Scientific Notation Calculator & Converter
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A scientific notation calculator writes any number as a × 10ⁿ, where the coefficient a is at least 1 and less than 10 and the exponent n counts how many places the decimal point moves. For example, 45,600,000 = 4.56 × 10⁷ (the point moves 7 places left) and 0.00042 = 4.2 × 10⁻⁴ (4 places right). In E notation these are 4.56E7 and 4.2E-4.
Show steps
- Place the decimal point after the first non-zero digitThe first non-zero digit is 4, so the coefficient is 4.5600000, a number from 1 up to (but not including) 10.
- Count the places the point movedThe decimal point moved 7 places to the left, so the exponent is positive: n = 7. Moving left makes the coefficient smaller, so 10⁷ makes up for it.
- Keep the significant digitsThe number has 3 significant figures (up to 8), so the coefficient keeps 3 digits: 4.56. The trailing zeros are placeholders, so they are dropped. If they were measured, choose up to 8 significant figures above.
- Write the answerScientific notation: 4.56 × 107. E notation: 4.56E7. Engineering notation (exponent a multiple of 3): 45.6 × 106.
- Check4.56 × 10,000,000 = 45,600,000.
Related: significant figures calculator · rounding calculator · exponent calculator
How to use the scientific notation calculator
- To scientific: type a number in standard form (45,600,000), E notation (4.56e7) or as a × 10^n (53 x 10^3). You get scientific, E and engineering notation, the standard form, the order of magnitude and the number in words, plus a picture of the decimal point moving.
- To standard: type the coefficient and the exponent separately, for example 3.456 and 4, to get 34,560 with every step.
- Calculate: add, subtract, multiply, divide, raise to a power or take a square root. The answer is shown exactly and rounded by significant-figure rules, so it works for both math and science classes.
Choose a number of significant figures to round the coefficient, or keep the number’s own digits. The steps update as you type.
What is scientific notation?
Scientific notation is a short way to write very large and very small numbers as a coefficient times a power of 10: a × 10ⁿ. The coefficient (also called the significand or mantissa) has exactly one non-zero digit before the decimal point, so 1 ≤ |a| < 10. The exponent n is a whole number. A positive exponent means a large number, a negative exponent means a number between 0 and 1, and n = 0 means the number is already between 1 and 10, because 10⁰ = 1.
Writing numbers this way does not change their value. It makes the size easy to read (the exponent is the order of magnitude), keeps long strings of zeros out of the way and shows clearly how many digits were measured.
How to convert a number to scientific notation
- Find the first non-zero digit and put the decimal point right after it. That gives the coefficient.
- Count how many places the decimal point moved. That count is the exponent.
- If the point moved left (a large number), the exponent is positive. If it moved right (a small number), the exponent is negative.
- Keep the significant digits in the coefficient and drop placeholder zeros.
Example 1: 357,096. The point moves from after the 6 to after the 3, which is 5 places left, so 357,096 = 3.57096 × 10⁵.
Example 2: 0.005600. The first non-zero digit is 5. The point moves 3 places right, so the exponent is −3. The two trailing zeros come after a decimal point, so they are significant and stay: 0.005600 = 5.600 × 10⁻³.
Converting scientific notation to standard form
Read the exponent and move the decimal point that many places: right for a positive exponent, left for a negative one. Write zeros where you run out of digits.
- 3.456 × 10⁴: move 4 places right. Three digits follow the point, so add one zero: 34,560.
- 3.9 × 10⁴: move 4 places right and add three zeros: 39,000.
- 5.6 × 10⁻³: move 3 places left, writing two zeros right after the decimal point: 0.0056.
The sign of the exponent changes everything: 6.7 × 10³ = 6,700, but 6.7 × 10⁻³ = 0.0067. A negative exponent never makes the number negative; only a minus sign in front of the coefficient does that.
Scientific notation examples (quick reference)
| Standard form | Scientific notation | E notation | Decimal point moves |
|---|---|---|---|
| 300 | 3 × 10² | 3E2 | 2 left |
| 127.5 | 1.275 × 10² | 1.275E2 | 2 left |
| 422,000 | 4.22 × 10⁵ | 4.22E5 | 5 left |
| 6,020,000 | 6.02 × 10⁶ | 6.02E6 | 6 left |
| 0.0071 | 7.1 × 10⁻³ | 7.1E-3 | 3 right |
| 0.00042 | 4.2 × 10⁻⁴ | 4.2E-4 | 4 right |
E notation and engineering notation
E notation is how calculators and spreadsheets show scientific notation: the letter E (or e) means “times ten to the power of”. So 4.22E5 = 4.22 × 10⁵ and 7.1E-3 = 7.1 × 10⁻³. The E has nothing to do with the number e ≈ 2.718.
