To write a number in scientific notation, put the decimal point after the first non-zero digit, count how many places it moved, and use that count as the exponent on 10 — positive if the point moved left (a big number), negative if it moved right (a small number). For example, 299,792,458 = 2.99792458 × 10^8 and 0.0000000045 = 4.5 × 10^-9.
Scientific notation is simply a compact way of writing numbers that are too big or too small for everyday form. Scientists use it for the speed of light, the mass of an electron, and the distance to the Sun, because writing out all the zeros invites exactly one mistake: miscounting them. Below is the three-step method, worked examples for big and small numbers, the trap that catches most students, and a shortcut for counting the places.
What “Scientific Notation” Actually Means
A number in scientific notation has two parts: a coefficient and a power of 10. It is written as a × 10^n, where the coefficient a is at least 1 and less than 10, and the exponent n is a whole number — positive, negative, or zero. So 4,500,000 becomes 4.5 × 10^6, and 0.00025 becomes 2.5 × 10^-4. The coefficient carries the digits; the exponent carries the size. That split is the whole point: you can read the size of a number from a single digit, the exponent, instead of counting a wall of zeros.
Check any value instantly with a free scientific notation calculator, which shows the decimal-point moves step by step.
The 3-Step Method
The method is the same for every number. Learn it once and it works for a population figure, a wavelength, or a bank balance.
Step 1: Put the decimal point after the first non-zero digit.
Look at the number and find its first digit that is not zero. Move the decimal point to sit right after it. In 4,500,000 the first non-zero digit is 4, so the point goes after the 4: 4.5. In 0.0000000045 the first non-zero digit is also 4, so the point goes after the 4: 4.5 again.
Step 2: Count how many places the point moved.
Count the jumps. For 4,500,000 the point moved 6 places left; for 0.0000000045 it moved 9 places right. The count becomes the exponent — but the direction decides the sign, which is step 3.
Step 3: Set the sign — left means positive, right means negative.
If the original number was bigger than 1, the point moved left and the exponent is positive. If the original number was between 0 and 1, the point moved right and the exponent is negative. So 4,500,000 = 4.5 × 10^6 and 0.0000000045 = 4.5 × 10^-9. If you ever forget the sign, do the size check: 4.5 × 10^9 would be 4,500,000,000 — a billion times too big for 0.0000000045 — so the exponent has to be negative.
Worked Example: Big Numbers
Worked example: write 299,792,458 in scientific notation — the speed of light.
- Find the first non-zero digit. It is the 2 at the front.
- Put the point after it. 299,792,458 becomes 2.99792458.
- Count the moves. The point moved 8 places left — count the digits after the 2: there are 8 of them.
- Set the sign. The number is bigger than 1, so the exponent is positive: +8.
- Write the answer: 299,792,458 = 2.99792458 × 10^8 m/s.
More big numbers, worked the same way: 92,000,000 → the point moves 7 places left → 9.2 × 10^7. 86,400 (the seconds in one day) → the point moves 4 places left → 8.64 × 10^4. 377,000 → 3.77 × 10^5. 52,000 → 5.2 × 10^4. And 93,000 → 9.3 × 10^4. In every case the exponent equals the number of digits after the first digit — a shortcut worth memorising: count the digits after the first one, and that is your positive exponent.
Worked Example: Small Numbers (and the Trap)
Worked example: write 0.0000000045 in scientific notation — the number most people get wrong in 5 seconds.
- Find the first non-zero digit. It is the 4 — after 8 zeros.
- Put the point after it. 0.0000000045 becomes 4.5.
- Count the moves. The point moved 9 places right.
- Set the sign. The number is smaller than 1, so the exponent is negative: −9.
- Write the answer: 0.0000000045 = 4.5 × 10^-9.
The trap is step 4. Most people write 4.5 × 10^9 — the right digits, the wrong sign — and never notice, because 4.5 × 10^9 looks plausible until you check the size: it is 4,500,000,000, while the number you started with is smaller than a hundred-millionth. The size check takes two seconds and catches the single most common scientific-notation mistake in exams.
More small numbers: 0.00025 → the point moves 4 places right → 2.5 × 10^-4. 0.0061 → 6.1 × 10^-3. 0.0000078 → 7.8 × 10^-6. 0.00032 → 3.2 × 10^-4. 0.000101 → 1.01 × 10^-4. And 0.000145 → 1.45 × 10^-4. The shortcut for small numbers: count the zeros before the first non-zero digit and add one — 0.0000000045 has 8 zeros, so the exponent is −(8 + 1) = −9.
