1. Scientific notation

Why Scientists Use Scientific Notation

Science often deals with numbers that are extremely large or extremely small.

For example:

• The speed of light is approximately 300,000,000 m/s.
• The mass of an electron is approximately 0.000000000000000000000000000000911 kg.
• The distance from Earth to the Sun is approximately 150,000,000,000 m.

Writing numbers like these repeatedly is difficult and increases the chance of making mistakes.

Scientific notation provides a shorter and clearer way to represent them.

---

The Basic Form

A number written in scientific notation has the form:

a × 10ⁿ

where:

• a is a number from 1 up to, but not including, 10.
• 10 is the base.
• n is an integer called the exponent.

For example:

4.5 × 10⁶

The coefficient is:

4.5

The exponent is:

6

Understanding Powers of Ten

Powers of ten tell us how many times 10 is multiplied or divided.

10¹ = 10

10² = 100

10³ = 1,000

10⁴ = 10,000

10⁵ = 100,000

10⁶ = 1,000,000

Negative exponents represent values smaller than 1:

10⁻¹ = 0.1

10⁻² = 0.01

10⁻³ = 0.001

10⁻⁴ = 0.0001

10⁻⁵ = 0.00001

10⁻⁶ = 0.000001

---

Converting Large Numbers to Scientific Notation

Consider:

5,600,000

Move the decimal point until there is only one non-zero digit before it:

5.6

The decimal point moved 6 places to the left.

Therefore:

5,600,000 = 5.6 × 10⁶

For numbers greater than 10, the exponent will normally be positive.

---

Worked Example: 72,000

Start with:

72,000

Move the decimal point:

72,000 → 7.2

It moves 4 places left.

Therefore:

72,000 = 7.2 × 10⁴

Check:

10⁴ = 10,000

so:

7.2 × 10,000 = 72,000

---

Worked Example: 458,000,000

Move the decimal point until the coefficient is between 1 and 10:

458,000,000 → 4.58

The decimal point moves 8 places left.

Therefore:

458,000,000 = 4.58 × 10⁸

---

Converting Small Numbers to Scientific Notation

Consider:

0.00034

Move the decimal point until the coefficient is between 1 and 10:

0.00034 → 3.4

The decimal point moves 4 places to the right.

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

Therefore:

0.00034 = 3.4 × 10⁻⁴

Worked Example: 0.0062

Move the decimal:

0.0062 → 6.2

The decimal moves 3 places right.

Therefore:

0.0062 = 6.2 × 10⁻³

---

Worked Example: 0.00000081

Move the decimal:

0.00000081 → 8.1

The decimal moves 7 places right.

Therefore:

0.00000081 = 8.1 × 10⁻⁷

---

A Useful Rule

When converting decimal notation to scientific notation:

Large Numbers

Move the decimal left.

The exponent is positive.

Example:

83,000 = 8.3 × 10⁴

Small Numbers

Move the decimal right.

The exponent is negative.

Example:

0.000083 = 8.3 × 10⁻⁵

Notice that the two numbers are very different even though their coefficients are both 8.3.

---

Converting Scientific Notation Back to Decimal Form

The exponent tells you how to move the decimal point.

Positive Exponent

Move the decimal point right.

Example:

3.2 × 10⁵

Move 5 places:

3.2 → 320,000

Therefore:

3.2 × 10⁵ = 320,000

---

Negative Exponent

Move the decimal point left.

Example:

4.7 × 10⁻⁴

Move 4 places:

4.7 → 0.00047

Therefore:

4.7 × 10⁻⁴ = 0.00047

---

Zero Exponent

Remember:

10⁰ = 1

Therefore:

6.3 × 10⁰ = 6.3

Any non-zero number raised to the power zero equals 1.

---

Is It Proper Scientific Notation?

The coefficient must satisfy:

1 ≤ coefficient < 10

Therefore:

4.7 × 10⁵

is correctly written in scientific notation.

But:

47 × 10⁴

is not in standard scientific notation because 47 is greater than 10.

We can rewrite it:

47 × 10⁴ = 4.7 × 10⁵

Similarly:

0.52 × 10⁸

should be written:

5.2 × 10⁷

---

Comparing Numbers in Scientific Notation

Scientific notation makes very large and very small numbers easier to compare.

Consider:

3.2 × 10⁷

and:

8.9 × 10⁵

Compare the exponents first:

10⁷ > 10⁵

Therefore:

3.2 × 10⁷ > 8.9 × 10⁵

The coefficient does not need to be compared because the exponents are different.

---

Comparing Numbers with the Same Exponent

Consider:

4.8 × 10⁶

and:

7.1 × 10⁶

The exponents are the same.

