Standard Form and Significant Figures: Stop Losing Easy Marks

Quick answer: Standard form writes any number as a x 10^n, where a is at least 1 but less than 10, and n is a whole number. So 4,500,000 becomes 4.5 x 10^6 and 0.00032 becomes 3.2 x 10^-4. Significant figures are counted from the first non-zero digit, and most GCSE, IGCSE, WAEC and Cambridge science answers should be rounded to 2 or 3 significant figures - giving more can actually cost you a mark.

I dropped a mark in a Year 11 physics mock for writing 26.111 m/s when the mark scheme wanted 26 m/s. My working was perfect. My arithmetic was perfect. But I'd handed in six significant figures when the data only justified two, and the examiner took the "appropriate accuracy" mark straight off. Standard form and significant figures feel like fussy admin, but they are some of the cheapest marks in maths and science - and the easiest to throw away. Here's everything I wish someone had drilled into me earlier.

The one rule of standard form people break

Standard form (your American AP friends call it scientific notation) is just a x 10^n. The rule that catches everyone is the size of a: it must be at least 1 and less than 10. One digit before the decimal point, nothing more.

So 36 x 10^7 is not in standard form, even though it equals the right number - a is 36, which is too big. You have to rewrite it as 3.6 x 10^8. Examiners at AQA, Edexcel, OCR and Cambridge all treat "a not between 1 and 10" as wrong, even if the value underneath is correct - so check it on every answer.

Writing numbers in standard form

For a big number, count how many places the decimal point moves left until one digit sits in front of it. That count is your positive power.

• 4,500,000 becomes 4.5 x 10^6 (the point moved 6 places left)
• 92,000 becomes 9.2 x 10^4

For a small number, count how many places the point moves right until it sits just after the first non-zero digit. That count is your negative power.

• 0.00032 becomes 3.2 x 10^-4 (moved 4 places right)
• 0.0000067 becomes 6.7 x 10^-6

The classic slip is the sign of the power. Small numbers (less than 1) always take a negative power; big numbers take a positive one. Write 0.00032 as 3.2 x 10^4 and you've turned a tiny number into 32,000 - a factor of a hundred million out.

Calculating in standard form

Multiplying and dividing is genuinely fast once you see the trick. Deal with the numbers and the powers separately.

• Multiply: multiply the a-values, add the powers. (7.2 x 10^4) x (5.0 x 10^3) = 36 x 10^7. Then renormalise: 3.6 x 10^8.
• Divide: divide the a-values, subtract the powers. (8.4 x 10^6) divided by (2.0 x 10^2) = 4.2 x 10^4.

Notice the multiply example landed on 36 x 10^7 - not standard form yet. Always do the renormalise step at the end.

Adding and subtracting is where people quietly lose marks, because you cannot just add the a-values unless the powers already match. Make the powers equal first.

• (3.0 x 10^5) + (4.0 x 10^4). Rewrite the second as 0.4 x 10^5, then add: 3.4 x 10^5.

If you'd carelessly added 3.0 and 4.0 to get 7.0 x 10^5, you'd be off by nearly double.

Significant figures vs decimal places

These two get mixed up constantly, and the question always tells you which one it wants - so read it. Significant figures are counted from the first non-zero digit.

• 0.00408 has 3 significant figures (4, 0, 8). Leading zeros never count; the zero between the 4 and 8 does.
• 20.0 has 3 significant figures - trailing zeros after a decimal point count.
• 4500 is ambiguous (2, 3 or 4 sig figs?), which is exactly why writing it as 4.5 x 10^3 is better: that is unmistakably 2 significant figures.

Decimal places just count digits after the point. 3.14159 to 3 decimal places is 3.142; to 3 significant figures it is 3.14. But for 0.00408, three decimal places gives 0.004 (only 1 sig fig) while 3 significant figures gives 0.00408. Same number, completely different answers - never assume they are interchangeable.

