How many zeros are in a trillion? How many places after the decimal point does the width of a virus have? Writing numbers like 6,020,000,000,000,000,000,000,000 or 0.0000000001 is tiring and mistakes are easy to make. Scientists, engineers, programmers and students therefore use a compact system called scientific notation.
Scientific notation writes any number as a short decimal multiplied by a power of ten. It is neat, comparable at a glance and easy to calculate with. In this guide you will learn what the format is, how to convert numbers in both directions, how to add, subtract, multiply, divide and raise to powers, and how calculators display it. We also cover significant figures, real-world examples, common mistakes and practice questions. To convert numbers instantly, use our free scientific notation calculator.
Quick Answer: What Is Scientific Notation?
Scientific notation writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number.
Examples: 4,500 = 4.5 × 10³, and 0.00032 = 3.2 × 10⁻⁴. A positive exponent means a large number; a negative exponent means a number smaller than 1.
The Format of Scientific Notation
Every number in scientific notation has two parts:
- The coefficient (or mantissa), a, which is a number at least 1 and less than 10.
- The power of ten, 10ⁿ, where the exponent n is a positive or negative whole number (or zero).
For example, in 3.6 × 10⁵, the coefficient is 3.6 and the exponent is 5. Written out, 10⁵ = 100,000, so 3.6 × 100,000 = 360,000.
The rule “1 ≤ a < 10” is what makes the notation standard. Writing 36 × 10⁴ is mathematically correct but not proper scientific notation, because 36 is greater than 10. You would rewrite it as 3.6 × 10⁵.
If exponents feel unfamiliar, review our guide on square roots and exponents first. The idea of a power of ten is simply 10 multiplied by itself n times, or divided n times if n is negative.
| Power | Value | Name |
|---|---|---|
| 10⁻³ | 0.001 | one thousandth |
| 10⁻² | 0.01 | one hundredth |
| 10⁻¹ | 0.1 | one tenth |
| 10⁰ | 1 | one |
| 10¹ | 10 | ten |
| 10² | 100 | hundred |
| 10³ | 1,000 | thousand |
| 10⁶ | 1,000,000 | million |
| 10⁹ | 1,000,000,000 | billion |
| 10¹² | 1,000,000,000,000 | trillion |
Converting Large Numbers to Scientific Notation
Steps
- Find the decimal point. In a whole number, it sits at the right end.
- Move it to the left until there is exactly one non-zero digit in front of it.
- Count how many places you moved. That count is the exponent (positive).
- Write the new number times 10 raised to that exponent. Drop any trailing zeros.
Examples
- 4,500 → move the decimal 3 places left → 4.5 × 10³
- 45,000,000 → move 7 places left → 4.5 × 10⁷
- 602,000,000,000,000,000,000,000 → move 23 places left → 6.02 × 10²³ (Avogadro’s number)
- 299,792,458 → move 8 places left → 2.99792458 × 10⁸, often rounded to 3.0 × 10⁸ (speed of light in metres per second)
Converting Small Numbers to Scientific Notation
For numbers smaller than 1, move the decimal point to the right, and the exponent becomes negative.
Steps
- Move the decimal point right until it sits just after the first non-zero digit.
- Count the places moved. The exponent is that count with a minus sign.
- Write the number times 10 to the negative exponent.
Examples
- 0.00032 → move 4 places right → 3.2 × 10⁻⁴
- 0.0000072 → move 6 places right → 7.2 × 10⁻⁶
- 0.5 → move 1 place right → 5 × 10⁻¹
- 0.0000000001 → move 10 places right → 1 × 10⁻¹⁰ (about the size of an atom, in metres)
A memory aid: big number, positive exponent; tiny number, negative exponent. If you are unsure about the direction, test the result mentally. 3.2 × 10⁻⁴ must be much smaller than 3.2, so 0.00032 makes sense.
Converting Back to Standard (Decimal) Form
To reverse the process, move the decimal point by the number of places shown in the exponent.
- Positive exponent: move the decimal point right, adding zeros if needed.
- Negative exponent: move the decimal point left, adding leading zeros if needed.
Examples
- 3.8 × 10⁻³ → move 3 places left → 0.0038
- 7.1 × 10⁴ → move 4 places right → 71,000
- 2.05 × 10⁶ → move 6 places right → 2,050,000
- 9 × 10⁻² → 0.09
If you want to see these tiny decimals as fractions, our guide on fractions to decimals explains how to move between the two forms.
Multiplying and Dividing in Scientific Notation
Multiplication and division are the easiest operations, because you handle the coefficients and the powers of ten separately.
Multiplication Rule
(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10^(m+n)
Example 1: (3 × 10⁴) × (2 × 10⁵). Multiply coefficients: 3 × 2 = 6. Add exponents: 4 + 5 = 9. Answer: 6 × 10⁹.
