal) form.
- Round numbers to a given number of significant figures.
Success looks like
- I can write 4 300 000 as 4.3 × 10⁶ and 0.000 52 as 5.2 × 10⁻⁴.
- I can write 3.07 × 10⁵ as an ordinary number.
- I can round a number such as 0.004 718 to 2 significant figures.
The big idea
Scientists work with huge numbers (the distance to the Sun is about 150 000 000 km) and tiny ones (a bacterium is about 0.000 002 m long). Writing all those zeros is slow and easy to get wrong, so we use scientific notation: a number from 1 up to (but not including) 10, multiplied by a power of 10.
Powers of 10 build on place value: 10¹ = 10, 10² = 100, 10³ = 1000, and so on. For small numbers we use negative indices: 10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001.
To write a large number in scientific notation:
- Place the decimal point after the first non-zero digit: 4 300 000 → 4.3.
- Count how many places the decimal point moved to get there (6 places), and use that as a positive index: 4.3 × 10⁶.
For a small number, the decimal point moves to the right, so the index is negative. 0.000 52 → 5.2, moving 4 places right, so 0.000 52 = 5.2 × 10⁻⁴. A quick check: a negative index means the number is less than 1.
To convert back to ordinary form, move the decimal point the number of places shown by the index — right for a positive index (the number gets bigger) and left for a negative index (the number gets smaller). Fill gaps with zeros.
Significant figures (s.f.) count from the first non-zero digit. Zeros at the start of a decimal (like the zeros in 0.0047) are not significant; zeros between other digits are; zeros at the end of a decimal are (0.4050 has 4 s.f.). To round to a number of significant figures, look at the next digit: 5 or more rounds up. When rounding a whole number, keep its size by filling with zeros: 86 451 to 3 s.f. is 86 500, not 865.
Calculators often show scientific notation with an E, so 3.2E8 means 3.2 × 10⁸.
Worked examples
Large and small numbers
Write (a) 4 300 000 and (b) 0.000 52 in scientific notation.
- (a) Put the point after the 4: 4.3. The point moves 6 places to the left, so the index is 6: 4.3 × 10⁶.
- (b) Put the point after the 5: 5.2. The point moves 4 places to the right, so the index is −4: 5.2 × 10⁻⁴.
Answer: (a) 4.3 × 10⁶; (b) 5.2 × 10⁻⁴
Back to ordinary form
Write (a) 3.07 × 10⁵ and (b) 8.1 × 10⁻³ as ordinary numbers.
- (a) Positive index: move the point 5 places right. 3.07 → 307 000.
- (b) Negative index: move the point 3 places left. 8.1 → 0.0081.
Answer: (a) 307 000; (b) 0.0081
Significant figures
Round (a) 0.004 718 to 2 significant figures and (b) 86 451 to 3 significant figures.
- (a) The first significant figure is 4. The first two are 4 and 7; the next digit is 1, so round down: 0.0047.
- (b) The first three significant figures are 8, 6 and 4; the next digit is 5, so round up to 865, then fill with zeros: 86 500.
Answer: (a) 0.0047; (b) 86 500
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Write 52 000 in scientific notation.
5.2 × 10⁴ — The first part must be from 1 to less than 10, so use 5.2. The point moves 4 places: 5.2 × 10⁴. (52 × 10³ and 0.52 × 10⁵ have the right value but are not in scientific notation.)
Write 0.0009 in scientific notation.
9 × 10⁻⁴ — The point moves 4 places to the right to give 9, so the index is −4.
Write 6.1 × 10⁻² as an ordinary number.
0.061 — A negative index of −2 means move the point 2 places left: 0.061.
Round 3.14159 to 3 significant figures.
3.14 — The first three significant figures are 3, 1 and 4. The next digit is 1, so round down: 3.14.
Write 7.25 × 10³ as an ordinary number.
7250 — Move the point 3 places right: 7.25 → 7250.
Write 640 000 in scientific notation.
6.4 × 10⁵ — Put the point after the 6 to get 6.4. The point moves 5 places, so 6.4 × 10⁵.
Round 0.006 382 to 2 significant figures.
0.0064 — The first two significant figures are 6 and 3. The next digit is 8, so round up to 64: 0.0064.
How many significant figures does 0.04050 have?
4 — Start at the first non-zero digit (4). The digits 4, 0, 5 and the final 0 are all significant: 4 significant figures.
Watch out for
- Writing a first part that is 10 or more (such as 52 × 10³). The number in front must be at least 1 and less than 10.
- Mixing up the sign of the index. Remind the learner: big numbers have positive indices; numbers between 0 and 1 have negative indices.
- Dropping place-holder zeros when rounding whole numbers, such as writing 86 451 to 3 s.f. as 865. The rounded number must be about the same size as the original.
Tips for parents
- Ask the learner to say the number aloud after converting (“three hundred and seven thousand”) to check it is sensible.
- Look up real measurements together — the mass of the Earth, the width of a human hair — and convert them both ways.
- Show how your calculator displays very large results (for example 99 999 × 99 999 999) and what the E means.
Go further
- Make a “powers of ten” poster showing objects from the size of an atom (about 10⁻¹⁰ m) to the size of a galaxy.
- Multiply numbers in scientific notation by multiplying the front numbers and adding the indices, such as (2 × 10³) × (4 × 10⁵) = 8 × 10⁸.
- Research how many significant figures a kitchen scale or a measuring tape can reliably give, and why.
About 45 minutes
You'll need
- Scientific calculator
- Paper and pencil
- A place-value chart (optional)
Key words
- Scientific notation
- A way of writing a number as a number from 1 up to (but not including) 10, multiplied by a power of 10, such as 4.3 × 10⁶.
- Power of 10
- 10 raised to an index, such as 10³ = 1000 or 10⁻² = 0.01.
- Negative index
- An index below zero, used for small numbers: 10⁻ⁿ = 1 ÷ 10ⁿ, so 10⁻³ = 0.001.
- Ordinary form
- A number written out in full with ordinary digits, such as 307 000.
- Significant figures
- The digits in a number that carry meaning about its size, starting from the first non-zero digit.
Still tricky?
For parents
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