Writing Numbers in Scientific Notation

Learn how to convert large and small numbers into scientific notation and understand when to use this format.

Intermediate25 minLesson
Printable worksheet

Definition

Scientific notation is a way to write very large or very small numbers in a compact form:
Where:
  • is a number between and (including , but not ):
  • is an integer (positive, negative, or zero)
Examples:
  • (Avogadro's number)
  • (one billionth)
The exponent tells you how many places to move the decimal point.

Try it now

Which number is written correctly in scientific notation?

Worked Examples

Write in scientific notation.

1

Place the decimal after the first non-zero digit

Move the decimal to get a number between 1 and 10: →

2

Count decimal places moved

Original: → means we moved 7 places left → 7 places

3

Write the power of 10

Moving left = positive exponent →

4

Combine into scientific notation

→

Common Mistakes

Writing instead of

Why it's wrong: The coefficient must be between 1 and 10. The number 12.5 is greater than 10.

Correct: Always adjust so the coefficient satisfies . If , move the decimal left and increase the exponent.

Using the wrong sign for the exponent

Why it's wrong: Students confuse which direction gives positive vs negative exponents.

Correct: Large numbers (move decimal LEFT) → positive exponent. Small numbers less than 1 (move decimal RIGHT) → negative exponent.

Forgetting to count all the zeros

Why it's wrong: When counting places, especially with many zeros, it's easy to miscount.

Correct: Write out each digit position or use place value: has 5 places to move (count: ).

Writing instead of

Why it's wrong: The coefficient 0.82 is less than 1, which violates the rule .

Correct: The coefficient must be at least 1. Move the decimal right to get , then adjust the exponent to .

Interactive Visual

Place Value Chart

Enter a number:
0.0
Ones
1
Tenths
0.1
00

Expanded Form:

Click on any place value to highlight it and see its value.

2^3 = 8
3

See how powers of a number grow on the number line. Change the base and exponent.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

15 problems
Problem 1 of 15
Easy

Which number is written correctly in scientific notation?

Why It Matters

Scientists, engineers, and mathematicians work with numbers that would be impossible to write out fully:
  • Astronomy: The distance to the nearest star is about km. In scientific notation: km
  • Biology: A human cell is about meters wide. In scientific notation: m
  • Computing: A modern computer can perform calculations per second: operations/sec
  • Chemistry: The mass of a hydrogen atom is kg: kg
Without scientific notation, calculations with these numbers would be extremely error-prone!

Real World Applications

Astronomy: Distances in Space

Astronomers use scientific notation to express vast cosmic distances that would be impractical to write in standard form.

Example:

The distance to the Andromeda Galaxy is approximately meters, written as m.

1Try It Yourself

A light-year is approximately meters.

Write this distance in scientific notation.

Step 1: Write the mathematical expression

Move the decimal to get a number between 1 and 10:

Microbiology: Sizes of Organisms

Biologists use scientific notation to describe organisms too small to see with the naked eye.

Example:

A typical bacterium is about meters long, written as m (2 micrometers).

2Try It Yourself

A virus particle measures approximately meters in diameter.

Express this measurement in scientific notation.

Step 1: Write the mathematical expression

Convert the decimal:

Computer Science: Data Storage

Tech companies measure data in bytes, with prefixes representing powers of 10.

Example:

A terabyte is bytes, or bytes.

3Try It Yourself

Global internet traffic in 2024 is estimated at bytes per year.

Write this in scientific notation.

Step 1: Write the mathematical expression

Convert to standard form:

Key Takeaways

  • 1Scientific notation expresses numbers as where
  • 2For large numbers, move decimal LEFT and use POSITIVE exponent
  • 3For small numbers (less than 1), move decimal RIGHT and use NEGATIVE exponent
  • 4Count the decimal places moved to determine the exponent value
  • 5Always check that your coefficient is between 1 and 10

Frequently Asked Questions

If , you can always rewrite it by moving the decimal and adjusting the exponent. For example, . Having one standard form prevents confusion.
If , you can always rewrite it by moving the decimal and adjusting the exponent. For example, . Having one standard form prevents confusion.
Write it as . The coefficient 1 satisfies .
Yes! Since , this equals 3. Any number between 1 and 10 can be written with exponent 0.

Glossary

Scientific notation
A way to write numbers as where
Coefficient
The number in scientific notation, which must be between 1 and 10
Exponent
The power in , indicating how many places to move the decimal
Standard form
Another name for scientific notation, commonly used in the UK
Order of magnitude
The power of 10 closest to a number, used to compare sizes

Formula Card

Scientific Notation Format

Where $a$ is the coefficient ($1 \le a < 10$) and $n$ is the exponent. Positive $n$ for large numbers, negative $n$ for small numbers.

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