Scientific Notation and Significant Figures

Scientific notation is a convenient way to write very large or very small numbers while making their significant figures clear.

It is widely used in science, mathematics, chemistry, physics, engineering, and other fields where measurements can contain many zeros or span a large range of values.

For example, the number:

0.000450

can be written as:

4.50 × 10⁻⁴

Both forms represent the same numerical value, but scientific notation makes the three significant figures immediately visible.

If you are learning this topic for the first time, our guide explaining What Significant Figures Tell You About a Number provides the foundation for understanding why certain digits matter.

What Is Scientific Notation?

Scientific notation expresses a number in the form:

a × 10ⁿ

where:

• a is a number greater than or equal to 1 and less than 10

• 10ⁿ represents a power of ten

• n is an integer

For example:

5,000 = 5 × 10³

0.00072 = 7.2 × 10⁻⁴

Scientific notation separates the significant digits from the position of the decimal point.

This makes it particularly useful when working with significant figures.

Why Is Scientific Notation Useful for Significant Figures?

Scientific notation removes much of the ambiguity that can occur with zeros.

Consider:

5000

Depending on the context, it may not be clear how many significant figures are intended.

Compare this with:

• 5 × 10³ → 1 significant figure

• 5.0 × 10³ → 2 significant figures

• 5.00 × 10³ → 3 significant figures

• 5.000 × 10³ → 4 significant figures

The coefficient tells you exactly which digits are significant.

This is one reason scientific notation is useful when reporting measured values.

For more examples of how zeros are treated, see our Significant Figures Rules guide.

How to Count Significant Figures in Scientific Notation

When a number is written in scientific notation, all digits in the coefficient are significant.

The exponent does not count as a significant figure.

Example 1

3.45 × 10⁶

The coefficient is 3.45.

It contains:

3 significant figures

The 10⁶ only determines the magnitude of the number.

Example 2

7.0 × 10⁻³

The coefficient is 7.0.

It contains:

2 significant figures

Example 3

1.250 × 10⁸

The coefficient contains four digits:

1, 2, 5, and 0

Therefore:

1.250 × 10⁸ has 4 significant figures.

Converting a Number to Scientific Notation

To convert a number to scientific notation:

1. Move the decimal point until only one non-zero digit remains to its left.

2. Count how many places the decimal point moved.

3. Use that number as the exponent of 10.

4. Preserve the significant digits of the original number.

Example 1: Large Number

Convert:

45,600

If the intended precision is four significant figures:

4.560 × 10⁴

The decimal point moved four places to the left.

The coefficient 4.560 contains four significant figures.

Example 2: Small Number

Convert:

0.000725

Move the decimal point four places to the right:

7.25 × 10⁻⁴

The negative exponent indicates that the original number was less than 1.

The coefficient 7.25 contains three significant figures.

Why Does the Exponent Become Negative?

When converting a number smaller than 1 into scientific notation, the decimal point moves to the right.

This produces a negative exponent.

For example:

0.0056 = 5.6 × 10⁻³

The decimal point moves three places to the right.

Therefore, the exponent is:

−3

Another example:

0.000034 = 3.4 × 10⁻⁵

The decimal point moves five places to the right.

Converting Scientific Notation to Standard Form

To convert scientific notation back into ordinary decimal notation, move the decimal point according to the exponent.

Positive Exponent

A positive exponent moves the decimal point to the right.

Example:

3.25 × 10⁴

Move the decimal point four places right:

32,500

Negative Exponent

A negative exponent moves the decimal point to the left.

Example:

4.5 × 10⁻³

Move the decimal point three places left:

0.0045

The significant figures remain the same.

4.5 × 10⁻³ has 2 significant figures.

0.0045 also has 2 significant figures.

Scientific Notation and Zeros

Scientific notation is particularly useful when a number contains many zeros.

Consider:

0.0004500

The leading zeros are not significant, while the trailing zeros after the decimal point are significant.

The number contains:

4 significant figures

In scientific notation:

4.500 × 10⁻⁴

The coefficient makes those four significant figures explicit.

