The Quick Answer
Scientific notation is a way of expressing very large or very small numbers as a coefficient between 1 and 10 multiplied by a power of 10, such as 3.0 * 10^8 for the speed of light.
The general format is:
a * 10^n
where 1 <= |a| < 10 and n is an integer (positive, negative, or zero).
Instead of writing 93,000,000, you write 9.3 * 10^7. Instead of 0.000042, you write 4.2 * 10^-5. The notation compresses the number into a compact form that is easier to read, compare, and compute with.
Try our scientific notation converter to convert any number instantly.
How to Convert a Number to Scientific Notation
Large numbers (positive exponents)
Move the decimal point left until you have a number between 1 and 10. Count how many places you moved it -- that count becomes the positive exponent.
Example: 93,000,000
- Start: 93,000,000 (decimal is implicitly after the last zero)
- Move the decimal 7 places to the left: 9.3000000
- Drop trailing zeros: 9.3
- Result: 9.3 * 10^7
Example: 602,200,000,000,000,000,000,000
- Move the decimal 23 places left: 6.022
- Result: 6.022 * 10^23 (Avogadro's number)
Small numbers (negative exponents)
Move the decimal point right until you have a number between 1 and 10. Count how many places you moved it -- that count becomes the negative exponent.
Example: 0.000042
- Start: 0.000042
- Move the decimal 5 places to the right: 4.2
- Result: 4.2 * 10^-5
Example: 0.000000000000000000000000000000000662607
- Move the decimal 34 places right: 6.62607
- Result: 6.62607 * 10^-34 (Planck's constant in joule-seconds)
The rule
- Decimal moves left --> positive exponent (number is large)
- Decimal moves right --> negative exponent (number is small)
- The exponent equals the number of places the decimal moved
Converting Back to Standard Form
Reverse the process:
- Positive exponent: Move the decimal right by the exponent value, adding zeros as needed.
- 9.3 * 10^7 --> move right 7 places --> 93,000,000
- Negative exponent: Move the decimal left by the exponent's absolute value, adding leading zeros.
- 4.2 * 10^-5 --> move left 5 places --> 0.000042
Arithmetic in Scientific Notation
Multiplication
Multiply the coefficients and add the exponents:
(a * 10^m) * (b * 10^n) = (a * b) * 10^(m+n)
Example: (3 * 10^4) * (2 * 10^5)
- Coefficients: 3 * 2 = 6
- Exponents: 4 + 5 = 9
- Result: 6 * 10^9 (six billion)
Example: (4.5 * 10^3) * (2.0 * 10^6)
- Coefficients: 4.5 * 2.0 = 9.0
- Exponents: 3 + 6 = 9
- Result: 9.0 * 10^9
If the coefficient product is 10 or greater, adjust:
(5 * 10^3) * (4 * 10^2) = 20 * 10^5 = 2.0 * 10^6
Division
Divide the coefficients and subtract the exponents:
(a * 10^m) / (b * 10^n) = (a / b) * 10^(m-n)
Example: (8 * 10^6) / (4 * 10^3)
- Coefficients: 8 / 4 = 2
- Exponents: 6 - 3 = 3
- Result: 2 * 10^3 (two thousand)
Example: (6.0 * 10^8) / (3.0 * 10^-2)
- Coefficients: 6.0 / 3.0 = 2.0
- Exponents: 8 - (-2) = 10
- Result: 2.0 * 10^10
Addition and Subtraction
These require the same exponent before you can combine:
(a * 10^n) + (b * 10^n) = (a + b) * 10^n
Example with same exponent: (5.2 * 10^3) + (3.1 * 10^3)
- Same exponent: add coefficients: 5.2 + 3.1 = 8.3
- Result: 8.3 * 10^3
Example with different exponents: (5.2 * 10^3) + (3.1 * 10^4)
- Convert to the same exponent (use the larger one): 5.2 * 10^3 = 0.52 * 10^4
- Add: 0.52 * 10^4 + 3.1 * 10^4 = 3.62 * 10^4
- Result: 3.62 * 10^4 (36,200)
Subtraction works the same way -- match exponents first, then subtract coefficients.
Significant Figures and Precision
Scientific notation makes significant figures explicit. The coefficient shows exactly which digits are significant:
- 3 * 10^8 -- 1 significant figure
- 3.0 * 10^8 -- 2 significant figures
- 3.00 * 10^8 -- 3 significant figures
- 2.998 * 10^8 -- 4 significant figures
In standard form, it is ambiguous whether trailing zeros are significant: does 300,000,000 have 1 significant figure or 9? Scientific notation eliminates this confusion (NIST Guide for the Use of the International System of Units).
When doing calculations, your result should have no more significant figures than your least precise input.
Engineering Notation
Engineering notation is a variant where the exponent is always a multiple of 3 (... -6, -3, 0, 3, 6, 9 ...). This aligns with metric prefixes:
| Exponent | Prefix | Symbol | Example |
|---|---|---|---|
| 10^12 | tera | T | 1.21 TW (global nuclear power capacity) |
| 10^9 | giga | G | 2.4 GHz (Wi-Fi frequency) |
| 10^6 | mega | M | 15 MW (large wind turbine) |
| 10^3 | kilo | k | 47 k-ohms (common resistor value) |
| 10^-3 | milli | m | 20 mA (LED current) |
| 10^-6 | micro | u | 100 uF (capacitor value) |
| 10^-9 | nano | n | 7 nm (modern CPU transistor size) |
In engineering notation, 47,000 is written as 47 * 10^3 (not 4.7 * 10^4). The coefficient can range from 1 to 999.
