Scientific Notation: Definition, Formula & Example

Scientific notation expresses very large or very small numbers as a coefficient multiplied by a power of 10.

The standard form is:

a × 10ⁿ

where:

1 ≤ |a| < 10

and:

n is an integer

For example:

6,300,000 = 6.3 × 10⁶

because moving the decimal point six places to the left produces:

6.3

Similarly:

0.00042 = 4.2 × 10^-4

because moving the decimal point four places to the right produces:

4.2

Scientific notation makes magnitude easier to see and simplifies calculations involving extreme values.

What Is Scientific Notation?

Scientific notation separates a number into:

a coefficient

and:

a power of 10.

The coefficient contains the significant digits.

The exponent records the decimal-place shift needed to recover the original number.

For example:

3.75 × 10⁸

has coefficient:

3.75

and exponent:

8

Expanding it gives:

375,000,000

Scientific Notation Formula

The standard form is:

N = a × 10ⁿ

where:

1 ≤ |a| < 10

and:

n ∈ Z

For positive numbers:

1 ≤ a < 10

For negative numbers:

-10 < a ≤ -1

Examples:

2.4 × 10⁵

-7.1 × 10^-3

are both normalized scientific notation.

Why the Coefficient Must Be Between 1 and 10

Consider:

42 × 10⁵

This represents a valid numerical expression but is not normalized scientific notation because:

42 ≥ 10

Move the coefficient decimal one place left:

42 = 4.2 × 10¹

Then:

42 × 10⁵

= 4.2 × 10¹ × 10⁵

= 4.2 × 10⁶

Therefore the normalized form is:

4.2 × 10⁶

Converting a Large Number to Scientific Notation

Convert:

8,450,000

Move the decimal point until only one nonzero digit remains to its left:

8.45

The decimal moved:

6 places left

Therefore the exponent is:

+6

So:

8,450,000 = 8.45 × 10⁶

Converting a Small Number to Scientific Notation

Convert:

0.0000072

Move the decimal point to obtain:

7.2

The decimal moved:

6 places right

Therefore the exponent is:

-6

So:

0.0000072 = 7.2 × 10^-6

Positive vs. Negative Exponents

A positive exponent typically represents a number with magnitude at least 10.

For example:

3.2 × 10⁴

= 32,000

A negative exponent represents division by a power of 10.

For example:

3.2 × 10^-4

= 3.2/10⁴

= 0.00032

The sign of the exponent controls the scale, not the sign of the number itself.

Negative Numbers in Scientific Notation

A negative value keeps its negative coefficient.

For example:

-52,000

becomes:

-5.2 × 10⁴

The exponent is positive because the magnitude is large.

Likewise:

-0.00031

becomes:

-3.1 × 10^-4

The negative sign belongs to the coefficient.

Scientific Notation and Powers of Ten

The scientific functions used on calculators include powers such as:

10ˣ

These directly support scientific notation.

Examples:

10³ = 1,000

10⁰ = 1

10^-1 = 0.1

10^-4 = 0.0001

Multiplying by these powers shifts the decimal point systematically.

Powers of Ten Table

Every decrease of 1 in the exponent divides the value by 10.

Why 10⁰ = 1

Using exponent rules:

10¹/10¹ = 10^(1-1)

Left side:

10/10 = 1

Right side:

10⁰

Therefore:

10⁰ = 1

This allows ordinary values from 1 through just under 10 to be written with exponent zero.

For example:

6.2 = 6.2 × 10⁰

Why Negative Exponents Produce Decimals

Use:

10^-n = 1/10ⁿ

For example:

10^-3 = 1/1000

= 0.001

Therefore:

4.6 × 10^-3

means:

4.6 × 0.001

= 0.0046

Decimal-Point Rule

To convert ordinary decimal notation into scientific notation:

For a number with magnitude:

≥ 10

move the decimal left and use a positive exponent.

For a nonzero number with magnitude:

< 1

move the decimal right and use a negative exponent.

The exponent’s magnitude equals the number of places moved.

