Scientific Notation Converter Guide: How to Read and Write Powers of 10
Quick Answer
- *Scientific notation writes numbers as a coefficient (1–10) × a power of 10.
- *Move the decimal left for large numbers (positive exponent) and right for small numbers (negative exponent).
- *E notation (like 3.2E5) is the calculator/programming equivalent of 3.2 × 10&sup5;.
- *Used across astronomy, chemistry, physics, and computing — anywhere numbers get extremely large or small.
What Is Scientific Notation?
Scientific notation is a standardized way of writing numbers that are too large or too small to be convenient in decimal form. It expresses any number as a coefficient between 1 and 10 multiplied by 10 raised to an integer power.
The format is: a × 10^n, where 1 ≤ a < 10 and n is an integer.
For example, the distance from Earth to the Sun is about 93,000,000 miles. In scientific notation: 9.3 × 10&sup7;. A hydrogen atom's radius is about 0.000000000053 meters, or 5.3 × 10¹¹ m.
How to Convert to Scientific Notation
Step-by-Step Process
- Find the decimal point in your number (if there isn't one visible, it's at the end)
- Move the decimal point until you have a number between 1 and 10
- Count how many places you moved the decimal
- If you moved left, the exponent is positive. If right, it's negative.
Worked Examples
| Standard Form | Decimal Moves | Scientific Notation |
|---|---|---|
| 5,280 | 3 places left | 5.28 × 10³ |
| 300,000,000 | 8 places left | 3.0 × 10&sup8; |
| 0.0072 | 3 places right | 7.2 × 10³ |
| 0.00000015 | 7 places right | 1.5 × 10&sup7; |
| 42 | 1 place left | 4.2 × 10¹ |
Converting Back to Standard Form
Reverse the process: move the decimal point by the number of places indicated by the exponent. Positive exponent means move right (making the number larger). Negative exponent means move left (making it smaller).
6.022 × 10²³ = move decimal 23 places right = 602,200,000,000,000,000,000,000
That's Avogadro's number — the number of atoms in one mole of a substance. Good luck writing that without scientific notation.
E Notation: The Calculator Version
On calculators and in programming languages, scientific notation is often written with an "E" or "e" instead of "× 10^". The number after E is the exponent.
| E Notation | Scientific Notation | Standard Form |
|---|---|---|
| 3.2E5 | 3.2 × 10&sup5; | 320,000 |
| 1.6E-19 | 1.6 × 10¹&sup9; | 0.00000000000000000016 |
| 9.81E0 | 9.81 × 10&sup0; | 9.81 |
In Python, you write 3.2e5. In JavaScript and most C-family languages, it's the same syntax. Over 95% of programming languages support E notation natively (IEEE 754 standard).
Arithmetic with Scientific Notation
Multiplication
Multiply the coefficients and addthe exponents. Example: (4 × 10³) × (2 × 10&sup5;) = 8 × 10&sup8;.
Division
Divide the coefficients and subtractthe exponents. Example: (8 × 10&sup6;) ÷ (4 × 10²) = 2 × 10&sup4;.
Addition and Subtraction
You must first make the exponents match. Convert one number so both have the same power of 10, then add or subtract the coefficients.
Real-World Applications
Scientific notation isn't just a classroom exercise. Here are real quantities that demand it:
| Quantity | Value | Field |
|---|---|---|
| Speed of light | 3.0 × 10&sup8; m/s | Physics |
| Avogadro's number | 6.022 × 10²³ | Chemistry |
| Planck's constant | 6.626 × 10³&sup4; J·s | Quantum physics |
| Distance to Andromeda | 2.537 × 10²² m | Astronomy |
| Mass of an electron | 9.109 × 10³¹ kg | Particle physics |
NASA uses scientific notation in virtually all mission calculations. The James Webb Space Telescope orbits at 1.5 × 10&sup6; km from Earth — a distance that would be unwieldy in standard form.
Common Mistakes to Avoid
Coefficient Outside 1–10 Range
Writing 32 × 10&sup4; instead of 3.2 × 10&sup5; is technically not proper scientific notation. The coefficient must always be at least 1 and less than 10.
Mixing Up Positive and Negative Exponents
A positive exponent means a large number. A negative exponent means a small number (less than 1). The number 5 × 10³ is 0.005, not 5,000.
Forgetting to Adjust After Arithmetic
If you multiply (5 × 10³) × (4 × 10²) and get 20 × 10&sup5;, you need to normalize to 2.0 × 10&sup6;.
Convert any number instantly
Use our free Scientific Notation Converter →Frequently Asked Questions
What is scientific notation?
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. For example, 93,000,000 becomes 9.3 × 10&sup7; and 0.00045 becomes 4.5 × 10&sup4;.
How do you convert a number to scientific notation?
Move the decimal point until you have a number between 1 and 10. Count how many places you moved it. If you moved left, the exponent is positive. If you moved right, the exponent is negative. Example: 5,280 becomes 5.28 × 10³ (moved decimal 3 places left).
What does E mean in scientific notation?
The letter E (or e) stands for "times ten raised to the power of." It is used on calculators and in programming. So 3.2E5 means 3.2 × 10&sup5;, which equals 320,000. This is sometimes called E notation or exponential notation.
How do you multiply numbers in scientific notation?
Multiply the coefficients and add the exponents. For example: (3 × 10&sup4;) × (2 × 10³) = 6 × 10&sup7;. If the resulting coefficient is 10 or greater, adjust by moving the decimal and incrementing the exponent.
When is scientific notation used in real life?
Scientific notation is used in astronomy (distances between stars), chemistry (Avogadro's number: 6.022 × 10²³), physics (speed of light: 3 × 10&sup8; m/s), computer science (memory sizes), and any field dealing with extremely large or small measurements.