The Counting Shortcut (Both Directions)
- Big number (≥ 1): the exponent is the number of digits after the first digit. 299,792,458 has 8 digits after the 2 → 10^8. 4,500,000 has 6 digits after the 4 → 10^6.
- Small number (< 1): the exponent is minus (the zeros before the first non-zero digit, plus one). 0.0000000045 has 8 zeros → −9. 0.00025 has 3 zeros → −4.
Both shortcuts are the same rule counted two ways — the exponent is always the number of places the decimal point moved, and the sign follows the direction. A free scientific notation calculator counts the moves for you and shows them, so you can check your hand-counted exponent in one click.
What About Numbers Between 1 and 10, or Exactly 10?
Numbers already between 1 and 10 need no move at all: 7.0782 is 7.0782 × 10^0, and 707.82 is 7.0782 × 10^2 (the point moved 2 places left). The number 10 is 1 × 10^1, and 100 is 1 × 10^2. And 1 itself is 1 × 10^0, because any number to the power 0 is 1 — the same rule that makes x^0 = 1 for any base, which the exponent calculator shows for any value you enter.
Rounding to Significant Figures
Textbooks rarely want all eight digits of 2.99792458. Rounded to 3 significant figures, the speed of light is 3.00 × 10^8 m/s — and the trailing zeros matter: 3.00 shows three significant figures of precision, while 3 × 10^8 shows only one. When you round, round the coefficient and keep the exponent exactly as it is: 6.02 × 10^23 is Avogadro’s number to 3 significant figures, and 1.496 × 10^8 km is the Earth–Sun distance to 4. Never round the exponent itself — it is a count of places, not a measurement.
4 Mistakes That Cost Marks
1. Flipping the exponent sign. The #1 trap, covered above: 0.0000000045 = 4.5 × 10^-9, never 4.5 × 10^9. Always run the size check — does your answer look bigger or smaller than 1, the way the original was?
2. Letting the coefficient leave the 1-to-10 range. 45 × 10^-10 equals 0.0000000045, but it is not scientific notation — the coefficient must be between 1 and 10. Move the point one place and adjust the exponent: 4.5 × 10^-9.
3. Counting the moves wrong by one. 0.0000000045 needs 9 moves, not 8 — count the zeros (8) and add one. For 299,792,458 the exponent is 8, not 9 — count the digits after the first digit (8). An off-by-one error changes the answer by a factor of 10.
4. Writing the exponent as a decimal. The exponent must be a whole number. 4.5 × 10^6.5 is not scientific notation — and it is not equal to 4,500,000 either. If your count comes out fractional, you miscounted the moves.
Frequently Asked Questions
How do you write numbers in scientific notation?
Move the decimal point after the first non-zero digit, count how many places it moved, and use that count as the exponent on 10 — positive if the point moved left (a number bigger than 1), negative if it moved right (a number between 0 and 1). For example, 299,792,458 = 2.99792458 × 10^8 and 0.0000000045 = 4.5 × 10^-9.
What is 0.0000000045 in scientific notation?
4.5 × 10^-9. The decimal point moves 9 places to the right to reach 4.5, and because the number is smaller than 1, the exponent is negative. The common wrong answer, 4.5 × 10^9, is 4,500,000,000 — a billion times too big.
How do you write 299,792,458 in scientific notation?
2.99792458 × 10^8. The decimal point moves 8 places to the left to reach 2.99792458, and because the number is larger than 1, the exponent is positive. This is the speed of light in metres per second, usually rounded to 3.00 × 10^8 m/s.
When is the exponent negative in scientific notation?
Whenever the original number is between 0 and 1. The decimal point has to move right to reach a coefficient between 1 and 10, and moving right means a negative exponent: 0.00025 = 2.5 × 10^-4, 0.0061 = 6.1 × 10^-3.
How do you write a number like 707.82 in scientific notation?
Put the point after the first digit: 7.0782. It moved 2 places left, so the exponent is +2: 707.82 = 7.0782 × 10^2. Numbers already between 1 and 10 take the exponent 0.
Why do scientists use scientific notation?
Because the exponent carries the size where it cannot be miscounted. 6.02 × 10^23 and 6.626 × 10^-34 say at a glance “about 10^23” and “about 10^-34”, while the full numbers are walls of zeros where one miscounted zero changes the answer tenfold.
Keep Practicing
Work these by hand, then verify each with the free scientific notation calculator: 4,500,000 = 4.5 × 10^6; 0.00025 = 2.5 × 10^-4; 92,000,000 = 9.2 × 10^7; 0.0061 = 6.1 × 10^-3; 86,400 = 8.64 × 10^4; and 0.0000078 = 7.8 × 10^-6.