What Is Scientific Notation?

Scientific notation is a compact way to write very large and very small numbers.

Instead of writing:

450,000,000

we can write:

4.5 × 10⁸

Instead of writing:

0.0000032

we can write:

3.2 × 10⁻⁶

Scientific notation is widely used in mathematics, science, engineering, computing, medicine, and technology because it makes extreme quantities easier to read, compare, and calculate.

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The Structure of Scientific Notation

A number in scientific notation has two main parts:

a × 10ⁿ

where:

• a is the coefficient
• 10 is the base
• n is the exponent

For proper scientific notation:

1 ≤ |a| < 10

For positive quantities, this simply means the coefficient must be at least 1 but less than 10.

Examples:

3.4 × 10⁵

7.25 × 10⁻⁸

1.02 × 10¹²

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The Coefficient

The coefficient contains the significant digits of the number.

For example:

6.25 × 10⁷

The coefficient is:

6.25

Since:

1 ≤ 6.25 < 10

this is correctly written in scientific notation.

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The Exponent

The exponent tells us the scale of the number.

Consider:

4.2 × 10⁶

The exponent is:

6

This tells us that the coefficient is multiplied by:

10⁶ = 1,000,000

Therefore:

4.2 × 10⁶ = 4,200,000

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Positive Exponents

Positive exponents are commonly used for numbers greater than or equal to 10.

For example:

5.3 × 10⁴

Since:

10⁴ = 10,000

then:

5.3 × 10,000 = 53,000

Therefore:

5.3 × 10⁴ = 53,000

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Negative Exponents

Negative exponents are commonly used for positive numbers between 0 and 1.

For example:

4.7 × 10⁻³

Since:

10⁻³ = 0.001

then:

4.7 × 0.001 = 0.0047

Therefore:

4.7 × 10⁻³ = 0.0047

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Understanding the Direction

A useful pattern is:

Positive exponent → large number

Negative exponent → small positive number

For example:

3.5 × 10⁶ = 3,500,000

while:

3.5 × 10⁻⁶ = 0.0000035

The sign of the exponent completely changes the scale.

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Visualizing Scientific Notation

The connection between the coefficient, exponent, and ordinary decimal notation can be visualized directly:

The coefficient contains the important digits, while the exponent describes their place-value scale.

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Writing Large Numbers in Scientific Notation

Consider:

72,000,000

First place the decimal so the coefficient is between 1 and 10:

7.2

The original decimal point moved:

7 places

Therefore:

72,000,000 = 7.2 × 10⁷

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A Reliable Method for Large Numbers

To convert a large number to scientific notation:

Step 1: Locate the decimal point.

For a whole number, it is understood to be at the end.

72,000,000.

Step 2: Move the decimal until exactly one nonzero digit is before it.

7.2

Step 3: Count how many places it moved.

7 places

Step 4: Use that number as the exponent.

7.2 × 10⁷

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Example: 560,000

Write:

560,000

in scientific notation.

Move the decimal:

560,000 → 5.6

The decimal moved:

5 places

Therefore:

560,000 = 5.6 × 10⁵

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Example: 8,430,000,000

Move the decimal until the coefficient is:

8.43

Count the places:

9

Therefore:

8,430,000,000 = 8.43 × 10⁹

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Writing Small Numbers in Scientific Notation

Consider:

0.00052

Move the decimal until the coefficient is:

5.2

The decimal moves:

4 places to the right

Because the original number is smaller than 1, the exponent is negative.

Therefore:

0.00052 = 5.2 × 10⁻⁴

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A Reliable Method for Small Numbers

To convert a small decimal to scientific notation:

Step 1: Find the first nonzero digit.

Step 2: Place the decimal immediately after that digit.

Step 3: Count how many places the decimal moved.

Step 4: Use a negative exponent because the original number is smaller than 1.

For example:

0.0000078

becomes:

7.8 × 10⁻⁶

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Example: 0.0034

Move the decimal:

0.0034 → 3.4

The decimal moves three places.

Since the original number is less than 1:

0.0034 = 3.4 × 10⁻³

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Example: 0.00000091

Move the decimal to produce:

9.1

Count:

7 places

Therefore:

0.00000091 = 9.1 × 10⁻⁷

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Standard Form and Scientific Notation

In these notes, standard form means the ordinary decimal form of a number.

