Scientific Notation Calculator
Convert between decimal numbers and scientific notation. Enter a decimal to get scientific notation, or enter a coefficient and exponent to get the decimal form. See also Exponent Calculator and Log Calculator.
How to Convert to Scientific Notation
Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10. To convert a decimal number: move the decimal point until you have a number between 1 and 10, then count how many places you moved it. That count becomes the exponent. Moving left gives a positive exponent (large numbers); moving right gives a negative exponent (small numbers).
Scientific Notation Formula
Scientific notation: a × 10^n
where 1 ≤ |a| < 10 and n is an integer
E-notation: aE+n or aE-n
(used in programming and calculators)
To convert decimal → scientific:
Move decimal point to get 1 ≤ |a| < 10
n = number of places moved (+ for left, − for right)
Example
Convert 123,456,789 to scientific notation
Move decimal 8 places left: 1.23456789
= 1.23456789 × 10^8
E-notation: 1.23456789e+8
Frequently Asked Questions
What is E-notation?
E-notation is a compact way to write scientific notation used in programming and calculators. Instead of "× 10^", it uses "E" or "e". For example, 3.14 × 10^8 is written as 3.14E+8 or 3.14e8.
When should I use scientific notation?
Scientific notation is useful for very large numbers (like the speed of light: 3 × 10^8 m/s) or very small numbers (like atomic sizes: 1 × 10^-10 m). It makes these numbers easier to read, compare, and calculate with.
What is the coefficient in scientific notation?
The coefficient (also called the significand or mantissa) is the number before the "× 10^n" part. In standard scientific notation, it must be between 1 (inclusive) and 10 (exclusive). For example, in 6.022 × 10^23, the coefficient is 6.022.
How do I multiply numbers in scientific notation?
Multiply the coefficients and add the exponents. For example: (2 × 10^3) × (3 × 10^4) = 6 × 10^7. If the resulting coefficient is ≥ 10, adjust by moving the decimal and incrementing the exponent.
Solved Examples — Scientific Notation
Example: Convert 0.00000742 to scientific notation
Solution:
Step 1: Move decimal right until we get a number between 1 and 10
Step 2: 0.00000742 → 7.42 (moved 6 places right)
Step 3: Moving right gives a negative exponent
Answer: 7.42 × 10⁻⁶
Example: Multiply (3.2 × 10⁴) × (5.0 × 10³)
Solution:
Step 1: Multiply coefficients: 3.2 × 5.0 = 16.0
Step 2: Add exponents: 4 + 3 = 7
Step 3: Result = 16.0 × 10⁷
Step 4: Normalize: 1.6 × 10⁸ (moved decimal left, increased exponent by 1)
Answer: 1.6 × 10⁸
Example: The Earth is 1.496 × 10⁸ km from the Sun. Express this in meters.
Solution:
Step 1: 1 km = 10³ meters
Step 2: 1.496 × 10⁸ km × 10³ m/km = 1.496 × 10¹¹ m
Answer: 1.496 × 10¹¹ meters
Practice Questions
Try these on your own:
- Convert 93,000,000 to scientific notation (Answer: 9.3 × 10⁷)
- Convert 4.56 × 10⁻³ to standard decimal (Answer: 0.00456)
- Divide (8.4 × 10⁶) by (2.1 × 10²) (Answer: 4.0 × 10⁴)
- Add 3.2 × 10⁵ + 4.7 × 10⁴ (Answer: 3.67 × 10⁵)
- The mass of a proton is 1.67 × 10⁻²⁷ kg. Express in grams. (Answer: 1.67 × 10⁻²⁴ g)
- Which is larger: 8.9 × 10⁶ or 3.2 × 10⁷? (Answer: 3.2 × 10⁷)
Common Mistakes to Avoid
The most common error is getting the sign of the exponent wrong. Remember: large numbers (greater than 10) have positive exponents, and small numbers (less than 1) have negative exponents. For example, 0.005 = 5 × 10⁻³ (not 10³). Another frequent mistake is having a coefficient that is not between 1 and 10 — writing 34.5 × 10⁶ is not proper scientific notation (it should be 3.45 × 10⁷). When adding or subtracting numbers in scientific notation, both numbers must have the same exponent first — you cannot simply add coefficients if exponents differ. When comparing numbers, always look at the exponent first: 2.0 × 10⁸ is larger than 9.9 × 10⁷ because 10⁸ > 10⁷.
Key Takeaways
- Scientific notation format: a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer.
- Positive exponents = large numbers (moved decimal left). Negative exponents = small numbers (moved decimal right).
- Multiplication: multiply coefficients, add exponents. Division: divide coefficients, subtract exponents.
- Addition/subtraction: first make exponents equal, then add/subtract coefficients.
- E-notation (3.14e8) is the programming equivalent of 3.14 × 10⁸.
- Used in science for extremely large/small quantities: distances in space, atomic masses, speed of light, etc.