Example: 73,900,000

Move the decimal:

73,900,000 → 7.39

Movement:

7 places left

Therefore:

73,900,000 = 7.39 × 10⁷

Example: 0.000000891

Move:

0.000000891 → 8.91

The decimal moves:

7 places right

Therefore:

0.000000891 = 8.91 × 10^-7

Convert Scientific Notation to Standard Form

To expand:

4.75 × 10⁵

move the decimal:

5 places right

giving:

475,000

For:

4.75 × 10^-5

move the decimal:

5 places left

giving:

0.0000475

Example: Expand 6.02 × 10²³

The exponent:

23

means move the decimal 23 places to the right.

This creates an extremely large integer.

In most calculations, retaining:

6.02 × 10²³

is much easier to read and manipulate than writing every zero.

That compactness is one of the principal advantages of scientific notation.

Multiplying Scientific Notation

To multiply:

(a × 10ᵐ)(b × 10ⁿ)

multiply coefficients and add exponents:

ab × 10^(m+n)

Then normalize the coefficient if necessary.

Example: Multiply 3 × 10⁴ by 2 × 10⁵

Calculate coefficients:

3 × 2 = 6

Add exponents:

4 + 5 = 9

Therefore:

(3 × 10⁴)(2 × 10⁵) = 6 × 10⁹

The coefficient is already normalized.

Example Requiring Normalization

Calculate:

(6 × 10³)(4 × 10⁵)

Multiply coefficients:

6 × 4 = 24

Add exponents:

3 + 5 = 8

Initial result:

24 × 10⁸

Normalize:

24 = 2.4 × 10¹

Therefore:

24 × 10⁸

= 2.4 × 10⁹

So:

Result = 2.4 × 10⁹

Dividing Scientific Notation

To divide:

(a × 10ᵐ)/(b × 10ⁿ)

divide coefficients and subtract exponents:

(a/b) × 10^(m-n)

with:

b ≠ 0

Then normalize if needed.

Example: Divide 8 × 10⁷ by 2 × 10³

Divide coefficients:

8/2 = 4

Subtract exponents:

7 – 3 = 4

Therefore:

(8 × 10⁷)/(2 × 10³) = 4 × 10⁴

Division Requiring Normalization

Calculate:

(2 × 10⁵)/(8 × 10²)

Coefficients:

2/8 = 0.25

Exponents:

5 – 2 = 3

Initial result:

0.25 × 10³

The coefficient is below 1.

Normalize:

0.25 × 10³

= 2.5 × 10²

Therefore:

Result = 2.5 × 10²

Raising Scientific Notation to a Power

For:

(a × 10ⁿ)ᵏ

apply the exponent to both factors:

aᵏ × 10^(nk)

Then normalize if necessary.

For example:

(2 × 10³)²

= 2² × 10⁶

= 4 × 10⁶

Therefore:

(2 × 10³)² = 4 × 10⁶

Example: Cube of 3 × 10²

Calculate:

(3 × 10²)³

Coefficient:

3³ = 27

Power of 10:

10^(2×3) = 10⁶

Initial result:

27 × 10⁶

Normalize:

2.7 × 10⁷

Therefore:

(3 × 10²)³ = 2.7 × 10⁷

Roots of Scientific Notation

Roots can be especially simple when the exponent is divisible by the root index.

For example:

√(9 × 10⁸)

Take each square root:

√9 = 3

√(10⁸) = 10⁴

Therefore:

√(9 × 10⁸) = 3 × 10⁴

Cube Root Example

Evaluate:

∛(8 × 10¹²)

Since:

∛8 = 2

and:

12/3 = 4

we get:

∛(8 × 10¹²) = 2 × 10⁴

The exponent division works cleanly because 12 is divisible by 3.

Roots With Nondivisible Exponents

Consider:

√(4 × 10⁷)

Rewrite:

4 × 10⁷ = 40 × 10⁶

Then:

√(40 × 10⁶)

= √40 × 10³

Alternatively:

√4 × 10^(7/2)

= 2 × 10^3.5

Both are mathematically valid, but the result may need further transformation depending on the desired form.

Adding Scientific-Notation Numbers

Addition is easiest when the powers of 10 match.