For example:

Scientific notation:

6.3 × 10⁵

Standard form:

630,000

Scientific notation:

6.3 × 10⁻⁵

Standard form:

0.000063

Being able to move confidently between these forms is an important skill.

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Scientific Notation to Standard Form: Positive Exponents

Consider:

2.7 × 10⁴

The positive exponent means the number becomes larger.

Multiply by:

10,000

Therefore:

2.7 × 10⁴ = 27,000

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Scientific Notation to Standard Form: Negative Exponents

Consider:

2.7 × 10⁻⁴

Now:

10⁻⁴ = 0.0001

Therefore:

2.7 × 10⁻⁴ = 0.00027

The negative exponent produces a small positive number.

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Place Value, Not Just "Moving the Decimal"

A common shortcut says to "move the decimal point."

This works, but a deeper understanding comes from place value.

When multiplying by:

10³

each digit becomes:

1,000 times greater in value

When multiplying by:

10⁻³

each digit becomes:

1,000 times smaller in value

Scientific notation is therefore an application of the base-ten place-value system.

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Checking the Coefficient

Not every expression involving a power of ten is proper scientific notation.

For example:

42 × 10⁵

is not in proper scientific notation because:

42 > 10

Rewrite:

42 = 4.2 × 10

So:

42 × 10⁵

becomes:

4.2 × 10⁶

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Another Correction

Consider:

0.73 × 10⁸

This is not proper scientific notation because the coefficient is less than 1.

Rewrite:

0.73 = 7.3 × 10⁻¹

Therefore:

0.73 × 10⁸ = 7.3 × 10⁷

Correct scientific notation:

7.3 × 10⁷

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Normalizing Scientific Notation

The process of adjusting an answer so that the coefficient is between 1 and 10 is sometimes called normalizing.

For example:

25 × 10⁶

becomes:

2.5 × 10⁷

Another example:

0.48 × 10⁻³

becomes:

4.8 × 10⁻⁴

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Comparing Numbers in Scientific Notation

Scientific notation makes large and small numbers easier to compare.

Consider:

4.2 × 10⁷

and:

6.8 × 10⁵

Compare the exponents first.

Since:

7 > 5

we know:

4.2 × 10⁷ > 6.8 × 10⁵

The exponent determines the overall order of magnitude.

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Comparing Numbers with the Same Exponent

Consider:

3.4 × 10⁸

and:

7.1 × 10⁸

The exponents are the same.

Compare the coefficients:

3.4 < 7.1

Therefore:

3.4 × 10⁸ < 7.1 × 10⁸

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Comparing Small Numbers

Compare:

5.2 × 10⁻⁴

and:

8.1 × 10⁻⁶

Remember:

10⁻⁴ > 10⁻⁶

Therefore:

5.2 × 10⁻⁴ > 8.1 × 10⁻⁶

With negative exponents, an exponent closer to zero represents the larger positive value.

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Comparing Negative Exponents Carefully

Compare:

2 × 10⁻³

and:

9 × 10⁻⁵

Write approximately:

0.002

and:

0.00009

Therefore:

2 × 10⁻³ > 9 × 10⁻⁵

Even though 9 is greater than 2, the exponent has a much greater effect on the scale.

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Ordering Scientific Notation

Order from least to greatest:

4.5 × 10³

7.2 × 10⁵

3.1 × 10²

9.4 × 10³

Compare exponents first:

10² < 10³ < 10⁵

For the two values with exponent 3, compare coefficients.

Therefore:

3.1 × 10² < 4.5 × 10³ < 9.4 × 10³ < 7.2 × 10⁵

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Scientific Notation and Orders of Magnitude

Scientific notation clearly shows the approximate scale of a number.

For example:

3.2 × 10⁶

is on the scale of millions.

7.4 × 10⁹

is on the scale of billions.

2.5 × 10⁻⁶

is on the scale of millionths.

This is one reason scientific notation is so useful in science.

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Multiplying Numbers in Scientific Notation

Consider:

(2 × 10³)(4 × 10⁵)

Multiply the coefficients:

2 × 4 = 8

Multiply the powers of ten:

10³ × 10⁵ = 10⁸

Therefore:

(2 × 10³)(4 × 10⁵) = 8 × 10⁸

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Multiplication Rule

When multiplying numbers in scientific notation:

Step 1: Multiply the coefficients.

Step 2: Add the exponents.

Step 3: Normalize the answer if necessary.

