If you have been handed a specific number and told to "write this in scientific notation," you do not need another lecture on the rules. You need to see that exact type of number worked through, step by step, so you can check your answer or finish the assignment. That is what this page is: a bank of solved conversions for the numbers people search for most, each showing the answer and the two or three steps to get there. If you would rather type in your own number and get every notation form at once, the Scientific Notation Calculator is the companion tool to this page.
Table of Contents
- The short version: one rule
- How to convert any number in three steps
- Powers of ten: the pattern behind the exponent
- Whole numbers, worked
- Decimals smaller than 1, worked
- Named magnitudes: thousand, million, billion
- Standard form vs scientific notation
- Converting back to standard form
- Real-world numbers, worked
- More numbers to practice
- Worked examples
- Common mistakes to avoid
- Frequently asked questions
- References
The Short Version: One Rule
Every number in scientific notation looks like a × 10n, where a is at least 1 but less than 10, and n tells you how far, and which way, the decimal point moved. That is the whole idea. Two directions cover every case:
- Big number (10 or more): the decimal moves left, so the exponent is positive.
- Small number (less than 1): the decimal moves right, so the exponent is negative.
Everything below is that one rule applied to real numbers. If you understand this much, the rest is counting.
How to Convert Any Number in Three Steps
Before the worked list, here is the method every example follows. It works the same whether the number is enormous or tiny.
- Place the decimal after the first non-zero digit. Read the number left to right, find the first digit that is not zero, and put the decimal point right after it. That gives you the coefficient a, a value between 1 and 10.
- Count how many places the decimal moved. That count is the size of your exponent n.
- Set the sign. If the original number was 10 or bigger (decimal moved left), the exponent is positive. If it was less than 1 (decimal moved right), the exponent is negative.
Here is why the sign flips. A big number needs the decimal dragged back toward the front, moving left, so you are recording how many powers of ten you removed: a positive count. A small number needs the decimal pushed forward past its leading zeros, moving right, so you are recording powers of ten you owe back: a negative count. Keep the direction and the sign paired in your head and you will not mix them up.
Powers of Ten: The Pattern Behind the Exponent
The exponent is not a mystery symbol. It is just a count of how many times you multiply or divide by ten, and every value in scientific notation is built on that ladder. Reading the table below out loud once makes the whole system click: each step down the exponent adds a zero to a big number, and each step into the negatives adds a leading zero to a small one.
| Power of ten | Standard value | In words |
|---|---|---|
| 10−3 | 0.001 | one thousandth |
| 10−2 | 0.01 | one hundredth |
| 10−1 | 0.1 | one tenth |
| 100 | 1 | one |
| 101 | 10 | ten |
| 102 | 100 | one hundred |
| 103 | 1,000 | one thousand |
| 106 | 1,000,000 | one million |
| 109 | 1,000,000,000 | one billion |
Two rows are worth pausing on. Anything to the power of zero is 1, which is why the coefficient itself sits at the 100 line before you shift anything. And the negative side is a mirror of the positive side: 10−3 is as far below 1 as 103 is above it. Once the ladder is in your head, a conversion is just a question of which rung the number sits on.
Whole Numbers, Worked
These are the friendliest cases: the decimal starts at the far right, and every move is to the left, so every exponent is positive. Count the moves and you have the answer.
10 in scientific notation. Move the decimal one place left and a single digit sits at the front. → 1 × 101
118,000 in scientific notation. Move the decimal 5 places left: 118,000 becomes 1.18. → 1.18 × 105
384,400 in scientific notation (the average Earth to Moon distance in km). Move the decimal 5 places left: 384,400 becomes 3.844. → 3.844 × 105
673.5 in scientific notation. Move the decimal 2 places left: 673.5 becomes 6.735. → 6.735 × 102
9,228,000 in scientific notation. Move the decimal 6 places left: 9,228,000 becomes 9.228. → 9.228 × 106
48,200 in scientific notation. Move the decimal 4 places left: 48,200 becomes 4.82. → 4.82 × 104
23,000 in scientific notation. Move the decimal 4 places left: 23,000 becomes 2.3. → 2.3 × 104
700,000 in scientific notation. Move the decimal 5 places left: 700,000 becomes 7. The trailing zeros disappear into the exponent. → 7 × 105
9,000,000 in scientific notation. Move the decimal 6 places left: 9,000,000 becomes 9. → 9 × 106
80,023 in scientific notation. Move the decimal 4 places left: 80,023 becomes 8.0023. The internal zeros stay because they sit between non-zero digits. → 8.0023 × 104
Decimals Smaller Than 1, Worked
These trip people up the most, because it is easy to miscount the leading zeros. The trick: count every digit after the decimal point up to and including the first non-zero digit. That count is the size of the negative exponent.
