The Sun is about 150,000,000,000 metres away. A water molecule is about 0.000000000275 metres wide. Learn the short, standard way to write, compare and calculate with numbers like these.
Light from the Sun takes about 500 seconds to reach us. A red blood cell is about 0.000007 metres wide. Both numbers are hard to write, hard to compare and easy to get wrong. Scientific notation is the short, standard way to write them, and once you know it, you can multiply, divide and compare huge and tiny numbers quickly and safely.
Look at these two numbers:
To read the first number, you must count eleven zeros. To read the second, you must count nine zeros after the decimal point, and one slip makes the answer ten times too big or too small. In real work, this is dangerous. A scientist who is wrong by one zero is wrong by a factor of ten.
Scientific notation removes this problem. It writes every number in the same short pattern, so the size of the number is easy to see at a glance. In many O-Level papers the same idea is called standard form. The two names mean exactly the same thing, so use whichever your teacher uses.
A number is in scientific notation when it is written as a number between 1 and 10, multiplied by a power of 10.
a × 10n
The number a must be at least 1 and less than 10 (written 1 ≤ a < 10). The power n must be a whole number, positive or negative.
The power of 10 tells you how many places the decimal point has moved:
Remember that 103 = 1,000 and 10-3 = 0.001. A negative power does not make the number negative. It makes the number small.
Use these three steps:
Write 4,560,000 in scientific notation.
Step 1: An ordinary whole number has its decimal point at the right end: 4,560,000.
Step 2: Move the point to just after the 4: 4.560000. Remove the useless zeros: 4.56.
Step 3: The point moved 6 places left, so the power is 6.
Answer: 4.56 × 106
Write 150,000,000,000 m (the Earth–Sun distance, rounded) in scientific notation.
Move the point after the 1: 1.50000000000. The point moved 11 places left.
Answer: 1.5 × 1011 m
Check: 1.5 × 1011 = 1.5 × 100,000,000,000 = 150,000,000,000. ✓
For a number smaller than 1, the decimal point moves to the right, so the power is negative. Count the places you move.
Write 0.00037 in scientific notation.
Step 1: Move the point to just after the first non-zero digit (the 3): 0.00037 → 3.7.
Step 2: Count the moves: 0.0003|7 needs 4 places to the right.
Step 3: The number is small, so the power is negative: −4.
Answer: 3.7 × 10-4
Check: 3.7 × 10-4 = 3.7 ÷ 10,000 = 0.00037. ✓
A helpful memory trick: big number, positive power; small number, negative power. If your answer breaks this rule, check your counting.
To reverse the process, move the decimal point the number of places shown by the power. A positive power moves it right (the number gets bigger). A negative power moves it left (the number gets smaller). Add zeros as needed.
(a) Write 5.3 × 105 as an ordinary number. Move the point 5 places right: 5.30000 → 530,000.
(b) Write 8.2 × 10-3 as an ordinary number. Move the point 3 places left: 0008.2 → 0.0082.
Sometimes a number looks like scientific notation but the first part is not between 1 and 10. Fix it by changing the first part and adjusting the power to match.
(a) 45 × 104: 45 = 4.5 × 10, so 45 × 104 = 4.5 × 10 × 104 = 4.5 × 105.
(b) 0.62 × 10-3: 0.62 = 6.2 ÷ 10 = 6.2 × 10-1, so 0.62 × 10-3 = 6.2 × 10-1 × 10-3 = 6.2 × 10-4.
Check (a): 45 × 10,000 = 450,000 and 4.5 × 105 = 450,000. ✓ Check (b): 0.62 × 0.001 = 0.00062 and 6.2 × 10-4 = 0.00062. ✓
Scientific notation is how scientists compare things of very different sizes. Every extra power of 10 means ten times bigger. Going from 10-10 m (an atom) to 1011 m (the Sun’s distance) is a jump of 21 powers of ten, so the distance to the Sun is about 1021 times the size of an atom.
Scientists also use SI prefixes, which are just names for powers of ten. You already know kilo (103) and milli (10-3). Here are the ones you need most:
| Prefix | Symbol | Power of 10 | Meaning |
|---|---|---|---|
| tera | T | 1012 | trillion |
| giga | G | 109 | billion |
| mega | M | 106 | million |
| kilo | k | 103 | thousand |
| milli | m | 10-3 | thousandth |
| micro | µ | 10-6 | millionth |
| nano | n | 10-9 | billionth |
So 3 nanometres is 3 × 10-9 m, and 64 gigabytes is 64 × 109 bytes, which is 6.4 × 1010 bytes in scientific notation. For a full list of official SI prefixes, see the link in Further Reading.
Scientific notation makes comparing easy. Follow this order:
Which is larger: 3.2 × 10-5 or 2.9 × 10-4?
The powers are −5 and −4. Since −4 is larger than −5, the second number is larger.
Check: 0.000032 and 0.00029. The second number has its first non-zero digit one place earlier, so it is bigger. ✓
Answer: 2.9 × 10-4
This is where scientific notation really helps. The rules come from the laws of indices.
(3.2 × 104) × (2.5 × 103)
First parts: 3.2 × 2.5 = 8. Powers: 4 + 3 = 7.
Answer: 8 × 107 (that is 80,000,000).
(4 × 105) × (6 × 103)
First parts: 4 × 6 = 24. Powers: 5 + 3 = 8. This gives 24 × 108, but 24 is not between 1 and 10.
Write 24 = 2.4 × 10, so 24 × 108 = 2.4 × 109.
Answer: 2.4 × 109 (that is 2,400,000,000).
(9.6 × 108) ÷ (3.2 × 103)
First parts: 9.6 ÷ 3.2 = 3. Powers: 8 − 3 = 5.
Answer: 3 × 105 (that is 300,000).
Powers work the same way. To square or cube a number in scientific notation, raise the first part to the power and multiply the power of 10 by that number. For example, (3 × 104)3 = 33 × 104×3 = 27 × 1012 = 2.7 × 1013.
You cannot add or subtract directly unless the powers of 10 are the same. First make them match, then add or subtract the first parts and keep the power.
3.4 × 105 + 6.0 × 104
Step 1: Match the powers. Change 3.4 × 105 to 34 × 104.
Step 2: Add the first parts: 34 + 6.0 = 40, so the sum is 40 × 104.
Step 3: Adjust: 40 × 104 = 4.0 × 105.
Answer: 4.0 × 105
Check: 340,000 + 60,000 = 400,000. ✓
7.2 × 10-3 − 5 × 10-4
Change 7.2 × 10-3 to 72 × 10-4. Then 72 − 5 = 67, so the result is 67 × 10-4 = 6.7 × 10-3.
Check: 0.0072 − 0.0005 = 0.0067. ✓
Answer: 6.7 × 10-3
Scientific notation also shows how accurate a measurement is. The digits in the first part are the significant figures. A final zero in the first part is kept on purpose, because it tells us the measurement was that precise.
To round, look at the digit after the place you need. For example, 3.14159 × 105 to 3 significant figures is 3.14 × 105, because the next digit (1) is less than 5. In exam answers, always follow the accuracy that the question asks for.
Most scientific calculators have a key marked ×10x or EXP. To enter 3.2 × 104, type 3.2, press that key, then type 4. Do not type × 10 yourself; use the key. Different models behave a little differently, so check your calculator’s manual or ask your teacher.
Spreadsheets and some calculators show very large or very small numbers with an E, for example 3.2E+04 or 7.2E-03. This E means “times ten to the power of” and has nothing to do with the mathematical constant e. When you write your answer in an exam, always convert it back to a × 10n.
Light travels at about 3 × 108 m/s. The Sun is about 1.5 × 1011 m away. Time = distance ÷ speed, so time = (1.5 × 1011) ÷ (3 × 108). Divide the first parts: 1.5 ÷ 3 = 0.5. Subtract the powers: 11 − 8 = 3. This gives 0.5 × 103, which adjusts to 5 × 102 seconds, or 500 s. That is 8 minutes and 20 seconds. When you look at the Sun, you see it as it was more than 8 minutes ago.
A phone has 64 gigabytes of storage. Storage makers count 1 GB as 109 bytes, so the storage is 64 × 109 = 6.4 × 1010 bytes. If one photo takes about 4 megabytes (4 × 106 bytes), then the number of photos is (6.4 × 1010) ÷ (4 × 106) = 1.6 × 104 = 16,000 photos. In real life the number is smaller, because the phone’s system and apps also use space.
A light year is the distance light travels in one year. One year has 365 × 24 × 3,600 = 31,536,000 seconds, which is about 3.15 × 107 s. Distance = speed × time = (3 × 108) × (3.1536 × 107) = 9.4608 × 1015 m, so one light year is about 9.46 × 1015 m. The nearest star after the Sun, Proxima Centauri, is about 4.24 light years away, which is about 4.0 × 1016 m. Scientific notation lets us write this in a few characters instead of 17 digits.
Chemistry: Avogadro’s constant, the number of particles in one mole, is about 6.022 × 1023. Biology: cell sizes and virus sizes are written in micrometres and nanometres. Computing: storage and speed use the prefixes mega, giga and tera. Finance and economics: very large amounts are often shown as, for example, 4.5 × 109. Astronomy: distances between planets and stars are written this way, because the numbers are enormous.
Write your answers on paper first, then open the answers to check.
1. Move the point 7 places left: 9.25 × 107.
2. Move the point 3 places right: 6.04 × 10-3.
3. First parts: 6 × 5 = 30. Powers: −3 + 5 = 2. So 30 × 102. Adjust: 3 × 103. (Check: 0.006 × 500,000 = 3,000. ✓)
These questions are in the style of O-Level papers. Try each one before opening the answers.
1. 5.08 × 10-5.
2. 67,000.
3. 3.6 ÷ 9 = 0.4 and 5 − 2 = 3, so 0.4 × 103 = 4 × 102.
4. 2.5 × 104 × 30 = 75 × 104 = 7.5 × 105 bolts.
5. Number of cells = 10-2 ÷ (7 × 10-6) = (1 ÷ 7) × 104 ≈ 0.1429 × 104 = 1.4 × 103 (about 1,400 cells).
Students often learn the “move the point” rule without understanding why it works. A good way to build understanding is to ask: “How many times bigger is 106 than 105?” and let them see that each step is ten times. Then ask them to estimate before they calculate. For example, ask whether 4.5 × 105 should be bigger or smaller than 1,000,000. Estimation catches most sign and counting errors.
Real contexts help: phone storage, distances in the solar system, and the size of cells and viruses make the topic meaningful. Watch for the two most common errors: adding the first parts but forgetting to match powers, and giving a first part such as 24 or 0.3 in the final answer. Encourage students to check each answer by converting back to an ordinary number.
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