For example:

3.2 × 10⁵ + 4.7 × 10⁵

Factor:

10⁵

Then:

(3.2 + 4.7) × 10⁵

= 7.9 × 10⁵

Therefore:

Result = 7.9 × 10⁵

Subtracting Scientific-Notation Numbers

If exponents match:

8.5 × 10⁶ – 3.1 × 10⁶

subtract coefficients:

(8.5 – 3.1) × 10⁶

= 5.4 × 10⁶

Therefore:

Result = 5.4 × 10⁶

Addition With Different Exponents

Consider:

3.2 × 10⁵ + 4.7 × 10⁴

First express both using:

10⁵

Since:

4.7 × 10⁴ = 0.47 × 10⁵

Then:

3.2 × 10⁵ + 0.47 × 10⁵

= 3.67 × 10⁵

Therefore:

Result = 3.67 × 10⁵

Do not simply add coefficients while ignoring unequal exponents.

Example With a Very Small Addend

Calculate:

6.00 × 10⁸ + 2.00 × 10³

Convert:

2.00 × 10³ = 0.00002 × 10⁸

Then:

(6.00 + 0.00002) × 10⁸

= 6.00002 × 10⁸

The second term is tiny relative to the first.

Depending on measurement precision, subsequent rounding may make its contribution disappear from the reported result.

Scientific Notation and Rounding

rounding rules often work naturally with the coefficient.

Suppose:

6.78341 × 10⁸

must be reported to four significant figures.

Round:

6.78341 → 6.783

because the next digit is 4.

Therefore:

6.783 × 10⁸

The exponent remains unchanged.

Rounding That Changes the Exponent

Round:

9.9987 × 10⁵

to three significant figures.

Coefficient:

9.9987 → 10.0

So:

10.0 × 10⁵

Normalize:

1.00 × 10⁶

The rounding carry changes the power of 10.

Significant Figures in Scientific Notation

Scientific notation makes significant figures easy to identify.

For:

4.20 × 10⁷

the coefficient:

4.20

has:

3 significant figures

The exponent does not affect the count.

Likewise:

8.000 × 10^-4

contains:

4 significant figures

Scientific Notation Removes Zero Ambiguity

Consider:

120,000

Without additional context, the number of significant trailing zeros may be unclear.

Scientific notation makes precision explicit:

1.2 × 10⁵ → 2 significant figures

1.20 × 10⁵ → 3 significant figures

1.2000 × 10⁵ → 5 significant figures

The numerical magnitude is the same, but the expressed precision differs.

Comparing Scientific-Notation Numbers

For positive numbers, compare exponents first.

Suppose:

A = 7.9 × 10⁸

B = 4.5 × 10⁹

Since:

10⁹ > 10⁸

we know immediately:

B > A

No coefficient comparison is needed.

Same-Exponent Comparison

Compare:

6.2 × 10⁵

and:

7.1 × 10⁵

The exponents are equal.

Compare coefficients:

6.2 < 7.1

Therefore:

6.2 × 10⁵ < 7.1 × 10⁵

Comparing Negative Scientific-Notation Values

For negative numbers, remember ordinary signed ordering.

For example:

-8 × 10⁶

is less than:

-2 × 10⁶

because it lies farther to the left on the real number line.

Scientific notation does not change the usual rules for negative values.

Order of Magnitude

The power of 10 gives a quick sense of a number’s scale.

For example:

3.4 × 10⁷

has magnitude in the tens of millions.

2.1 × 10^-6

has magnitude around millionths.

This allows values differing by many powers of ten to be compared quickly.

Ratio of Magnitudes

Consider:

5 × 10⁹

and:

5 × 10⁶

Their ratio is:

(5 × 10⁹)/(5 × 10⁶)

= 10³

= 1000

Therefore the first number is:

1000 times larger

than the second.

A difference of 3 in the base-10 exponent corresponds to a factor of:

10³

Scientific Notation in Sequence Sums

When terms in sequence sums have very different magnitudes, scientific notation makes scale differences clearer.

For example:

2 × 10⁶ + 3 × 10⁵ + 4 × 10⁴

rewrite using:

10⁶

Then:

2 × 10⁶ + 0.3 × 10⁶ + 0.04 × 10⁶

= 2.34 × 10⁶

Matching exponents makes the addition transparent.

Scientific Notation and Series Convergence

In numerical work with series convergence, later terms may become extremely small.

Suppose successive terms have magnitudes:

1 × 10^-4

1 × 10^-6

1 × 10^-8

1 × 10^-10

Scientific notation makes the decreasing scale immediately visible.

It can also help judge whether newly added terms materially affect a rounded partial sum.

Small Terms and Numerical Precision

Suppose a calculation adds:

1.000000 × 10⁸

and:

1 × 10^-2

Mathematically, the second term is not zero.

But in limited-precision numerical systems, adding quantities separated by many orders of magnitude may lose some low-order information.

Scientific notation reveals that scale difference before computation.

Engineering-Style Exponents

Some technical contexts prefer exponents that are multiples of:

3

For example, ordinary normalized scientific notation:

4.7 × 10⁵

might be expressed as:

470 × 10³

in an engineering-style representation.

That form is useful with prefixes such as kilo, mega, milli, and micro, but its coefficient no longer satisfies the strict scientific-notation condition:

1 ≤ |a| < 10

Therefore the two conventions should be distinguished.

Calculator E Notation

Calculators and software often display scientific notation using:

E

For example:

6.5E7

means:

6.5 × 10⁷

Likewise:

4.2E-6

means:

4.2 × 10^-6

This is a compact machine-friendly representation.

E Notation Is Not Euler’s Number

The display:

3E5

means:

3 × 10⁵

It does not mean:

3 × e⁵

where:

e ≈ 2.71828

These expressions have entirely different values.

Entering Scientific Notation

A scientific calculator may provide:

EXP

EE

or an exponent-entry control.

To enter:

4.5 × 10^-8

a calculator might accept:

4.5 EXP -8

and display:

4.5E-8

The exact key sequence varies by device, but the mathematical value remains:

4.5 × 10^-8

Standard Form vs. Scientific Notation

“Standard form” can mean different things in different educational systems.

In this article:

ordinary decimal notation

means forms such as:

450,000

or:

0.00045

Scientific notation means:

4.5 × 10⁵

or:

4.5 × 10^-4

When a problem uses the phrase “standard form,” its local convention should be checked.

Zero in Scientific Notation

The ordinary normalized form:

a × 10ⁿ

requires:

1 ≤ |a| < 10

Therefore zero cannot satisfy that coefficient condition.

Zero is simply written:

0

rather than having a unique normalized scientific-notation exponent.

Expressions such as:

0 × 10⁵

all equal zero but are not normalized scientific notation.

Numbers Between 1 and 10

A number such as:

7.2

can be written:

7.2 × 10⁰

because:

10⁰ = 1

This is valid normalized scientific notation, although ordinary notation is usually simpler for such values.

Numbers Between -10 and -1

Similarly:

-4.8

can be written:

-4.8 × 10⁰

The coefficient satisfies:

1 ≤ |-4.8| < 10

so the representation is normalized.

Scientific Notation and Percentages

Suppose:

0.000025

is converted to a percentage.

First scientific notation:

2.5 × 10^-5

Multiply by:

100 = 10²

Then:

2.5 × 10^-5 × 10²

= 2.5 × 10^-3 %

So:

0.000025 = 0.0025%

Powers of ten can make decimal-to-percentage shifts easier to track.

Scientific Notation and Ratios

Consider:

(6 × 10⁸)/(3 × 10⁵)

Divide coefficients:

6/3 = 2

Subtract exponents:

8 – 5 = 3

Therefore the ratio is:

2 × 10³

or:

2000

The first quantity is 2000 times the second.

Scientific Notation and Roots

A useful strategy when taking roots is to rewrite the exponent as a multiple of the root index.

For example:

√(2.5 × 10¹⁰)

can be separated as:

√2.5 × 10⁵

because:

√(10¹⁰) = 10⁵

Thus:

√(2.5 × 10¹⁰) = √2.5 × 10⁵

Approximately:

√2.5 ≈ 1.5811

so:

≈ 1.5811 × 10⁵

Common Mistake: Wrong Exponent Sign

Convert:

0.00063

to scientific notation.

The decimal must move:

4 places right

So the exponent is:

-4

Correct:

6.3 × 10^-4

Using +4 would represent:

63,000

rather than 0.00063.