0.00086 in scientific notation. Move the decimal 4 places right: 0.00086 becomes 8.6. → 8.6 × 10−4
0.0034 in scientific notation. Move the decimal 3 places right: 0.0034 becomes 3.4. → 3.4 × 10−3
0.00053 in scientific notation. Move the decimal 4 places right: 0.00053 becomes 5.3. → 5.3 × 10−4
0.00097 in scientific notation. Move the decimal 4 places right: 0.00097 becomes 9.7. → 9.7 × 10−4
0.00063 in scientific notation. Move the decimal 4 places right: 0.00063 becomes 6.3. → 6.3 × 10−4
0.00081 in scientific notation. Move the decimal 4 places right: 0.00081 becomes 8.1. → 8.1 × 10−4
0.00125 in scientific notation. Move the decimal 3 places right: 0.00125 becomes 1.25. → 1.25 × 10−3
0.00119 in scientific notation. Move the decimal 3 places right: 0.00119 becomes 1.19. → 1.19 × 10−3
0.0018 in scientific notation. Move the decimal 3 places right: 0.0018 becomes 1.8. → 1.8 × 10−3
0.000354 in scientific notation. Move the decimal 4 places right: 0.000354 becomes 3.54. → 3.54 × 10−4
0.000013 in scientific notation. Move the decimal 5 places right: 0.000013 becomes 1.3. → 1.3 × 10−5
0.042 in scientific notation. Only one leading zero here, so move the decimal 2 places right: 0.042 becomes 4.2. → 4.2 × 10−2
0.3643 in scientific notation. The first non-zero digit is right after the point, so move 1 place right: 0.3643 becomes 3.643. → 3.643 × 10−1
0.5401 in scientific notation. Move the decimal 1 place right: 0.5401 becomes 5.401. → 5.401 × 10−1
Named Magnitudes: Thousand, Million, Billion
Word problems love these because they test whether you understand place value, not just decimal-counting. The move is to write the full number out first, then convert it the same way as any other whole number.
427 thousand in scientific notation. Write it as 427,000, then move the decimal 5 places left to get 4.27. → 4.27 × 105
1 million in scientific notation. 1,000,000 has 6 zeros, and each zero is one power of ten. → 1 × 106
78 million in scientific notation. Write it as 78,000,000, then move the decimal 7 places left to get 7.8. → 7.8 × 107
150 million in scientific notation. Write it as 150,000,000, then move the decimal 8 places left to get 1.5. → 1.5 × 108
4.5 billion in scientific notation. Write it as 4,500,000,000, then move the decimal 9 places left to get 4.5. → 4.5 × 109
1,000,000,000 in scientific notation. One billion has 9 zeros. → 1 × 109
Notice the pattern on the scale: thousand is 103, million is 106, billion is 109. Every named jump adds 3 to the exponent, which is exactly why scientific notation makes these quantities easy to compare. Trying to line up 150,000,000 against 4,500,000,000 by counting zeros is error-prone. Comparing 108 against 109 is instant.
Standard Form vs Scientific Notation
Here is a quick side-by-side of five conversions from the lists above, so you can see the two formats next to each other.
| Standard Form | Scientific Notation |
|---|---|
| 673.5 | 6.735 × 102 |
| 384,400 | 3.844 × 105 |
| 0.0034 | 3.4 × 10−3 |
| 150,000,000 | 1.5 × 108 |
| 0.00086 | 8.6 × 10−4 |
Converting Back to Standard Form
Reading scientific notation back into a plain number is the same rule run in reverse. The exponent tells you how far to move the decimal, and the sign tells you which way.
- Positive exponent: move the decimal right that many places, padding with zeros. 3.844 × 105 becomes 384,400 (move 5 places right).
- Negative exponent: move the decimal left that many places, adding leading zeros. 8.6 × 10−4 becomes 0.00086 (move 4 places left).
The direction reverses because you are undoing the conversion. When you wrote the number in scientific notation you moved the decimal one way; to read it back you move it the opposite way by the same number of places. If a conversion and its reverse do not land on the original number, one of the two move counts is off.
Real-World Numbers, Worked
Scientific notation is not just a worksheet exercise. It is how measured quantities get written across science and geography, precisely because those numbers are awkward in standard form. A few worth practising:
9,851 in scientific notation. Move the decimal 3 places left: 9,851 becomes 9.851. → 9.851 × 103
5,008 km in scientific notation (roughly the length of the US Pacific coastline). Move the decimal 3 places left: 5,008 becomes 5.008. → 5.008 × 103
170 in scientific notation. Move the decimal 2 places left: 170 becomes 1.7. → 1.7 × 102
480 in scientific notation. Move the decimal 2 places left: 480 becomes 4.8. → 4.8 × 102
770 in scientific notation. Move the decimal 2 places left: 770 becomes 7.7. → 7.7 × 102
Geographic and astronomical figures like the 384,400 km Earth to Moon distance or the 5,008 km coastline above are exactly where this notation earns its keep. When you need the straight-line gap between two mapped points to feed into a calculation like these, the Distance Calculator returns it in plain form, ready to convert with the same left-shift rule.
More Numbers to Practice
Cover the answers and try these five yourself first. Each mixes a slightly different feature: a large number with a single significant digit, one with several, a small two-digit result, and one with many leading zeros. Work each one with the three-step method, then check.
6,400,000 in scientific notation. Move the decimal 6 places left: 6,400,000 becomes 6.4. → 6.4 × 106
2,030,000 in scientific notation. Move the decimal 6 places left: 2,030,000 becomes 2.03. The internal zero stays because it sits between the 2 and the 3. → 2.03 × 106
52 in scientific notation. Move the decimal 1 place left: 52 becomes 5.2. → 5.2 × 101
0.0906 in scientific notation. Move the decimal 2 places right, past the single leading zero to the 9: 0.0906 becomes 9.06. → 9.06 × 10−2
0.0000075 in scientific notation. Move the decimal 6 places right, past five leading zeros to the 7: 0.0000075 becomes 7.5. → 7.5 × 10−6
If any of these came out with the wrong sign or an off-by-one exponent, the two usual culprits are counting the leading zeros wrong or forgetting that a number below 1 always takes a negative exponent. Both are covered in the mistakes section below.
Worked Examples
The quick conversions above show the answer and the key move. These five go slower, spelling out every step, including the trickier normalizing cases and the two-number operations the Scientific Notation Calculator handles for you.
Example 1: A Whole Number With Internal Zeros (80,023)
Convert 80,023 to scientific notation.
- Find the first non-zero digit reading left to right: the 8. Place the decimal after it to get the coefficient 8.0023.
- Count the places the decimal moved from the right end of 80,023 to just after the 8: 4 places left.
- The number is bigger than 10, so the exponent is positive: +4.
- Result: 8.0023 × 104.
The two zeros between the 8 and the 23 stay in the coefficient because they sit between non-zero digits. Only the trailing structure gets folded into the exponent, never internal zeros.
Example 2: A Small Decimal With Leading Zeros (0.000354)
Convert 0.000354 to scientific notation.
- Scan right from the decimal point to the first non-zero digit: the 3. Place the decimal after it to get 3.54.
- Count every digit crossed, including the leading zeros: 0, 0, 0, then 3, which is 4 places right.
- The number is less than 1, so the exponent is negative: −4.
- Result: 3.54 × 10−4.
Miscounting one leading zero is the single most common error here. Counting to the first non-zero digit, not just counting the zeros, keeps you honest.
Example 3: A Named Magnitude (4.5 Billion)
Convert 4.5 billion to scientific notation.
- Write the words as digits: 4.5 billion is 4,500,000,000.
- Place the decimal after the first digit: 4.5 (it is already there).
- Count the places from the right end back to just after the 4: 9 places left.
- Result: 4.5 × 109.
Because a billion is 109, any "point-something billion" figure lands on an exponent of 9 with the leading digits as the coefficient. The same shortcut gives 78 million as 7.8 × 107 and 150 million as 1.5 × 108.
Example 4: Reading It Back to Standard Form (3.844 × 105)
Convert 3.844 × 105 back to a normal number.
- The exponent is a positive 5, so move the decimal 5 places to the right.
- 3.844 has three digits after the point, so the first three moves use them up: 384.4.
- Two moves remain, so pad with two zeros: 384,400.
- Result: 384,400, the Earth to Moon distance in km.
This is Example number 3 from the whole-number list run backwards, and it lands exactly on 384,400, which is the check that both directions agree.
Example 5: Multiplying Two Numbers in Scientific Notation
Calculate (1.18 × 105) × (3.0 × 102). This is what happens when you multiply 118,000 by 300.
- Multiply the coefficients: 1.18 × 3.0 = 3.54.