This is easier to interpret than the original decimal representation.

Trailing Zeros in Scientific Notation

Trailing zeros in the coefficient are significant.

Compare:

• 6 × 10⁴ → 1 significant figure

• 6.0 × 10⁴ → 2 significant figures

• 6.00 × 10⁴ → 3 significant figures

• 6.000 × 10⁴ → 4 significant figures

The exponent does not affect the number of significant figures.

Only the digits in the coefficient are counted.

Scientific Notation and Rounding

Scientific notation is also useful when rounding a number to a specified number of significant figures.

Suppose:

7.846 × 10⁵

needs to be rounded to 3 significant figures.

The first three significant digits are:

7, 8, and 4

The next digit is 6, so the 4 rounds up to 5.

Therefore:

7.85 × 10⁵

For a more detailed explanation of rounding procedures, see our Rounding Significant Figures guide.

Multiplication Using Scientific Notation

Scientific notation makes multiplication involving very large or very small numbers easier.

Consider:

(2.5 × 10³) × (4.0 × 10²)

First multiply the coefficients:

2.5 × 4.0 = 10.0

Then multiply the powers of ten:

10³ × 10² = 10⁵

So:

10.0 × 10⁵

Rewrite in standard scientific notation:

1.00 × 10⁶

The original values contain:

• 2.5 → 2 significant figures

• 4.0 → 2 significant figures

Therefore, the final answer should have 2 significant figures:

1.0 × 10⁶

The extra digits are removed when the result is rounded to the required precision.

Division Using Scientific Notation

For division, divide the coefficients and subtract the exponents.

Consider:

(6.0 × 10⁵) ÷ (2.0 × 10²)

Divide the coefficients:

6.0 ÷ 2.0 = 3.0

Subtract the exponents:

10⁵ ÷ 10² = 10³

Therefore:

3.0 × 10³

Both input values contain 2 significant figures, so the final result also contains 2 significant figures.

Answer:

3.0 × 10³

If you want to understand the complete multiplication and division precision rule, see Significant Figures in Multiplication and Division.

Addition and Subtraction in Scientific Notation

Addition and subtraction require more care.

The powers of ten should normally be made the same before adding or subtracting the coefficients.

Example

Calculate:

3.2 × 10⁴ + 4.5 × 10³

Rewrite the second value using the same power of ten:

4.5 × 10³ = 0.45 × 10⁴

Now add:

3.2 × 10⁴ + 0.45 × 10⁴

= 3.65 × 10⁴

The precision of the original values must then be considered when reporting the final result.

For a detailed explanation of the precision rule used for addition and subtraction, see Significant Figures in Addition and Subtraction.

Scientific Notation and Measurement Precision

Scientific notation does more than shorten a number. It can communicate the precision of a measured value.

For example:

2.4 × 10³ m

contains 2 significant figures.

While:

2.40 × 10³ m

contains 3 significant figures.

Both represent the same numerical magnitude, but they do not communicate the same reported precision.

This distinction matters when measurements are used in scientific calculations.

Scientific Notation in Chemistry and Physics

Scientific notation is particularly common in scientific disciplines.

Examples include:

• Atomic and molecular scales

• Distances in astronomy

• Electrical quantities

• Very small masses

• Large populations or particle counts

• Physical constants

• Laboratory measurements

For example, the speed of light in vacuum is commonly written as approximately:

3.00 × 10⁸ m/s

The coefficient 3.00 communicates three significant figures.

The exponent simply describes the magnitude.

When using values from scientific references, the precision should be interpreted according to the source and the purpose of the calculation rather than assuming that every displayed digit represents measurement uncertainty.

Exact Numbers and Scientific Notation

Scientific notation can also be used to express exact values, but the presence of scientific notation itself does not automatically make a value measured.

For example, a counted quantity can be written as:

3 × 10² students

if the quantity is exactly 300.

The distinction between an exact number and a measured value comes from how the number was obtained, not simply from how it is written.