Now compare the coefficients:

7.1 > 4.8

Therefore:

7.1 × 10⁶ > 4.8 × 10⁶

---

Comparing Numbers with Negative Exponents

Consider:

3.0 × 10⁻³

and:

3.0 × 10⁻⁶

Convert mentally:

3.0 × 10⁻³ = 0.003

3.0 × 10⁻⁶ = 0.000003

Therefore:

3.0 × 10⁻³ > 3.0 × 10⁻⁶

A more negative exponent represents a smaller positive number when the coefficients are comparable.

---

Multiplying in Scientific Notation

When multiplying numbers in scientific notation:

multiply the coefficients

and:

add the exponents

For example:

(2 × 10³)(4 × 10⁵)

Multiply the coefficients:

2 × 4 = 8

Add the exponents:

3 + 5 = 8

Therefore:

(2 × 10³)(4 × 10⁵) = 8 × 10⁸

---

Worked Example: Multiplication

Calculate:

(3.0 × 10⁴)(2.0 × 10³)

Multiply coefficients:

3.0 × 2.0 = 6.0

Add exponents:

4 + 3 = 7

Answer:

6.0 × 10⁷

---

When the Answer Needs Adjusting

Calculate:

(6 × 10⁴)(3 × 10⁵)

Multiply coefficients:

6 × 3 = 18

Add exponents:

4 + 5 = 9

This gives:

18 × 10⁹

But 18 is not between 1 and 10.

Rewrite:

18 × 10⁹ = 1.8 × 10¹⁰

Therefore:

(6 × 10⁴)(3 × 10⁵) = 1.8 × 10¹⁰

---

Dividing in Scientific Notation

When dividing:

divide the coefficients

and:

subtract the exponents

For example:

(8 × 10⁷) ÷ (2 × 10³)

Divide coefficients:

8 ÷ 2 = 4

Subtract exponents:

7 − 3 = 4

Therefore:

4 × 10⁴

---

Worked Example: Division

Calculate:

(9.0 × 10⁸) ÷ (3.0 × 10²)

Divide coefficients:

9.0 ÷ 3.0 = 3.0

Subtract exponents:

8 − 2 = 6

Therefore:

3.0 × 10⁶

---

Dividing with Negative Exponents

Calculate:

(6 × 10⁻³) ÷ (2 × 10⁻⁵)

Divide coefficients:

6 ÷ 2 = 3

Subtract exponents:

−3 − (−5)

= −3 + 5

= 2

Therefore:

3 × 10² = 300

Be especially careful when subtracting negative exponents.

---

Adding and Subtracting

Addition and subtraction require a different approach.

The powers of ten must be the same before the coefficients can be directly added or subtracted.

Consider:

3.2 × 10⁵ + 4.1 × 10⁵

The exponents already match.

Add the coefficients:

3.2 + 4.1 = 7.3

Therefore:

7.3 × 10⁵

---

Adding Different Powers of Ten

Calculate:

3.2 × 10⁵ + 4.0 × 10⁴

The exponents are different.

Rewrite:

4.0 × 10⁴ = 0.40 × 10⁵

Now:

3.2 × 10⁵ + 0.40 × 10⁵

Add:

3.60 × 10⁵

Therefore:

3.6 × 10⁵

depending on the required significant figures.

---

Subtraction Example

Calculate:

7.5 × 10⁶ − 2.0 × 10⁶

The exponents match.

Subtract coefficients:

7.5 − 2.0 = 5.5

Therefore:

5.5 × 10⁶

---

Scientific Notation on a Calculator

Many scientific calculators display scientific notation using:

EXP

or:

EE

or a form such as:

×10ˣ

For example, to enter:

6.02 × 10²³

you may enter:

6.02 EXP 23

depending on the calculator.

Do not enter:

6.02 × 10 × 10²³

That would represent a different number.

---

Calculator Displays

A calculator might display:

3.45E8

This means:

3.45 × 10⁸

Similarly:

7.2E−5

means:

7.2 × 10⁻⁵

The letter E in this context means:

× 10 raised to the power of

It does not mean the mathematical constant e.

---

Scientific Notation in Physics

Scientific notation is especially important in physics because physical quantities can cover enormous ranges.

For example:

speed of light ≈ 3.00 × 10⁸ m/s

elementary charge ≈ 1.60 × 10⁻¹⁹ C

gravitational constant ≈ 6.67 × 10⁻¹¹ N·m²/kg²

Earth's mass ≈ 5.97 × 10²⁴ kg

proton mass ≈ 1.67 × 10⁻²⁷ kg

Scientific notation makes these values much easier to use in calculations.

---

Worked Physics Example: Distance

Light travels at approximately:

3.00 × 10⁸ m/s

How far does it travel in:

2.00 s?

Use:

d = vt

Substitute:

d = (3.00 × 10⁸)(2.00)

Multiply:

d = 6.00 × 10⁸ m

Therefore, light travels approximately:

6.00 × 10⁸ m

in 2.00 seconds.