Worked example: standard form and sig figs together

A car travels 4.7 x 10^3 m in a time of 1.8 x 10^2 s. Find its average speed, giving your answer to 2 significant figures.

• Step 1 - write the formula. Average speed = distance divided by time.
• Step 2 - divide the numbers. 4.7 divided by 1.8 = 2.611...
• Step 3 - handle the powers. 10^3 divided by 10^2 = 10^(3-2) = 10^1.
• Step 4 - combine. 2.611... x 10^1 = 26.11... m/s.
• Step 5 - round last. The data is given to 2 significant figures, so the answer is 2 significant figures: 26 m/s.

The key habit lives in step 5: round only at the very end. If you'd rounded 2.611 to 2.6 back in step 2 and carried that through, small errors can push you over a rounding boundary and cost the final mark. Keep the full figure on your calculator until the last line. If you want to check your own steps on a real question, paste it into the math solver and it walks through the working in the same order.

Why exams are strict about significant figures

In Cambridge and AQA physics and chemistry, there is usually a specific mark for giving your answer to an appropriate number of significant figures. The unwritten rule: your answer should carry the same number of sig figs as the least precise piece of data in the question - almost always 2 or 3. Give 1 sig fig and it looks like a guess; give 8 and you are claiming an accuracy your measurements never had. Both get penalised - usually once per paper, but repeatedly across the year if you never fix the habit.

GCSE and IGCSE maths phrase it as "give your answer to an appropriate degree of accuracy" - same idea. WAEC and WASSCE examiner reports repeat it every year too: candidates lose marks for over-rounding mid-calculation and for forgetting to round at the end.

Common mistakes that cost easy marks

• Leaving a bigger than 10 (36 x 10^7 instead of 3.6 x 10^8).
• Putting the wrong sign on the power for small numbers.
• Copying a calculator's "3.2E-4" display literally - write it as 3.2 x 10^-4 in your answer.
• Rounding halfway through, then rounding again (double rounding).
• Confusing decimal places with significant figures.
• Adding or subtracting without matching the powers first.

Test yourself

1. Write 0.00069 in standard form.
2. Work out (6.0 x 10^5) x (4.0 x 10^-2), giving your answer in standard form.
3. The mass of a sample is 0.02058 g. Round it to 3 significant figures.

Quick answers: (1) 6.9 x 10^-4. (2) 24 x 10^3, which renormalises to 2.4 x 10^4. (3) 0.0206 g - the digits are 2, 0, 5, 8; keep the first three (2, 0, 5) and the next digit, 8, rounds the 5 up to 6.

Want more like these at your exact board and level? Generate a set with the quiz generator, or paste a photo of a textbook question into the math solver to see every step laid out. To find out precisely where a mark would be lost, run your written answer through Mark my answer.

FAQ

Do I use significant figures or decimal places in the exam? Whichever the question specifies. If it only says "give an appropriate degree of accuracy", match the significant figures of the data - usually 2 or 3. In science, sig figs are the default; in money questions, 2 decimal places.

Can I really lose marks for too many significant figures? Yes. In Cambridge and AQA science, writing an answer to 6 or 7 sig figs when the data only justifies 2 or 3 loses the appropriate-accuracy mark. More precision is not "safer" - it is simply wrong.

Is standard form the same as scientific notation? Yes. "Standard form" is the GCSE and IGCSE term; "scientific notation" is the AP and US term. Both mean a x 10^n with a between 1 and 10.

When do I round - during the calculation or at the end? At the very end, once. Keep the full number on your calculator through every step and only round the final answer to the required accuracy. Rounding partway through is the single most common way to drop that last mark.

In short: Keep a between 1 and 10, get the sign of the power right, and round only your final answer - to 2 or 3 significant figures unless the question says otherwise. These are among the easiest marks in GCSE, IGCSE, WAEC, CBSE and AP maths and science, so stop handing them back. Check your working step by step with the math solver and you will stop losing them for good.