For example:

(3 × 10⁴)(2 × 10³)

Coefficient:

3 × 2 = 6

Exponent:

4 + 3 = 7

Answer:

6 × 10⁷

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Multiplication Requiring Normalization

Calculate:

(4 × 10⁵)(3 × 10²)

Multiply coefficients:

4 × 3 = 12

Add exponents:

5 + 2 = 7

Initial result:

12 × 10⁷

But 12 is not a valid scientific-notation coefficient.

Rewrite:

12 × 10⁷ = 1.2 × 10⁸

Therefore:

1.2 × 10⁸

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Another Multiplication Example

Calculate:

(2.5 × 10⁶)(4 × 10³)

Multiply coefficients:

2.5 × 4 = 10

Add exponents:

6 + 3 = 9

Initial result:

10 × 10⁹

Normalize:

10 × 10⁹ = 1 × 10¹⁰

Therefore:

1 × 10¹⁰

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Dividing Numbers in Scientific Notation

Consider:

(8 × 10⁷) ÷ (2 × 10³)

Divide coefficients:

8 ÷ 2 = 4

Subtract exponents:

7 − 3 = 4

Therefore:

4 × 10⁴

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Division Rule

When dividing numbers in scientific notation:

Step 1: Divide the coefficients.

Step 2: Subtract the exponents.

Step 3: Normalize if necessary.

For example:

(9 × 10⁸) ÷ (3 × 10²)

Coefficient:

9 ÷ 3 = 3

Exponent:

8 − 2 = 6

Therefore:

3 × 10⁶

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Division Requiring Normalization

Calculate:

(4 × 10⁵) ÷ (8 × 10²)

Divide coefficients:

4 ÷ 8 = 0.5

Subtract exponents:

5 − 2 = 3

Initial result:

0.5 × 10³

This is not proper scientific notation.

Rewrite:

0.5 × 10³ = 5 × 10²

Therefore:

5 × 10²

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Adding Numbers in Scientific Notation

Addition requires more care.

Consider:

3 × 10⁵ + 4 × 10⁵

The powers of ten are the same.

Add the coefficients:

(3 + 4) × 10⁵

Therefore:

7 × 10⁵

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Addition with Different Exponents

Consider:

3 × 10⁵ + 4 × 10⁴

We cannot simply add the coefficients because the powers of ten are different.

Rewrite:

4 × 10⁴ = 0.4 × 10⁵

Then:

3 × 10⁵ + 0.4 × 10⁵

= 3.4 × 10⁵

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Subtracting in Scientific Notation

Consider:

8.5 × 10⁶ − 2.1 × 10⁶

The exponents match.

Subtract coefficients:

8.5 − 2.1 = 6.4

Therefore:

6.4 × 10⁶

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Why Addition Is Different

For multiplication:

exponents can be added

For division:

exponents can be subtracted

But for addition and subtraction:

the powers of ten must represent the same place-value scale first

This is similar to adding fractions: the units must match.

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Worked Example 1: Large Number

Write:

6,500,000

in scientific notation.

Coefficient:

6.5

Decimal moved:

6 places

Therefore:

6.5 × 10⁶

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Worked Example 2: Small Number

Write:

0.000084

in scientific notation.

Coefficient:

8.4

Decimal moved:

5 places

Therefore:

8.4 × 10⁻⁵

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Worked Example 3: Standard Form

Convert:

3.72 × 10⁷

to standard form.

Multiply by:

10,000,000

Therefore:

37,200,000

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Worked Example 4: Small Standard Form

Convert:

6.1 × 10⁻⁶

to standard form.

Answer:

0.0000061

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Worked Example 5: Comparison

Which is greater?

4.8 × 10⁶

or:

7.9 × 10⁵

Compare exponents:

6 > 5

Therefore:

4.8 × 10⁶ > 7.9 × 10⁵

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Worked Example 6: Multiplication

Calculate:

(3 × 10⁴)(5 × 10⁶)

Multiply:

3 × 5 = 15

Add exponents:

4 + 6 = 10

Initial result:

15 × 10¹⁰

Normalize:

1.5 × 10¹¹

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Worked Example 7: Division

Calculate:

(6 × 10⁹) ÷ (2 × 10⁴)

Divide coefficients:

6 ÷ 2 = 3

Subtract exponents:

9 − 4 = 5

Therefore:

3 × 10⁵

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Worked Example 8: Addition

Calculate:

2.5 × 10⁷ + 1.8 × 10⁷

Add coefficients:

2.5 + 1.8 = 4.3

Therefore:

4.3 × 10⁷

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Worked Example 9: Addition with Different Exponents

Calculate:

6 × 10⁴ + 3 × 10³

Rewrite:

3 × 10³ = 0.3 × 10⁴

Then:

6 × 10⁴ + 0.3 × 10⁴

= 6.3 × 10⁴

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Worked Example 10: Multi-Step Calculation

Calculate:

(2 × 10³)(3 × 10⁴) ÷ (6 × 10²)

First multiply:

(2 × 10³)(3 × 10⁴) = 6 × 10⁷

Then divide:

(6 × 10⁷) ÷ (6 × 10²)

= 1 × 10⁵

Therefore:

100,000

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Scientific Notation in Astronomy

Astronomy involves enormous distances.

For example, the average distance between Earth and the Sun is approximately:

1.5 × 10⁸ km

Writing:

1.5 × 10⁸ km

is often easier to interpret and calculate with than:

150,000,000 km

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Scientific Notation in Biology

Cells and microorganisms are extremely small.

A cell might have a size on the order of:

10⁻⁵ m

A bacterium may have dimensions on the order of:

10⁻⁶ m

Scientific notation makes these different scales easier to compare.

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Scientific Notation in Chemistry

Chemistry often involves extremely large numbers of particles.

One mole contains approximately:

6.02 × 10²³ particles

This is known as Avogadro's constant.

Writing the number in scientific notation makes its enormous scale immediately visible.

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Scientific Notation in Physics

Physics involves both extremely large and extremely small values.

Examples include:

• distances between astronomical objects
• masses of particles
• wavelengths
• electrical quantities
• very short time intervals
• frequencies
• speeds

Scientific notation allows the same mathematical system to describe all these scales.

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Scientific Notation in Technology

Modern technology operates across enormous ranges of scale.

Scientific notation can describe:

• billions of computer operations
• tiny electronic components
• data-transfer rates
• microscopic manufacturing tolerances
• short processing times

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Estimating with Scientific Notation

Scientific notation makes estimation easier.

Estimate:

(4.9 × 10⁶)(2.1 × 10³)

Round:

4.9 ≈ 5

2.1 ≈ 2

Then:

(5 × 10⁶)(2 × 10³)

= 10 × 10⁹

= 1 × 10¹⁰

So the answer should be around:

10¹⁰

This can help check a calculator result.

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Using a Calculator

Scientific calculators often display scientific notation using formats such as:

6.2E8

This means:

6.2 × 10⁸

Similarly:

3.5E−6

means:

3.5 × 10⁻⁶

The E represents "× 10 raised to the power of."

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Calculator Example

Suppose a calculator displays:

2.45E7

Interpret this as:

2.45 × 10⁷

Therefore:

24,500,000

Understanding calculator notation is important when working with very large or very small results.

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Significant Figures and Scientific Notation

Scientific notation also makes significant figures easier to identify.

Consider:

4.50 × 10⁶

The coefficient:

4.50

contains three significant figures.

Compare:

4.5 × 10⁶

which contains two significant figures.

Although both describe values around 4.5 million, the written precision is different.

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Common Mistakes

Mistake 1: Using a coefficient greater than or equal to 10

Incorrect:

45 × 10⁶

Correct:

4.5 × 10⁷

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Mistake 2: Using a coefficient below 1

Incorrect:

0.62 × 10⁵

Correct:

6.2 × 10⁴

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Mistake 3: Using a positive exponent for a small positive decimal

Incorrect:

0.00034 = 3.4 × 10⁴

Correct:

0.00034 = 3.4 × 10⁻⁴

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Mistake 4: Using a negative exponent for a large number

Incorrect:

72,000 = 7.2 × 10⁻⁴

Correct:

72,000 = 7.2 × 10⁴

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Mistake 5: Comparing only coefficients

For:

9 × 10⁴

and:

2 × 10⁶

2 is smaller than 9, but:

2 × 10⁶ > 9 × 10⁴

Compare exponents first.

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Mistake 6: Adding exponents during addition

Incorrect:

2 × 10³ + 3 × 10⁴ = 5 × 10⁷

Exponent rules for multiplication do not apply to addition.

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Error Analysis

A student writes:

0.000072 = 7.2 × 10⁵

The coefficient is correct, but the exponent sign is wrong.