- Add the exponents: 5 + 2 = 7.
- Result: 3.54 × 107 (which is 35,400,000).
Multiplying and dividing are actually easier than adding in this notation, because you never have to line up the exponents first: you multiply the fronts and add the powers. To divide instead, you would divide the coefficients and subtract the exponents. When a result needs comparing against another quantity as a share, the Percentage Calculator takes the two plain numbers directly.
Common Mistakes to Avoid
Leaving the coefficient at 10 or more. The coefficient must be at least 1 and less than 10. If a calculation gives you 13.5 × 106, that is not finished: shift the decimal one more place left and bump the exponent up by one to get 1.35 × 107.
Getting the sign backwards. Big numbers get positive exponents, small numbers get negative ones. If you write 0.0034 as 3.4 × 103, you have described a number in the thousands, not a number smaller than 1. Say the direction out loud, "small number, decimal moves right, negative," until it is automatic.
Miscounting leading zeros. For decimals, count every digit up to and including the first non-zero one. Counting only the zeros, or stopping one short, is what turns a correct −4 into a wrong −3.
Dropping internal zeros. In 80,023 the middle zeros are part of the coefficient (8.0023), not spare digits to fold into the exponent. Only the trailing zeros of a round number like 700,000 disappear.
Assuming more digits means more precision. Writing 7 × 105 as 7.00000 × 105 claims a precision the original 700,000 never had. Keep the coefficient to the significant digits actually given.
Frequently Asked Questions
What is scientific notation in simple terms?
It is a compact way to write very large or very small numbers as a coefficient times a power of ten, in the form a × 10n, where the coefficient is at least 1 but less than 10. For example, 384,400 becomes 3.844 × 105 and 0.00086 becomes 8.6 × 10−4.
How do you write a number in scientific notation step by step?
Place the decimal point just after the first non-zero digit to get a coefficient between 1 and 10, count how many places the decimal moved, and use that count as the exponent. The exponent is positive if the original number was 10 or larger, and negative if it was less than 1.
How do you know if the exponent is positive or negative?
Look at the size of the original number. A number of 10 or more needs the decimal moved left, which gives a positive exponent. A number less than 1 needs the decimal moved right past its leading zeros, which gives a negative exponent.
What is 0.00086 in scientific notation?
Move the decimal 4 places to the right, past the three leading zeros and up to the 8, to get 8.6. Because the number is less than 1, the exponent is negative, so 0.00086 = 8.6 × 10−4.
What is 384,400 in scientific notation?
Move the decimal 5 places left to get 3.844, giving 3.844 × 105. This is the average distance from the Earth to the Moon in kilometres, a classic example of why the notation is useful.
How do you convert scientific notation back to a normal number?
Move the decimal by the number of places shown in the exponent: right for a positive exponent, left for a negative one, padding with zeros as needed. For instance, 8.6 × 10−4 moves 4 places left to become 0.00086, and 3.844 × 105 moves 5 places right to become 384,400.
What is 1 million in scientific notation, and 4.5 billion?
One million is 1,000,000, which has 6 zeros, so it is 1 × 106. Written out, 4.5 billion is 4,500,000,000, so the decimal moves 9 places left to give 4.5 × 109.
Can the coefficient be 10 or more?
No. The coefficient must be at least 1 and less than 10. If a calculation produces something like 13.5 × 106, you normalize it by moving the decimal one more place left and adding 1 to the exponent, giving 1.35 × 107.
What is the difference between scientific and engineering notation?
Both use a coefficient times a power of ten, but engineering notation restricts the exponent to multiples of three (such as 103, 106, 10−3) so it lines up with SI prefixes like kilo, mega, and milli. Scientific notation allows any integer exponent.
How many significant figures does a number in scientific notation have?
Exactly the number of digits in the coefficient. Writing 700,000 as 7 × 105 states 1 significant figure, while 7.00 × 105 would state 3. That clarity is one of the main reasons the notation is used in science.
How can I check my scientific notation answer quickly?
Enter your original number into the Scientific Notation Calculator. It returns the scientific, engineering, and e-notation forms plus the significant figure count, and it also multiplies, divides, adds, and subtracts two numbers in scientific notation with the working shown.
References
- NIST Guide to the SI, Chapter 7: the authoritative US federal reference for expressing quantities with powers of ten.
- Khan Academy: Scientific Notation Review: a clear walkthrough of writing and reading numbers with positive and negative exponents.
- Math Is Fun: Scientific Notation: worked conversions in both directions with interactive examples.