Engineering notation only uses exponents that are multiples of 3, so the coefficient is from 1 up to 999. Each of those exponents matches an SI prefix, which makes units easy to read:
| Number | Scientific | Engineering | Prefix |
|---|---|---|---|
| 45,600,000 | 4.56 × 10⁷ | 45.6 × 10⁶ | mega (M) |
| 345,600,000,000 | 3.456 × 10¹¹ | 345.6 × 10⁹ | giga (G) |
| 0.00042 | 4.2 × 10⁻⁴ | 420 × 10⁻⁶ | micro (µ) |
| 0.0071 | 7.1 × 10⁻³ | 7.1 × 10⁻³ | milli (m) |
Check the coefficient and exponent
A number such as 53 × 10³ has the right value but is not in normalized form, because 53 is not less than 10. Rewrite 53 as 5.3 × 10¹ and add the exponents: 53 × 10³ = 5.3 × 10⁴. In the other direction, 0.25 × 10⁶ = 2.5 × 10⁵: making the coefficient 10 times bigger makes the exponent 1 smaller.
Zero is the one special case. No power of 10 turns 0 into a coefficient from 1 to 10, so zero is simply written as 0.
Adding, subtracting, multiplying and dividing in scientific notation
- Add or subtract: give both numbers the same exponent, then add the coefficients. 3.2 × 10⁴ + 5.1 × 10³ = 3.2 × 10⁴ + 0.51 × 10⁴ = 3.71 × 10⁴ (37,100).
- Multiply: multiply the coefficients and add the exponents. 6.0 × 10²³ × 2.5 × 10⁻³ = 15 × 10²⁰ = 1.5 × 10²¹.
- Divide: divide the coefficients and subtract the exponents. (4.8 × 10⁵) ÷ (1.6 × 10⁻²) = 3 × 10⁵⁻⁽⁻²⁾ = 3.0 × 10⁷.
- Powers: raise the coefficient to the power and multiply the exponents. (3.0 × 10⁸)² = 9.0 × 10¹⁶.
After each operation, check that the coefficient is still from 1 to under 10. If it is not, move the point and adjust the exponent, as in 15 × 10²⁰ = 1.5 × 10²¹.
Significant figures in scientific notation
Every digit in the coefficient is significant, which is why scientists like this form. 32,000 is ambiguous (2 to 5 significant figures), but 3.2 × 10⁴ has 2 and 3.2000 × 10⁴ has 5. For measured values, round the answer of a calculation like this:
- Multiplying or dividing: keep the fewest significant figures of the inputs. 1.23 × 10⁵ × 4.56 × 10⁻² = 5,608.8, which rounds to 5.61 × 10³.
- Adding or subtracting: keep the least precise place. 3.2 × 10⁴ is known to the thousands and 5.1 × 10³ to the hundreds, so 37,100 rounds to 3.7 × 10⁴.
The calculator shows the exact answer and the rounded one side by side, so you can use whichever your class expects. For more on counting digits, see the significant figures calculator below.
Homework-style examples in scientific notation
These follow the usual homework wording: write a number in scientific notation, write it in standard form, or give a quotient or sum in scientific notation.
Example 3: What is the quotient of 3.55 × 10⁶ and 7.1 × 10² expressed in scientific notation? Divide the coefficients: 3.55 ÷ 7.1 = 0.5. Subtract the exponents: 6 − 2 = 4. That gives 0.5 × 10⁴, but 0.5 is less than 1, so move the point one place right and lower the exponent by 1: 5 × 10³ (5.0 × 10³ to two significant figures).
Check: in standard form, 3,550,000 ÷ 710 = 5,000 = 5 × 10³.
Example 4: Find 0.00003 ÷ (2 × 10⁻⁵) and give the answer in scientific notation. First write 0.00003 as 3 × 10⁻⁵. Then 3 ÷ 2 = 1.5 and −5 − (−5) = 0, so the quotient is 1.5 × 10⁰, which is 1.5.