Scientific Notation in Calculations

When performing calculations with scientific notation, keep the following principles in mind:

Multiplication

Multiply the coefficients and add the exponents.

(a × 10ᵐ)(b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

Division

Divide the coefficients and subtract the exponents.

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Addition and subtraction

First express the values with compatible powers of ten, then perform the addition or subtraction while applying the appropriate precision rule.

Significant figures

After the calculation, report the result using the applicable significant-figure rule.

Common Mistakes With Scientific Notation

Mistake 1: Counting the exponent as a significant figure

In:

4.50 × 10⁶

there are 3 significant figures, not 4.

The exponent is not counted.

Mistake 2: Dropping a significant zero

Changing:

5.00 × 10³

to:

5 × 10³

changes the number of significant figures from 3 to 1.

Mistake 3: Moving the decimal point in the wrong direction

A positive exponent moves the decimal point to the right.

A negative exponent moves it to the left.

Mistake 4: Forgetting to normalize the coefficient

Standard scientific notation has one non-zero digit to the left of the decimal point.

For example:

45 × 10³

should be written as:

4.5 × 10⁴

Mistake 5: Applying the wrong calculation rule

Multiplication and division generally use the fewest significant figures.

Addition and subtraction generally use the fewest decimal places.

Scientific Notation Quick Reference

The examples involving whole-number trailing zeros illustrate why scientific notation is useful: it makes the intended precision explicit.

A Simple Method to Remember

When working with scientific notation and significant figures, remember:

The coefficient tells you the significant figures.

The exponent tells you the magnitude.

For example:

6.20 × 10⁻⁷

• 6.20 → 3 significant figures

• 10⁻⁷ → determines the magnitude

• Final number → 3 significant figures

This simple distinction prevents many common errors.

Check Your Calculation

If you are working with a numerical problem involving significant figures, scientific notation can make the intended precision easier to identify.

You can also use our Significant Figures Calculator to check supported calculations and verify the reported result.

For a broader explanation of how the calculator handles calculations, see How the Significant Figures Calculator Works.

Frequently Asked Questions

How many significant figures are in scientific notation?

All digits in the coefficient are significant. The exponent is not counted.

For example, 3.40 × 10⁵ has 3 significant figures.

Does the exponent count as a significant figure?

No. Only the digits in the coefficient count as significant figures.

Why is scientific notation useful for significant figures?

It makes the intended significant digits clear, especially when a number contains trailing zeros.

How many significant figures are in 5.00 × 10⁴?

There are 3 significant figures because the coefficient 5.00 contains three significant digits.

How many significant figures are in 0.000450?

There are 3 significant figures: 4, 5, and the final 0.

It can be written as:

4.50 × 10⁻⁴

How do you convert a number to scientific notation?

Move the decimal point until one non-zero digit remains to the left of the decimal point. The number of positions moved becomes the exponent of 10.

What does a negative exponent mean?

A negative exponent indicates that the number is smaller than 1.

For example:

5 × 10⁻³ = 0.005

What does a positive exponent mean?

A positive exponent indicates a number larger than or equal to 10.

For example:

5 × 10³ = 5,000

How do you multiply numbers in scientific notation?

Multiply the coefficients and add the exponents. Then apply the appropriate significant-figure rule to the final result.

How do you divide numbers in scientific notation?

Divide the coefficients and subtract the exponents. Then report the final result with the appropriate significant-figure precision.

How do addition and subtraction work in scientific notation?

First express the numbers using compatible powers of ten. Then add or subtract the coefficients and apply the appropriate decimal-place precision rule.

Does scientific notation change the value of a number?

No. Converting a number to scientific notation changes how it is written, not its numerical value.

Is 500 always one significant figure?

Not necessarily. The written number 500 can be ambiguous without additional context. Scientific notation can make the intended precision clear, such as 5 × 10², 5.0 × 10², or 5.00 × 10².

Can scientific notation be used with measured values?

Yes. Scientific notation is commonly used to express measured values while making their significant figures explicit.