---

Worked Physics Example: Force

Suppose:

m = 2.0 × 10³ kg

and:

a = 3.0 × 10² m/s²

Use:

F = ma

Substitute:

F = (2.0 × 10³)(3.0 × 10²)

Multiply coefficients:

2.0 × 3.0 = 6.0

Add exponents:

3 + 2 = 5

Therefore:

F = 6.0 × 10⁵ N

---

Worked Physics Example: Density

Suppose an object has:

m = 6.0 × 10⁻³ kg

and:

V = 2.0 × 10⁻⁶ m³

Use:

ρ = m/V

Substitute:

ρ = (6.0 × 10⁻³) / (2.0 × 10⁻⁶)

Divide coefficients:

6.0 / 2.0 = 3.0

Subtract exponents:

−3 − (−6) = 3

Therefore:

ρ = 3.0 × 10³ kg/m³

or:

ρ = 3000 kg/m³

---

Worked Physics Example: Electric Charge

Suppose an object has:

5.0 × 10¹² electrons

and the magnitude of the charge of one electron is approximately:

1.60 × 10⁻¹⁹ C

Magnitude of total charge:

Q = (5.0 × 10¹²)(1.60 × 10⁻¹⁹)

Multiply coefficients:

5.0 × 1.60 = 8.0

Add exponents:

12 + (−19) = −7

Therefore:

Q = 8.0 × 10⁻⁷ C

If the particles are electrons and the question asks for signed charge, the charge would be negative:

Q = −8.0 × 10⁻⁷ C

---

Orders of Magnitude

Scientific notation can also help us compare the order of magnitude of quantities.

Consider:

2.0 × 10³ m

and:

2.0 × 10⁶ m

The exponents differ by:

6 − 3 = 3

The second distance is:

10³ = 1000

times larger.

Powers of ten allow scientists to compare scales very quickly.

---

From Atoms to Space

Scientific notation allows the same mathematical system to describe extremely different scales.

For example:

An atomic-scale distance might be approximately:

1 × 10⁻¹⁰ m

Earth's radius is approximately:

6.4 × 10⁶ m

Large astronomical distances can reach:

10²⁰ m and beyond.

Scientific notation lets physicists move between these scales without writing long strings of zeros.

---

Scientific Notation and SI Prefixes

Scientific notation is closely connected to SI prefixes.

Some common examples are:

For example:

1 km = 1 × 10³ m

1 mm = 1 × 10⁻³ m

1 µm = 1 × 10⁻⁶ m

1 nm = 1 × 10⁻⁹ m

---

Worked Example: Nanometres

Convert:

450 nm

to metres.

Since:

1 nm = 1 × 10⁻⁹ m

then:

450 nm = 450 × 10⁻⁹ m

Convert to proper scientific notation:

450 = 4.50 × 10²

Therefore:

450 nm = 4.50 × 10⁻⁷ m

---

Worked Example: Kilometres

Convert:

72 km

to metres.

Since:

1 km = 10³ m

then:

72 km = 72 × 10³ m

Convert to proper scientific notation:

72 × 10³ = 7.2 × 10⁴

Therefore:

72 km = 7.2 × 10⁴ m

---

Significant Figures

Scientific notation makes significant figures easier to identify.

Consider:

4.50 × 10⁶

The coefficient:

4.50

contains 3 significant figures.

The zeros contained in the power of ten do not determine the number of significant figures.

Compare:

4 × 10⁶ → 1 significant figure

4.5 × 10⁶ → 2 significant figures

4.50 × 10⁶ → 3 significant figures

These values communicate different levels of measurement precision.

---

Interpreting Scientific Notation

Consider:

6.2 × 10⁻⁸ m

The negative exponent tells us immediately that this is a very small distance.

Consider:

6.2 × 10⁸ m

The positive exponent tells us that this is a very large distance.

The sign of the exponent is therefore extremely important.

Changing:

10⁸

to:

10⁻⁸

does not make a small change.

It changes the scale enormously.

---

How Enormous Is the Difference?

Compare:

1 × 10⁸

with:

1 × 10⁻⁸

Their ratio is:

10⁸ / 10⁻⁸ = 10¹⁶

The first value is:

10,000,000,000,000,000

times larger than the second.

A missing negative sign can therefore completely change a scientific answer.

---

Estimating Answers

Scientific notation makes estimation easier.

Suppose:

(4.8 × 10⁶)(2.1 × 10³)

Before calculating exactly, estimate:

5 × 10⁶ × 2 × 10³

Approximately:

10 × 10⁹

which is:

1 × 10¹⁰

So the final answer should be around:

10¹⁰

If your calculator gives:

1.008 × 10⁴

you immediately know something has gone wrong.