Example 2: (4 × 10³) × (5 × 10⁶). Multiply coefficients: 4 × 5 = 20. Add exponents: 3 + 6 = 9. That gives 20 × 10⁹, but 20 is too big, so adjust: 2 × 10¹ × 10⁹ = 2 × 10¹⁰.
Example 3: (6 × 10⁴) × (3 × 10²) = 18 × 10⁶ = 1.8 × 10⁷.
Division Rule
(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10^(m−n)
Example 1: (8 × 10⁹) ÷ (2 × 10³) = 4 × 10⁶.
Example 2: (9 × 10⁸) ÷ (3 × 10³) = 3 × 10⁵.
Example 3: (1.5 × 10²) ÷ (3 × 10⁻⁴). Coefficients: 1.5 ÷ 3 = 0.5. Exponents: 2 − (−4) = 6. That gives 0.5 × 10⁶, so adjust: 5 × 10⁵.
The Adjustment Step
After multiplying or dividing, always check whether the coefficient is between 1 and 10. If it is 10 or more, divide it by 10 and add 1 to the exponent. If it is less than 1, multiply it by 10 and subtract 1 from the exponent.
Adding and Subtracting in Scientific Notation
Addition and subtraction need one extra step: the exponents must match before you combine the coefficients.
Steps
- Rewrite one number so both have the same power of ten.
- Add or subtract the coefficients.
- Keep the common power of ten.
- Adjust the result to proper scientific notation.
Example 1: Addition
3.2 × 10⁴ + 4.5 × 10³. Rewrite 4.5 × 10³ as 0.45 × 10⁴. Add: 3.2 + 0.45 = 3.65. Answer: 3.65 × 10⁴. Check: 32,000 + 4,500 = 36,500. ✓
Example 2: Subtraction
5 × 10⁶ − 2 × 10⁵. Rewrite 2 × 10⁵ as 0.2 × 10⁶. Subtract: 5 − 0.2 = 4.8. Answer: 4.8 × 10⁶. Check: 5,000,000 − 200,000 = 4,800,000. ✓
Example 3: Different Signs of Exponent
2.5 × 10³ + 7.5 × 10². Rewrite 7.5 × 10² as 0.75 × 10³. Add: 2.5 + 0.75 = 3.25. Answer: 3.25 × 10³ (that is, 2,500 + 750 = 3,250).
Powers and Roots
To raise a number in scientific notation to a power, raise the coefficient to that power and multiply the exponent of ten by it.
(a × 10ⁿ)ᵖ = aᵖ × 10^(n×p)
- (2 × 10³)³ = 2³ × 10⁹ = 8 × 10⁹
- (3 × 10⁻²)² = 9 × 10⁻⁴ = 9 × 10⁻⁴
- (4 × 10⁶)² = 16 × 10¹² = 1.6 × 10¹³
For roots, first make the exponent of ten divisible by the root index. For √(9 × 10⁶), the root of 9 is 3 and the root of 10⁶ is 10³, so the answer is 3 × 10³.
E Notation on Calculators and Computers
Calculators, spreadsheets and programming languages cannot display small raised exponents, so they use E notation. The letter E stands for “times ten to the power of”.
| E notation | Scientific notation | Standard number |
|---|---|---|
| 1.5E6 | 1.5 × 10⁶ | 1,500,000 |
| 2E3 | 2 × 10³ | 2,000 |
| 7.2E-5 | 7.2 × 10⁻⁵ | 0.000072 |
| 9.1E-31 | 9.1 × 10⁻³¹ | mass of an electron in kg (approximately) |
Note that E in this context is not the mathematical constant e (about 2.718). It simply means “× 10 to the power”. If a calculator shows 1.2E+9 after a big multiplication, it means 1,200,000,000. Spreadsheets such as Excel and Google Sheets also switch to this format when a number is too wide to show.
Significant Figures and Scientific Notation
Scientific notation is also the clearest way to show how precise a measurement is. The digits in the coefficient are the significant figures.
- 4.5 × 10³ has 2 significant figures (about 4,500).
- 4.50 × 10³ has 3 significant figures, which tells the reader the measurement is precise to the nearest 10.
- 0.00450 = 4.50 × 10⁻³ has 3 significant figures; the leading zeros only show the position of the decimal point.
When you multiply or divide measured values, round the answer to the same number of significant figures as the least precise input. This habit is common in physics and chemistry classes.
Comparing Numbers in Scientific Notation
Comparing is easy once the numbers are in standard scientific form.
- Compare the exponents first. The larger exponent means the larger number (for positive numbers).
- If the exponents are equal, compare the coefficients.