Engineers and technicians often prefer this because it maps directly to the prefixes printed on components and used in specifications.
Reference Table: Important Constants in Scientific Notation
| Quantity | Value | Source |
|---|---|---|
| Speed of light (c) | 2.998 * 10^8 m/s | NIST CODATA |
| Earth-Sun distance | 1.496 * 10^11 m | NASA |
| Avogadro's number (N_A) | 6.022 * 10^23 mol^-1 | NIST CODATA |
| Planck's constant (h) | 6.626 * 10^-34 J*s | NIST CODATA |
| Diameter of hydrogen atom | ~1.2 * 10^-10 m | Textbook estimate |
| Mass of electron (m_e) | 9.109 * 10^-31 kg | NIST CODATA |
| Boltzmann constant (k_B) | 1.381 * 10^-23 J/K | NIST CODATA |
| Gravitational constant (G) | 6.674 * 10^-11 m^3 kg^-1 s^-2 | NIST CODATA |
| US national debt (approx 2025) | ~3.6 * 10^13 dollars | US Treasury |
| Grains of sand on Earth (est) | ~7.5 * 10^18 | University of Hawaii estimate |
The range from 10^-34 to 10^23 -- a span of 57 orders of magnitude -- illustrates why scientific notation exists. Writing these numbers in standard form would be impractical.
Common Mistakes
1. Coefficient outside the 1-to-10 range. Writing 12.5 * 10^3 is not proper scientific notation. Adjust to 1.25 * 10^4.
2. Forgetting to adjust the exponent after arithmetic. If multiplication gives 15 * 10^7, you must normalize: 1.5 * 10^8.
3. Adding/subtracting without matching exponents. You cannot add 3 * 10^4 and 5 * 10^3 directly. Convert first: 30 * 10^3 + 5 * 10^3 = 35 * 10^3 = 3.5 * 10^4.
4. Sign confusion with negative exponents. 10^-3 = 0.001, not -1000. A negative exponent means a small positive number, not a negative number.
5. Using the wrong calculator key. The EE or EXP button enters the entire "times 10 to the power" part. Typing 3 * 10 ^ 8 as separate keystrokes can cause errors on some calculators.
Try It Yourself
Use the scientific notation converter to convert any number to or from scientific notation instantly. The exponent calculator handles arbitrary powers and roots, and the base converter converts between decimal, binary, octal, and hexadecimal.
Frequently Asked Questions
What is scientific notation?
Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10. For example, 93,000,000 becomes 9.3 * 10^7. It makes very large or very small numbers easier to read, compare, and calculate with. It is the standard format in science and engineering (National Science Foundation educational materials).
What is 10^6 in standard form?
10^6 equals 1,000,000 (one million). The exponent 6 tells you to multiply 1 by 10 six times: 1 * 10 * 10 * 10 * 10 * 10 * 10 = 1,000,000. Similarly, 10^9 is one billion and 10^3 is one thousand.
How do I multiply numbers in scientific notation?
Multiply the coefficients together and add the exponents. For example, (3 * 10^4) * (2 * 10^5) = 6 * 10^9. If the coefficient result is 10 or greater, shift one power: (5 * 10^3) * (4 * 10^2) = 20 * 10^5 = 2.0 * 10^6.
How do I divide numbers in scientific notation?
Divide the coefficients and subtract the exponents. For example, (8 * 10^6) / (4 * 10^3) = 2 * 10^3. If the coefficient result is less than 1, adjust: (2 * 10^5) / (8 * 10^3) = 0.25 * 10^2 = 2.5 * 10^1.
What is the difference between scientific and engineering notation?
In scientific notation, the exponent can be any integer, and the coefficient is between 1 and 10. In engineering notation, the exponent must be a multiple of 3, and the coefficient ranges from 1 to 999. Engineering notation aligns with metric prefixes (kilo = 10^3, mega = 10^6, etc.).
How do I add numbers in scientific notation?
Convert both numbers to the same power of 10, then add the coefficients. For example, 5.2 * 10^3 + 3.1 * 10^4 becomes 0.52 * 10^4 + 3.1 * 10^4 = 3.62 * 10^4.
What does a negative exponent mean?
A negative exponent means the number is between 0 and 1. The expression 10^-3 equals 1/10^3 = 1/1000 = 0.001. In scientific notation, 4.2 * 10^-5 equals 0.000042.
How many significant figures should I use?
Match the precision of your least precise measurement. If one input has 2 significant figures and another has 5, your result should have 2 significant figures. The coefficient in scientific notation shows significant figures explicitly.
How do I enter scientific notation on a calculator?
Use the EE or EXP button. To enter 3.0 * 10^8: type 3.0, press EE, type 8. Do not manually type "* 10 ^" as separate operations. On computer software, the notation is often 3.0E8 or 3.0e8.
Is 10 * 10^5 valid scientific notation?
No. In proper scientific notation, the coefficient must be at least 1 and strictly less than 10. The expression 10 * 10^5 should be rewritten as 1.0 * 10^6. Similarly, 0.5 * 10^3 should be 5.0 * 10^2.