Check: 2 × 10⁻⁵ = 0.00002, and 0.00003 ÷ 0.00002 = 1.5.
Example 5: Write 605,000 in scientific notation. Put the point after the first digit, 6: the coefficient is 6.05 (the trailing zeros are placeholders). The point moved 5 places left, so 605,000 = 6.05 × 10⁵.
Check: 6.05 × 100,000 = 605,000.
Example 6: Write 3.2 × 10⁻⁴ in standard form. The exponent is −4, so move the point 4 places left, writing three zeros after the decimal point: 0.00032.
Check: 0.00032 is 3.2 ten-thousandths, and one ten-thousandth is 10⁻⁴.
Example 7: Express 4.9 × 10⁻⁵ + 0.0005 in scientific notation. Write 0.0005 with the same power of 10: 0.0005 = 50 × 10⁻⁵. Add the coefficients: 4.9 + 50 = 54.9, so the sum is 54.9 × 10⁻⁵. Normalize: 5.49 × 10⁻⁴.
Check: 0.000049 + 0.0005 = 0.000549 = 5.49 × 10⁻⁴.
Common mistakes
- Getting the sign backwards: 0.0071 has a negative exponent (−3) because it is smaller than 1.
- Leaving the coefficient too big or too small, such as 45.6 × 10⁶ when the question asks for scientific notation (that form is engineering notation).
- Adding exponents when adding numbers. Only multiplication adds exponents; addition needs matching exponents first.
- Dropping significant zeros: 5.600 × 10⁻³ and 5.6 × 10⁻³ have the same value but not the same precision.
Scientific Notation practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Put the point after the 8: it moves 7 places left, so the exponent is +7.
The point moves 4 places right to land after the 5, so the exponent is −4. 50.6 × 10⁻⁵ has the right value but its coefficient is not below 10.
Move the point 5 places right: 7.08 → 708,000 (three zeros are added).
The coefficient must be at least 1 and less than 10. Only 9.9 qualifies.
64 = 6.4 × 10¹, so 64 × 10⁵ = 6.4 × 10⁶: a smaller coefficient needs a larger exponent.
Multiply 3 × 4 = 12 and add the exponents 4 + (−7) = −3, giving 12 × 10⁻³ = 1.2 × 10⁻².
Rewrite 2.3 × 10⁴ as 0.23 × 10⁵, then add 6.1 + 0.23 = 6.33. Exponents are not added when adding numbers.
Quiz complete
Frequently asked questions
How do I convert a number into scientific notation?
Move the decimal point until exactly one non-zero digit is to its left. The number of places you moved it is the exponent: positive if you moved left, negative if you moved right. 53,000 becomes 5.3 × 10⁴.
How do you convert scientific notation to standard form?
Move the decimal point right by the exponent when it is positive, filling empty places with zeros. 6.7 × 10³ = 6,700 and 3.456 × 10⁴ = 34,560.
How do you convert scientific notation with a negative exponent into a decimal?
Move the decimal point left by the size of the exponent and write leading zeros. 6.7 × 10⁻³ = 0.0067 and 5.6 × 10⁻³ = 0.0056.
Does scientific notation change the value of a number?
No. 45,600,000 and 4.56 × 10⁷ are the same number written two ways. Precision only changes if you round the coefficient, for example to 4.6 × 10⁷.
What does E mean in scientific notation?
E stands for "times ten to the power of". Calculators show 4.22 × 10⁵ as 4.22E5 and 7.1 × 10⁻³ as 7.1E-3.
How do you find a quotient in scientific notation?
Divide the coefficients, subtract the exponents, then normalize so the coefficient is from 1 to under 10. For 3.55 × 10⁶ ÷ 7.1 × 10², 3.55 ÷ 7.1 = 0.5 and 6 − 2 = 4, so 0.5 × 10⁴ = 5 × 10³.
Can you add numbers in scientific notation?
Yes. Rewrite them with the same power of 10, add the coefficients, then normalize: 3.2 × 10⁴ + 5.1 × 10³ = 3.71 × 10⁴, or 3.7 × 10⁴ when rounded by significant-figure rules.
What is the difference between scientific and engineering notation?
Scientific notation keeps one digit before the point (4.2 × 10⁻⁴). Engineering notation uses exponents that are multiples of 3 so they match SI prefixes (420 × 10⁻⁶, or 420 micro).
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