---

Common Mistakes

Coefficient Is Too Large

Incorrect:

45 × 10⁶

Correct:

4.5 × 10⁷

The coefficient should be at least 1 but less than 10.

---

Using the Wrong Sign

Incorrect:

0.00052 = 5.2 × 10⁴

Correct:

0.00052 = 5.2 × 10⁻⁴

Small positive numbers require negative exponents.

---

Counting Zeros Instead of Decimal Moves

Focus on how many places the decimal point moves.

Example:

0.0036 → 3.6

The decimal moves 3 places.

Therefore:

3.6 × 10⁻³

---

Adding Exponents During Addition

Incorrect:

2 × 10⁵ + 3 × 10⁵ = 5 × 10¹⁰

For addition with matching exponents:

2 × 10⁵ + 3 × 10⁵ = 5 × 10⁵

Exponents are added when multiplying, not when adding.

---

Forgetting to Normalize the Answer

If multiplication gives:

24 × 10⁷

rewrite it as:

2.4 × 10⁸

---

Forgetting Units

In physics:

3.0 × 10⁸

is incomplete if the quantity requires units.

For the speed of light:

3.0 × 10⁸ m/s

is meaningful.

---

Calculator Entry Errors

Entering scientific notation incorrectly can change an answer by many orders of magnitude.

Always estimate the expected size of the answer before trusting the calculator.

---

Check Your Understanding

1. Write 65,000 in scientific notation.

2. Write 7,200,000 in scientific notation.

3. Write 0.0048 in scientific notation.

4. Write 0.00000092 in scientific notation.

5. Convert 3.4 × 10⁵ to decimal notation.

6. Convert 7.2 × 10⁻⁴ to decimal notation.

7. Which is larger: 5.2 × 10⁷ or 9.8 × 10⁶?

8. Which is smaller: 4.0 × 10⁻³ or 4.0 × 10⁻⁶?

9. Calculate: (2 × 10³)(4 × 10⁵).

10. Calculate: (3 × 10⁻²)(2 × 10⁴).

11. Calculate: (8 × 10⁷) / (2 × 10³).

12. Calculate: (6 × 10⁻⁴) / (3 × 10⁻⁶).

13. Calculate: 4.2 × 10⁵ + 3.1 × 10⁵.

14. Calculate: 7.8 × 10⁶ − 2.3 × 10⁶.

15. Rewrite 38 × 10⁷ in proper scientific notation.

16. Rewrite 0.72 × 10⁵ in proper scientific notation.

17. Write 350 nm in metres using scientific notation.

18. A vehicle has a mass of 2.5 × 10³ kg and accelerates at 4.0 × 10¹ m/s². Calculate the resultant force.

19. Light travels at 3.0 × 10⁸ m/s. Calculate the distance travelled in 5.0 s.

20. Explain why scientific notation is particularly useful in physics.

---

Key Terms

• Scientific notation – a method of writing numbers using a coefficient multiplied by a power of ten.
• Coefficient – the number placed before the power of ten; in standard scientific notation it is at least 1 but less than 10.
• Base – the number being raised to a power; scientific notation uses base 10.
• Exponent – number showing the power to which the base is raised.
• Positive exponent – commonly represents large values when scientific notation is normalized.
• Negative exponent – commonly represents values between 0 and 1 when scientific notation is normalized.
• Power of ten – number such as 10³, 10⁶ or 10⁻⁹.
• Order of magnitude – scale of a quantity expressed in terms of powers of ten.
• Decimal notation – ordinary representation of a number using digits and a decimal point.
• Significant figures – digits used to communicate the precision of a measured or calculated value.
• SI prefix – prefix such as kilo, milli, micro or nano representing a power-of-ten multiplier.

---

Key Takeaways

• Scientific notation is used to represent very large and very small numbers efficiently.
• Scientific notation has the form a × 10ⁿ.
• The coefficient must be at least 1 and less than 10.
• Large numbers generally have positive exponents.
• Small positive numbers between 0 and 1 have negative exponents.
• A positive exponent moves the decimal right when converting back to decimal notation.
• A negative exponent moves the decimal left.
• When multiplying scientific notation, multiply the coefficients and add the exponents.
• When dividing, divide the coefficients and subtract the exponents.
• When adding or subtracting, first express the numbers using the same power of ten.
• Answers should be rewritten into proper scientific notation when necessary.
• Scientific notation is especially important in physics because physical quantities range from subatomic scales to astronomical scales.
• SI prefixes such as kilo, milli, micro and nano are closely related to powers of ten.
• Scientific notation makes significant figures easier to communicate.
• Estimating the expected order of magnitude is an excellent way to catch calculator errors.

A useful summary is:

large number → positive exponent

small number → negative exponent

multiply → add exponents

divide → subtract exponents

add/subtract → match exponents first