The original value is smaller than 1.

Therefore, the exponent must be negative:

0.000072 = 7.2 × 10⁻⁵

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Another Error Analysis

A student calculates:

(5 × 10⁴)(4 × 10³)

and writes:

20 × 10⁷

The multiplication is mathematically equivalent, but the result is not yet in proper scientific notation.

Normalize:

20 × 10⁷ = 2 × 10⁸

Therefore, the final answer should be:

2 × 10⁸

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A Reliable Conversion Strategy

To convert standard form to scientific notation:

1. Position the decimal so exactly one nonzero digit is before it.
2. Count the number of place-value positions.
3. Use a positive exponent for a large number.
4. Use a negative exponent for a small positive number.
5. Check that the coefficient is at least 1 but less than 10.

To convert scientific notation to standard form:

1. Identify the exponent.
2. Use the power of ten to determine the scale.
3. Write the digits in their correct place values.
4. Add placeholder zeros where necessary.
5. Check whether the answer should be large or small.

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A Reliable Calculation Strategy

For multiplication:

Multiply coefficients → add exponents → normalize

For division:

Divide coefficients → subtract exponents → normalize

For addition and subtraction:

Match powers of ten → operate on coefficients → normalize

Always estimate the approximate scale of the answer before finishing.

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Did You Know?

Scientific notation allows scientists to describe an extraordinary range of sizes using the same mathematical language.

For example, scientific notation can describe quantities associated with:

• subatomic particles
• atoms
• molecules
• cells
• humans
• buildings
• planets
• stars
• galaxies
• the observable universe

Without powers of ten and scientific notation, comparing such dramatically different scales would be much more difficult.

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Key Terms

• Scientific notation: Method for expressing a number using a coefficient multiplied by a power of ten.
• Standard form: Ordinary decimal representation of a number.
• Coefficient: Number multiplying the power of ten.
• Base: The number 10 in scientific notation.
• Exponent: Number indicating the power of ten.
• Positive exponent: Usually indicates a large value when the coefficient is in standard scientific-notation form.
• Negative exponent: Usually indicates a positive value between 0 and 1.
• Power of ten: Expression such as 10⁶ or 10⁻⁴.
• Normalize: Rewrite an expression so the coefficient is at least 1 but less than 10.
• Order of magnitude: Approximate scale of a quantity represented by a power of ten.
• Significant figures: Digits communicating the precision of a measured or stated quantity.

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Key Rules

Proper scientific notation has the form:

a × 10ⁿ

where:

1 ≤ |a| < 10

For large positive numbers:

positive exponent

For small positive numbers:

negative exponent

Multiplication:

(a × 10ᵐ)(b × 10ⁿ) = (ab) × 10ᵐ⁺ⁿ

Division:

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

For addition and subtraction:

first express the numbers using the same power of ten

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Key Takeaways

• Scientific notation provides a compact way to write very large and very small numbers.
• A scientific-notation number consists of a coefficient and a power of ten.
• The coefficient must have exactly one nonzero digit before the decimal point.
• The absolute value of the coefficient must be at least 1 and less than 10.
• Large positive numbers normally use positive exponents.
• Small positive numbers between 0 and 1 normally use negative exponents.
• The exponent describes the place-value scale of the number.
• Scientific notation is based directly on powers of ten.
• Large numbers can be converted by identifying the coefficient and counting place-value positions.
• Small numbers can be converted in the same way but use negative exponents.
• Scientific notation can always be converted back into ordinary decimal form.
• When comparing numbers with different exponents, compare the exponents first.
• When exponents are equal, compare the coefficients.
• With negative exponents, an exponent closer to zero represents the larger positive scale.
• To multiply numbers in scientific notation, multiply coefficients and add exponents.
• To divide numbers in scientific notation, divide coefficients and subtract exponents.
• Multiplication or division may produce an answer that must be normalized.
• Addition and subtraction require the powers of ten to match before coefficients can be combined.
• Scientific notation makes estimation and order-of-magnitude checks easier.
• Calculator E notation is another way of displaying scientific notation.
• Scientific notation can communicate measurement precision through significant figures.
• Scientific notation is used throughout astronomy, biology, chemistry, physics, engineering, computing, and technology.
• It allows quantities spanning enormous ranges of scale to be represented and compared using the same mathematical system.
• Understanding scientific notation builds directly on place value, very large numbers, very small numbers, powers of ten, and exponent rules.