Common Mistake: Counting Decimal Moves Incorrectly

Convert:

52,800

The decimal location can be viewed as:

52,800.

Move to:

5.28

Count:

4 positions

Therefore:

52,800 = 5.28 × 10⁴

not:

5.28 × 10⁵

Common Mistake: Leaving an Unnormalized Coefficient

The expression:

15.2 × 10⁶

is numerically meaningful but not normalized scientific notation.

Move decimal one place left:

1.52

Increase exponent by 1:

1.52 × 10⁷

Common Mistake: Adding Exponents During Addition

Consider:

2 × 10⁴ + 3 × 10⁴

Do not calculate:

5 × 10⁸

Adding exponents is a multiplication rule.

For addition with matching exponents:

(2+3) × 10⁴

= 5 × 10⁴

Common Mistake: Multiplying Without Normalizing

Suppose:

(8 × 10³)(5 × 10⁴)

Initial product:

40 × 10⁷

Normalize:

4 × 10⁸

Therefore:

Result = 4 × 10⁸

Stopping at 40 × 10⁷ leaves the coefficient outside the normalized range.

Common Mistake: Treating E as Multiplication by e

The calculator value:

2.1E6

means:

2.1 × 10⁶

not:

2.1 × e⁶

Always distinguish exponent-entry notation from Euler’s constant.

Common Mistake: Dropping Significant Zeros

The forms:

4.5 × 10⁶

and:

4.500 × 10⁶

have the same numerical value but communicate different precision.

If trailing zeros are significant to the measurement, removing them loses information.

How to Check a Scientific-Notation Conversion

Suppose:

0.000072 = 7.2 × 10^-5

Check:

10^-5 = 0.00001

Then:

7.2 × 0.00001

= 0.000072

The original value is recovered.

Therefore the conversion is correct.

Frequently Asked Questions

What is scientific notation?

Scientific notation represents a number as:

a × 10ⁿ

where:

1 ≤ |a| < 10

and n is an integer.

How do you write a large number in scientific notation?

Move the decimal left until one nonzero digit remains before it. The number of places moved becomes a positive exponent.

How do you write a small decimal in scientific notation?

Move the decimal right until one nonzero digit remains before it. The number of places moved becomes a negative exponent.

Write 45,000 in scientific notation.

4.5 × 10⁴

Write 0.00045 in scientific notation.

4.5 × 10^-4

What does 10⁰ equal?

1

What does 10^-3 equal?

0.001

How do you multiply scientific notation?

Multiply coefficients and add powers of ten:

(a × 10ᵐ)(b × 10ⁿ) = ab × 10^(m+n)

then normalize.

How do you divide scientific notation?

Divide coefficients and subtract exponents:

(a × 10ᵐ)/(b × 10ⁿ) = (a/b) × 10^(m-n)

then normalize.

How do you add scientific-notation values?

First rewrite them with the same power of 10, then add the coefficients.

What does 6.02E23 mean?

6.02 × 10²³

Does E mean Euler’s number?

Not in calculator exponent notation.

Can zero be written in normalized scientific notation?

Zero does not have a unique normalized a × 10ⁿ representation satisfying 1 ≤ |a| < 10.

Final Example

Calculate:

(4.8 × 10⁷)(2.5 × 10^-3)

First multiply the coefficients:

4.8 × 2.5

= 12

Now add the exponents:

7 + (-3)

= 4

Initial result:

12 × 10⁴

Normalize the coefficient:

12 = 1.2 × 10¹

Therefore:

12 × 10⁴

= 1.2 × 10⁵

So:

(4.8 × 10⁷)(2.5 × 10^-3) = 1.2 × 10⁵

Check in ordinary notation:

4.8 × 10⁷ = 48,000,000

2.5 × 10^-3 = 0.0025

Multiply:

48,000,000 × 0.0025

= 120,000

and:

120,000 = 1.2 × 10⁵

The result agrees.

The central scientific-notation form is:

N = a × 10ⁿ, where 1 ≤ |a| < 10

Positive exponents represent movement toward larger powers of ten, negative exponents represent smaller decimal scales, and arithmetic follows the ordinary exponent rules once coefficients and powers of ten are handled separately.