What is scientific notation in math?
Scientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 103 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10−3.
Scientific Notation in a picture
Formal definition and notation
A nonzero number is in scientific notation when it is written as a × 10n, where 1 ≤ |a| < 10 and n is an integer. The factor a is sometimes called the coefficient or mantissa. Every nonzero real number has exactly one such form; negative numbers keep their sign on a, as in −2.5 × 104.
| Symbol | Read it as | Meaning |
|---|---|---|
| a × 10n | a times ten to the n | the general form, with 1 ≤ a < 10 and n an integer |
| 10−3 | ten to the negative third | 11000, or 0.001 |
| 3.86E5 | 3.86 E 5 | calculator display for 3.86 × 105 |
Key facts
- The first factor must be at least 1 and less than 10. 45 × 103 and 0.45 × 105 are not in scientific notation; 4.5 × 104 is.
- Each step of 1 in the exponent is a factor of 10: 3 × 106 is 10 times 3 × 105.
- To multiply, multiply the first factors and add the exponents: (2 × 103)(4 × 105) = 8 × 108.
- To divide, divide the first factors and subtract the exponents: 9 × 1073 × 102 = 3 × 105.
- After multiplying, fix a first factor that is 10 or more: 24 × 108 = 2.4 × 109.
How to convert to scientific notation
- a
- move the decimal point until exactly one nonzero digit is to its left
- n
- the number of places you moved the point
- sign of n
- positive if the original number is 10 or more; negative if it is between 0 and 1; n = 0 if it is already from 1 to 10
To go back to standard form, move the point n places right if n is positive, or left if n is negative, filling empty places with zeros.
Try it: powers of ten
Keep the base at 10 and change the exponent: each step multiplies or divides by 10. The readout writes the value in scientific notation.
Scientific Notation examples, step by step
Example 1: A large number
Write 52,000,000 in scientific notation.
- Place the decimal point after the first nonzero digit: 5.2.
- Count how far the point moved: from the end of 52,000,000 to after the 5 is 7 places left.
- Moving left for a large number gives a positive exponent: 107.
Answer: 52,000,000 = 5.2 × 10⁷.
Example 2: A small number
Write 0.00305 in scientific notation.
- The first nonzero digit is 3, so the first factor is 3.05.
- Count the moves: the point goes from after the first 0 to after the 3, which is 3 places right.
- The number is between 0 and 1, so the exponent is negative: 10−3.
- Check: 3.05 × 0.001 = 0.00305.
Answer: 0.00305 = 3.05 × 10⁻³.
Example 3: Multiply and fix the answer
Compute (6 × 105)(4 × 103). Write the answer in scientific notation.
- Multiply the first factors: 6 × 4 = 24.
- Add the exponents: 105 × 103 = 108. So far: 24 × 108.
- 24 is not less than 10, so rewrite it: 24 = 2.4 × 101.
- Combine: 2.4 × 101 × 108 = 2.4 × 109.
Answer: The product is 2.4 × 10⁹ (2,400,000,000), not 24 × 10⁸.
Common mistakes
The first factor must be at least 1 and less than 10. Moving the point one more place left makes 45 into 4.5 and adds 1 to the exponent.
6 × 103 is 6,000. A number between 0 and 1 needs a negative exponent.
The exponent counts places the decimal point moves, not zeros to tack on. The digit 2 already uses one of the four places.
104 × 105 is 10 used as a factor 4 + 5 = 9 times. Exponents add when you multiply powers of the same base.
Not to be confused with
In US schools, standard form is the usual way to write a number, such as 386,000; scientific notation is 3.86 × 105. (In the UK, standard form means scientific notation.)
An exponent is any power, such as the 3 in 23. Scientific notation always uses base 10 and a first factor from 1 to less than 10.
Quick check: 3 questions
Which number is written in scientific notation?
Answer: C. 4.08 is from 1 to less than 10, and 10−2 is an integer power of 10. Choice A has a first factor of 12 (too big), B has 0.5 (too small), and D uses base 2 instead of 10.Show answer and explanation
What is 7.06 × 10−4 in standard form?
Answer: A. A negative exponent moves the decimal point left: 4 places left from 7.06 gives 0.000706. Choice B moves 5 places, C moves right as if the exponent were positive, and D moves only 3 places.Show answer and explanation
What is (8 × 106) ÷ (2 × 102)?
Answer: D. Divide the first factors, 8 ÷ 2 = 4, and subtract the exponents, 6 − 2 = 4: 4 × 104. Choice B subtracts the first factors, C adds the exponents, and A subtracts one too many.Show answer and explanation
Scientific Notation: frequently asked questions
Why do we use scientific notation?
It makes very large and very small numbers easy to read, compare and calculate with. A number like 0.00000000016 is hard to read and easy to miscount, but 1.6 × 10−10 shows its size at a glance. Comparing two numbers starts with the exponents: 3 × 109 is bigger than 9 × 108.
Is 10 x 10^3 in scientific notation?
No. The first factor must be at least 1 and less than 10, so 10 is not allowed. Rewrite 10 × 103 as 1 × 104, which is 10,000. In the same way, 0.8 × 102 is not in scientific notation; write 8 × 101.
What does E mean on a calculator?
Calculators and spreadsheets show scientific notation with an E, which stands for times 10 to the power. 3.86E5 means 3.86 × 105, or 386,000, and 4.7E-4 means 4.7 × 10−4, or 0.00047. It does not mean the number e. Copy the answer with × 10 and the exponent.
Practice and tools
Sources
- Common Core State Standards, Grade 8 Expressions and Equations (8.EE.A.3, 8.EE.A.4)
- OpenStax, Prealgebra 2e, 10.5 Integer Exponents and Scientific Notation
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.A.3, CCSS 8.EE.A.4
Cite this entry
Effortless Math Team. “Scientific Notation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/scientific-notation/
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