To compare numbers in standard form, look at the power of ten first, then the decimal part. The power tells you the size class, and the decimal part only matters when two numbers share the same power. This skill appears in data questions about populations, distances, masses and sizes.
It builds on converting small measurements to standard form. Make sure both numbers are properly in standard form before you compare them.
How do I compare step by step?
- Check both numbers are in standard form, with the decimal part between 1 and 10. Rewrite any that are not.
- Compare the powers of ten. A higher power means a bigger number. Remember −4 is higher than −5.
- If the powers match, compare the decimal parts.
- To find how many times bigger, divide the decimal parts and subtract the powers.
- Check with ordinary numbers when the sizes are manageable.
Worked example
Part (a). Which is larger: 3.2 × 10−4 or 9.5 × 10−5?
The powers are −4 and −5. Since −4 is greater than −5, 3.2 × 10−4 is larger. Check: 0.00032 is bigger than 0.000095.
Part (b). How many times larger is it, correct to 2 significant figures?
Divide: (3.2 × 10−4) ÷ (9.5 × 10−5) = (3.2 ÷ 9.5) × 10−4 − (−5) = 0.3368… × 101 = 3.368…
To 2 significant figures this is 3.4 times larger. Check: 0.00032 ÷ 0.000095 = 3.368…, which matches.
Part (c). Put in ascending order: 58 × 102, 4.9 × 103, 0.51 × 104.
Two of these are not in standard form, so convert all three to ordinary numbers: 58 × 102 = 5800, 4.9 × 103 = 4900, 0.51 × 104 = 5100.
Ascending order: 4.9 × 103, 0.51 × 104, 58 × 102.
The mistake to watch for
The usual slip is to compare only the decimal parts and ignore the powers.
Mistaken answer to (a): 9.5 × 10−5 is larger, because 9.5 is bigger than 3.2.
The student compared 9.5 with 3.2 and never looked at the powers.
The correction: a power of ten changes the size far more than the decimal part does. 10−4 is ten times 10−5, so the decimal parts are compared only when the powers are equal. Write both numbers out as decimals if you are unsure: 0.00032 and 0.000095.
Check yourself
Try these without a calculator, then open each answer.
1. Which is larger: 7.8 × 10−3 or 1.2 × 10−2?
Show answer
−2 is greater than −3, so 1.2 × 10−2 is larger. Check: 0.012 is bigger than 0.0078.
2. How many times larger is 6 × 105 than 2 × 102?
Show answer
(6 ÷ 2) × 105−2 = 3 × 103, so 3000 times larger. Check: 600 000 ÷ 200 = 3000.
3. Put in ascending order: 2.5 × 10−3, 0.04, 1.8 × 10−2.
Show answer
As ordinary numbers: 0.0025, 0.04, 0.018. So the order is 2.5 × 10−3, 1.8 × 10−2, 0.04.
Where this leads next
Next, see how to tell whether an answer is the right size in estimating the size of a standard-form answer. The practice set for this module mixes all the skills together. The non-calculator working trainer can help you rehearse the routine by hand.
If comparisons go wrong only on timed questions, online one-to-one Mathematics tuition gives a teacher the chance to watch your routine as you work.