Scientific Notation Operations Calculator
Add, subtract, multiply, and divide numbers in scientific notation step by step. Lines up exponents, normalizes the result, and shows every format.
Calculating with scientific notation
Scientific notation writes a number as a mantissa times a power of ten, like 1.225 × 10⁵. Adding, subtracting, multiplying, and dividing such numbers each follow a short rule.
Add and subtract: match the exponents first
- Rewrite the number with the smaller exponent so both share the larger exponent.
- Add or subtract the mantissas, keeping that exponent.
- Normalize so the mantissa is between 1 and 10.
1.225 × 10⁵ + 3.655 × 10³: rewrite the second as 0.03655 × 10⁵, add the mantissas to get 1.26155, so the answer is 1.26155 × 10⁵.
Multiply and divide: work on parts
- Multiply — multiply mantissas, add exponents. 3 × 10⁴ times 2 × 10³ = 6 × 10⁷.
- Divide — divide mantissas, subtract exponents. 8 × 10⁶ ÷ 2 × 10² = 4 × 10⁴.
Normalizing
If the mantissa lands outside 1 to 10, shift the point and adjust the exponent. 9 × 10⁰ times 9 × 10⁰ gives 81 × 10⁰, which normalizes to 8.1 × 10¹.
More formats
The result is also shown as e-notation (1.26155e5), engineering notation (exponent a multiple of 3), and a plain number when it is not too large.
Common mistakes
- Adding mantissas without first matching the exponents.
- Adding exponents when you should multiply the mantissas, or vice versa.
- Forgetting to normalize a mantissa that ends up 10 or more.
To convert a single number to or from scientific notation, use the scientific notation converter.
Frequently asked questions
- How do you add numbers in scientific notation?
- Rewrite the number with the smaller exponent so both share the larger exponent, then add the mantissas and keep that exponent. For 1.225 times 10 to the 5th plus 3.655 times 10 to the 3rd, rewrite the second as 0.03655 times 10 to the 5th, then add to get 1.26155 times 10 to the 5th.
- Why do the exponents have to match to add or subtract?
- Adding mantissas only makes sense when they measure the same power of ten, just as fractions need a common denominator. Once the exponents match, the mantissas line up and can be added or subtracted directly.
- How do you multiply in scientific notation?
- Multiply the mantissas and add the exponents. For 3 times 10 to the 4th times 2 times 10 to the 3rd, multiply 3 by 2 to get 6 and add 4 plus 3 to get 7, giving 6 times 10 to the 7th.
- How do you divide in scientific notation?
- Divide the mantissas and subtract the exponents. For 8 times 10 to the 6th divided by 2 times 10 squared, divide 8 by 2 to get 4 and subtract 6 minus 2 to get 4, giving 4 times 10 to the 4th.
- What does normalizing the result mean?
- Scientific notation requires the mantissa to be at least 1 and less than 10. If a product like 9 times 9 gives 81, you shift the point and bump the exponent to write 8.1 times 10 to the 1st.
- What is engineering notation?
- Engineering notation is scientific notation with the exponent forced to a multiple of 3, matching metric prefixes like kilo, mega, and giga. It lines up with common unit steps.
- What is e-notation?
- E-notation writes a times 10 to the b as aEb or aeb. So 1.26155 times 10 to the 5th is 1.26155e5. It is how calculators and programming languages display scientific notation.