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:
a×10na \times 10^n
Where:
  • aa is a number between 11 and 1010 (including 11, but not 1010): 1≤a<101 \le a < 10
  • nn is an integer (positive, negative, or zero)
Examples:
  • 602,000,000,000,000,000,000,000=6.02×1023602{,}000{,}000{,}000{,}000{,}000{,}000{,}000 = 6.02 \times 10^{23} (Avogadro's number)
  • 0.000000001=1×10−90.000000001 = 1 \times 10^{-9} (one billionth)
The exponent nn tells you how many places to move the decimal point.

Try it now

Which number is written correctly in scientific notation?

Worked Examples

Write 45,600,00045{,}600{,}000 in scientific notation.
1

Place the decimal after the first non-zero digit

Move the decimal to get a number between 1 and 10: 4.564.56
2

Count how many places the decimal point moved

Original: 45600000.045600000.0 → 4.564.56 means we moved 7 places left → 7 places
3

Write the power of 10

Moving left = positive exponent → 10710^7
4

Combine into scientific notation

4.56×1074.56 \times 10^7

Common Mistakes

Writing 12.5×10312.5 \times 10^3 instead of 1.25×1041.25 \times 10^4

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 aa satisfies 1≤a<101 \le a < 10. If a≥10a \ge 10, 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: 0.0000450.000045 has 5 places to move (count: 0.00004.50.00004.5).

Writing 0.82×10−30.82 \times 10^{-3} instead of 8.2×10−48.2 \times 10^{-4}

Why it's wrong: The coefficient 0.82 is less than 1, which violates the rule 1≤a<101 \le a < 10.

Correct: The coefficient must be at least 1. Move the decimal right to get 8.28.2, then adjust the exponent to −4-4.

Interactive Visual

Place Value Chart

Enter a number:

45,600,000

Scientific notation: 4.56 × 10⁷

Ten Millions
10,000,000
Millions
1,000,000
Hundred Thousands
100,000
Ten Thousands
10,000
Thousands
1,000
Hundreds
100
Tens
10
Ones
1
45600000

Expanded Form:

4 × 10,000,000 + 5 × 1,000,000 + 6 × 100,000

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

Scientific notation

Standard form to scientific notation

Move the decimal point with the buttons or drag it. The digits stay where they are.

45600000.
45 600 000×100=45 600 00045\,600\,000 \times 10^{0} = 45\,600\,000

Each hop of the point changes the exponent by 1. The value stays the same.

The coefficient is 10 or more: move the point left.

Orange zeros are placeholders: they keep the digits in their places.

Interactive Sandbox

Expression Calculator

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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 40,000,000,000,00040{,}000{,}000{,}000{,}000 km. In scientific notation: 4×10134 \times 10^{13} km
  • Biology: A human cell is about 0.000010.00001 meters wide. In scientific notation: 1×10−51 \times 10^{-5} m
  • Computing: A modern computer can perform 1,000,000,000,0001{,}000{,}000{,}000{,}000 calculations per second: 1×10121 \times 10^{12} operations/sec
  • Chemistry: The mass of a hydrogen atom is 0.000000000000000000000000001670.00000000000000000000000000167 kg: 1.67×10−271.67 \times 10^{-27} 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 out in full.

Example:

The distance to the Andromeda Galaxy is approximately 24,000,000,000,000,000,000,00024{,}000{,}000{,}000{,}000{,}000{,}000{,}000 meters, written as 2.4×10222.4 \times 10^{22} m.

1Try It Yourself

A light-year is approximately 9,460,000,000,000,0009{,}460{,}000{,}000{,}000{,}000 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 0.0000020.000002 meters long, written as 2×10−62 \times 10^{-6} m (2 micrometers).

2Try It Yourself

A virus particle measures approximately 0.000000120.00000012 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 1,000,000,000,0001{,}000{,}000{,}000{,}000 bytes, or 1×10121 \times 10^{12} bytes.

3Try It Yourself

A large cloud data center can store about 5,300,000,000,000,000,0005{,}300{,}000{,}000{,}000{,}000{,}000 bytes.

Write this in scientific notation.

Step 1: Write the mathematical expression

Convert to scientific notation:

Key Takeaways

  • 1Scientific notation expresses numbers as a×10na \times 10^n where 1≤a<101 \le a < 10
  • 2For large numbers, move decimal LEFT and use POSITIVE exponent
  • 3For small numbers (less than 1), move decimal RIGHT and use NEGATIVE exponent
  • 4Count how many places the decimal point moves to find the exponent
  • 5Always check that your coefficient is between 1 and 10

Frequently Asked Questions

If a≥10a \ge 10, you can always rewrite it by moving the decimal and adjusting the exponent. For example, 10×105=1×10610 \times 10^5 = 1 \times 10^6. Having one agreed way to write each number prevents confusion.
If a≥10a \ge 10, you can always rewrite it by moving the decimal and adjusting the exponent. For example, 10×105=1×10610 \times 10^5 = 1 \times 10^6. Having one agreed way to write each number prevents confusion.
Write it as 1×1061 \times 10^6. The coefficient 1 satisfies 1≤a<101 \le a < 10.
Yes! Since 100=110^0 = 1, this equals 3. Any number between 1 and 10 can be written with exponent 0.

Glossary

Scientific notation
A way to write numbers as a×10na \times 10^n where 1≤a<101 \le a < 10
Coefficient
The number aa in scientific notation, which must be between 1 and 10
Exponent
The power nn in 10n10^n, 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

a×10na \times 10^n

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

Today's challengeOne problem, three tries

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