Example: Which is larger, 3.2 × 10⁵ or 9.8 × 10⁴? The exponents are 5 and 4, so 3.2 × 10⁵ is larger (320,000 versus 98,000), even though 9.8 is bigger than 3.2.
Example: Compare 6.1 × 10⁻³ and 2.4 × 10⁻³. The exponents match, so compare coefficients: 6.1 is bigger, so 6.1 × 10⁻³ is larger.
For negative exponents, remember that −3 is bigger than −5, so 10⁻³ (0.001) is larger than 10⁻⁵ (0.00001).
Engineering Notation
Engineers often use a variation where the exponent is always a multiple of 3, so it lines up with metric prefixes. For instance, 4.7 × 10³ ohms is written as 4.7 kΩ, and 2.2 × 10⁻⁶ farads is 2.2 µF.
| Prefix | Symbol | Power of ten |
|---|---|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
These prefixes tie directly into unit conversion. Converting 2.5 kilometres to metres is just 2.5 × 10³ metres.
Real-Life Examples
Astronomy
The average distance from Earth to the Sun is about 1.5 × 10⁸ kilometres. The nearest star system beyond the Sun is about 4 × 10¹³ kilometres away. Writing these in full would take a long line of zeros.
Chemistry and Physics
One mole contains about 6.02 × 10²³ particles, while the mass of an electron is roughly 9.11 × 10⁻³¹ kg. Scientific notation makes these numbers manageable.
Biology
A typical human hair is about 7 × 10⁻⁵ metres wide, and bacteria are often around 10⁻⁶ metres long. This helps biologists compare organisms of vastly different sizes.
Computers and Data
A terabyte is about 10¹² bytes. Our guide on data storage units explains how these sizes relate to each other.
Economics and Finance
National budgets and global trade are measured in trillions, which is 10¹². A country with a GDP of 3.5 × 10¹² is easier to compare with another at 1.2 × 10¹³ in this form. To see how growth percentages apply to such large figures, read our guide on percentage increase and decrease.
Medicine
Drug concentrations and virus particle counts are often tiny or enormous, so doctors and researchers use scientific notation to report them.
Common Mistakes
- Coefficient outside the range. 12 × 10³ must become 1.2 × 10⁴.
- Wrong sign on the exponent. Numbers smaller than 1 need a negative exponent; 0.003 is 3 × 10⁻³, not 3 × 10³.
- Counting the decimal moves incorrectly. Count carefully, or check by converting back.
- Adding exponents when adding numbers. You add exponents only when multiplying. For addition, make the exponents the same first.
- Forgetting the adjustment step after multiplication, such as leaving the answer as 20 × 10⁹.
- Confusing E with the constant e. 2E3 is 2,000, not 2 × 2.718³.
- Dropping significant zeros. 4.50 × 10³ and 4.5 × 10³ do not carry the same precision.
Practice Questions
- Write 45,000,000 in scientific notation.
- Write 0.0000072 in scientific notation.
- Write 3.8 × 10⁻³ in standard form.
- Multiply (6 × 10⁴) × (3 × 10²).
- Divide (9 × 10⁸) ÷ (3 × 10³).
- Add 2.5 × 10³ + 7.5 × 10².
Answers
- 4.5 × 10⁷
- 7.2 × 10⁻⁶
- 0.0038
- 18 × 10⁶ = 1.8 × 10⁷
- 3 × 10⁵
- 2,500 + 750 = 3,250 = 3.25 × 10³
Frequently Asked Questions
What is scientific notation?
It is a way of writing numbers as a × 10ⁿ, where a is at least 1 and less than 10, and n is a whole number.
How do I convert a number to scientific notation?
Move the decimal point until one non-zero digit is left in front of it. The number of places you moved is the exponent, positive for left moves and negative for right moves.
What does a negative exponent mean?
It means the number is less than 1. For example, 3.2 × 10⁻⁴ equals 0.00032.
How do I multiply in scientific notation?
Multiply the coefficients, add the exponents, then adjust so the coefficient lies between 1 and 10.
What does 1.5E6 mean?
It is E notation for 1.5 × 10⁶, which is 1,500,000.
Why is scientific notation useful?
It shortens extremely large or tiny numbers, makes them easy to compare, and reduces mistakes with long strings of zeros.
Is 0 written in scientific notation?
Zero has no standard scientific notation form because it has no non-zero digit. It is simply written as 0.
Conclusion
Scientific notation turns unwieldy numbers into neat, comparable expressions. Remember the format a × 10ⁿ with 1 ≤ a < 10, move the decimal point left for large numbers and right for small ones, add exponents when multiplying and subtract when dividing, and match exponents before adding or subtracting. Watch for E notation on calculators, and use significant figures to show precision.
When you need a quick conversion, try our free scientific notation calculator, and keep exploring with these related guides: