What Are Very Small Numbers?

Very small numbers are numbers that represent quantities much smaller than one whole unit.

Examples include:

0.5

0.08

0.003

0.00025

0.0000004

These numbers are especially important when measuring things that are extremely small, such as cells, microorganisms, wavelengths, tiny masses, short time intervals, and microscopic distances.

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Numbers Between Zero and One

There are infinitely many numbers between:

0 and 1

Examples include:

0.9

0.5

0.25

0.01

0.001

0.0001

As a positive decimal gets closer to zero, its value becomes smaller.

For example:

0.1 > 0.01 > 0.001 > 0.0001

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Decimal Place Value

The places to the right of the decimal point represent fractions of one whole.

0.1 = one tenth

0.01 = one hundredth

0.001 = one thousandth

0.0001 = one ten-thousandth

0.00001 = one hundred-thousandth

0.000001 = one millionth

Each position to the right represents a value 10 times smaller than the position before it.

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Extending the Place-Value System

The place-value system works in both directions.

Moving left:

1 → 10 → 100 → 1,000

Each step is:

× 10

Moving right:

1 → 0.1 → 0.01 → 0.001

Each step is:

÷ 10

This pattern continues indefinitely.

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Understanding Tenths

Divide one whole into 10 equal parts.

Each part represents:

1/10

or:

0.1

Examples:

0.2 = 2/10

0.5 = 5/10

0.9 = 9/10

Tenths are the first place to the right of the decimal point.

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Understanding Hundredths

Divide one whole into 100 equal parts.

Each part represents:

1/100

or:

0.01

For example:

0.25 = 25/100

0.08 = 8/100

0.50 = 50/100

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Understanding Thousandths

One thousandth is:

1/1000

or:

0.001

Examples:

0.004 = four thousandths

0.025 = twenty-five thousandths

0.372 = three hundred seventy-two thousandths

Thousandths are common in precise measurements.

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Beyond Thousandths

Scientific measurements may require even smaller place values.

0.0001 = one ten-thousandth

0.00001 = one hundred-thousandth

0.000001 = one millionth

For example:

0.000006

represents:

six millionths

As more zeros appear immediately after the decimal point, the positive value becomes smaller.

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Zeros Are Important

Compare:

0.5

0.05

0.005

0.0005

These numbers are very different.

Each additional zero immediately after the decimal point shifts the 5 one place to the right.

Therefore:

0.5 > 0.05 > 0.005 > 0.0005

Each value is one tenth of the previous value.

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Leading Zeros and Trailing Zeros

A leading zero after the decimal point can change the value.

For example:

0.4 ≠ 0.04

However, zeros added to the end of a decimal do not change its value.

For example:

0.4 = 0.40 = 0.400

Similarly:

0.025 = 0.0250

The location of the zero matters.

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Reading Small Decimal Numbers

Consider:

0.007

The 7 is in the thousandths place.

Therefore:

0.007 = seven thousandths

Consider:

0.0004

The 4 is in the ten-thousandths place.

Therefore:

0.0004 = four ten-thousandths

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Writing Small Numbers from Words

Write:

three hundredths

Answer:

0.03

Write:

six thousandths

Answer:

0.006

Write:

nine ten-thousandths

Answer:

0.0009

The zeros preserve the correct place value.

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Expanded Form

Small decimal numbers can be written in expanded form.

Consider:

0.284

This means:

0.2 + 0.08 + 0.004

or:

2/10 + 8/100 + 4/1000

Another example:

0.0307

means:

0.03 + 0.0007

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Comparing Very Small Numbers

Very small decimals can be compared using place value.

Compare:

0.03 and 0.05

Both have 0 ones and 0 tenths.

Compare the hundredths:

3 < 5

Therefore:

0.03 < 0.05

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Comparing Decimals of Different Lengths

Compare:

0.4 and 0.35

Add a trailing zero:

0.40 and 0.35

Now compare:

40 hundredths > 35 hundredths

Therefore:

0.4 > 0.35

Adding trailing zeros can make comparisons easier.

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Comparing Very Small Values

Compare:

0.006 and 0.009

Both have:

• 0 tenths
• 0 hundredths

Compare the thousandths:

6 < 9

Therefore:

0.006 < 0.009

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Be Careful with More Digits

A decimal with more digits is not automatically larger.

Compare:

0.2

and:

0.199

Write:

0.200

and:

0.199

Now the comparison is clear:

0.200 > 0.199

Therefore:

0.2 > 0.199

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Ordering Very Small Numbers

Order from least to greatest:

0.05, 0.5, 0.005, 0.15

Write with equal decimal places:

0.050

0.500

0.005

0.150

Now compare:

0.005 < 0.050 < 0.150 < 0.500

Therefore:

0.005 < 0.05 < 0.15 < 0.5

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Another Ordering Example

Order from greatest to least:

0.008, 0.08, 0.018, 0.8

Write:

0.008

0.080

0.018

0.800

Therefore:

0.8 > 0.08 > 0.018 > 0.008

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Small Numbers on a Number Line

Decimals can be located between other decimal values.

For example:

0.004

lies between:

0.003 and 0.005

Similarly:

0.035

lies between:

0.03 and 0.04

A number line can be repeatedly "zoomed in" to represent smaller and smaller quantities.

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Very Small Numbers and Fractions

Decimals and fractions are different ways to represent the same quantities.

For example:

0.1 = 1/10

0.01 = 1/100

0.001 = 1/1000

0.000001 = 1/1,000,000

This relationship helps explain the meaning of small decimals.

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Powers of Ten

Small place values can also be represented using negative powers of 10.

10⁻¹ = 0.1

10⁻² = 0.01

10⁻³ = 0.001

10⁻⁴ = 0.0001

10⁻⁵ = 0.00001

10⁻⁶ = 0.000001

Each decrease of 1 in the exponent represents division by 10.

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Why Negative Exponents?

Consider:

10³ = 1000

10² = 100

10¹ = 10

10⁰ = 1

Continue dividing by 10:

10⁻¹ = 0.1

10⁻² = 0.01

10⁻³ = 0.001

Negative powers provide a convenient way to describe very small numbers.

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Scientific Notation for Small Numbers

Scientific notation can make very small numbers easier to write.

For example:

0.001 = 1 × 10⁻³

0.00025 = 2.5 × 10⁻⁴

0.000006 = 6 × 10⁻⁶

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

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Writing a Small Number in Scientific Notation

Consider:

0.00042

Move the decimal until there is one nonzero digit before it:

4.2

The decimal moved four places to the right.

Therefore:

0.00042 = 4.2 × 10⁻⁴

The negative exponent indicates a number smaller than 1.

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Writing Scientific Notation as a Decimal

Consider:

3.7 × 10⁻⁵

The exponent −5 means the decimal value is very small.

Move the decimal five places to the left:

0.000037

Therefore:

3.7 × 10⁻⁵ = 0.000037

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Why Scientists Use Scientific Notation

Imagine repeatedly writing:

0.000000000000000000000001

A long string of zeros is difficult to:

• read
• copy
• compare
• calculate with
• check for errors

Scientific notation makes such values more manageable.

It also clearly communicates their order of magnitude.

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Small Measurements in Biology

Biologists frequently study structures much smaller than one metre.

A typical animal cell may have dimensions measured in micrometres.

A micrometre is:

0.000001 m

or:

1 × 10⁻⁶ m

This unit is written:

µm

Micrometres are useful because writing cell dimensions in metres would produce inconveniently small decimals.

---

Millimetres

One millimetre is:

1/1000 of a metre

Therefore:

1 mm = 0.001 m

or:

1 × 10⁻³ m

Objects measured in millimetres include:

• small insects
• seeds
• thin materials
• small mechanical parts

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Micrometres

One micrometre is:

1/1,000,000 of a metre

Therefore:

1 µm = 0.000001 m

or:

1 × 10⁻⁶ m

Micrometres are commonly used for:

• cells
• microorganisms
• fibres
• microscopic structures

---

Nanometres

A nanometre is even smaller.

1 nm = 0.000000001 m

or:

1 × 10⁻⁹ m

Nanometres are useful for describing:

• molecules
• very small biological structures
• wavelengths of visible light
• nanoscale materials
• electronic components

---

Comparing Metric Scales

Consider:

1 mm = 10⁻³ m

1 µm = 10⁻⁶ m

1 nm = 10⁻⁹ m

Therefore:

1 mm = 1,000 µm

and:

1 µm = 1,000 nm

So:

1 mm = 1,000,000 nm

Each step represents a major change in scale.

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Small Numbers in Microscopy

Suppose a cell is:

0.00002 m

long.

This can be written as:

2 × 10⁻⁵ m

It can also be expressed as:

20 µm

The measurement is the same.

Only the unit and notation have changed.

Scientists choose units that make measurements easier to communicate.

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Small Numbers in Chemistry

Atoms and molecules are extremely small.

Their dimensions are often measured in:

nanometres

or even smaller units.

For example, molecular dimensions can be fractions of a nanometre.

Writing these measurements directly in metres would require many zeros.

Scientific notation makes the scale much clearer.

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Small Numbers in Physics

Physics involves many extremely small quantities.

Examples include:

• atomic dimensions
• particle masses
• short time intervals
• tiny electrical currents
• wavelengths
• microscopic forces

Understanding decimal place value is therefore essential in scientific measurement.

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Wavelengths of Light

Visible light has wavelengths measured in hundreds of nanometres.

For example, a wavelength might be:

500 nm

Since:

1 nm = 10⁻⁹ m

then:

500 nm = 5 × 10⁻⁷ m

or:

0.0000005 m

Scientific notation and appropriate units make the value much easier to understand.

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Small Time Intervals

Technology often operates on extremely short timescales.

A millisecond is:

0.001 s

A microsecond is:

0.000001 s

A nanosecond is:

0.000000001 s

These tiny intervals are important in:

• computers
• communications
• electronics
• scientific instruments
• high-speed measurements

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Small Masses

Scientists may need to measure tiny masses.

For example:

1 milligram = 0.001 g

and:

1 microgram = 0.000001 g

Small masses are important in chemistry, biology, environmental science, and materials science.

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Why Small Measurements Matter

Suppose two samples have masses:

0.0042 g

and:

0.0048 g

The difference is:

0.0006 g

That may appear tiny.

But in a precise scientific investigation, this difference could be important.

Whether a difference matters depends on:

• the experiment
• the scale being studied
• the measuring instrument
• measurement uncertainty
• the precision required

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Precision in Scientific Measurements

A measurement such as:

0.4 g

does not communicate exactly the same precision as:

0.400 g

Numerically:

0.4 = 0.400

But in measurement contexts, the written precision may communicate information about the instrument or measurement.

For example:

0.400 g

suggests measurement to the nearest:

0.001 g

This is one reason decimal places matter in scientific work.

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Measurement Uncertainty

No physical measurement is perfectly exact.

Suppose a measuring instrument gives:

0.025 g

The measurement has a limited precision based on the instrument being used.

Scientists consider:

• resolution
• uncertainty
• repeated measurements
• experimental error
• significant figures

Very small differences should not automatically be treated as meaningful if they are smaller than the uncertainty of the measurement.

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Worked Example 1: Place Value

What is the value of 7 in:

0.0072?

The 7 is in the:

thousandths place

Therefore its value is:

0.007

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Worked Example 2: Writing a Decimal

Write:

four hundred-thousandths

One hundred-thousandth is:

0.00001

Four hundred-thousandths is:

0.00004

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Worked Example 3: Comparing

Compare:

0.006 and 0.06

Write:

0.006

0.060

Compare place values.

Therefore:

0.006 < 0.06

In fact, 0.06 is ten times 0.006.

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Worked Example 4: Ordering

Order from least to greatest:

0.003, 0.03, 0.013, 0.3

Write:

0.003

0.030

0.013

0.300

Therefore:

0.003 < 0.013 < 0.03 < 0.3

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Worked Example 5: Scientific Notation

Write:

0.000008

in scientific notation.

Move the decimal six places:

8 × 10⁻⁶

Therefore:

0.000008 = 8 × 10⁻⁶

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Worked Example 6: Decimal Form

Write:

4.2 × 10⁻⁴

as a decimal.

Move the decimal four places to the left:

0.00042

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Worked Example 7: Metric Scale

Convert:

5 µm

to metres.

Since:

1 µm = 0.000001 m

then:

5 µm = 0.000005 m

or:

5 × 10⁻⁶ m

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Worked Example 8: Comparing Scientific Measurements

Sample A has a thickness of:

0.0045 mm

Sample B has a thickness of:

0.0052 mm

Since:

0.0052 > 0.0045

Sample B is thicker.

Difference:

0.0052 − 0.0045 = 0.0007 mm

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Worked Example 9: Order of Magnitude

Compare:

0.001

and:

0.000001

Write as powers:

0.001 = 10⁻³

0.000001 = 10⁻⁶

Calculate the ratio:

10⁻³ ÷ 10⁻⁶ = 10³

Therefore:

0.001 is 1,000 times larger than 0.000001

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Worked Example 10: Scientific Context

A microscopic structure has a length of:

0.000025 m

Write it in scientific notation:

2.5 × 10⁻⁵ m

Since:

1 µm = 10⁻⁶ m

the measurement can also be written:

25 µm

Different forms can describe exactly the same measurement.

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Orders of Magnitude

Very small quantities can differ enormously even when both look "tiny."

Compare:

10⁻³

and:

10⁻⁹

These represent:

0.001

and:

0.000000001

The difference in exponent is:

6

Therefore:

10⁻³ is 10⁶, or 1,000,000 times, larger than 10⁻⁹

Orders of magnitude allow scientists to compare extremely different scales.

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Estimating Small Numbers

Just as large numbers can be estimated, small numbers can also be rounded.

For example:

0.004783

might be written approximately as:

0.0048

or:

4.8 × 10⁻³

The appropriate precision depends on the situation.

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Comparing Small Quantities Efficiently

When comparing small decimals:

Step 1: Line up the decimal points.

Step 2: Add trailing zeros if useful.

Step 3: Compare digits from left to right.

For example:

0.0048

and:

0.00435

Write:

0.00480

0.00435

Therefore:

0.0048 > 0.00435

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Why Units Matter

Consider:

0.5 mm

and:

0.5 cm

The numerical values look identical, but the units are different.

Since:

1 cm = 10 mm

then:

0.5 cm = 5 mm

Therefore:

0.5 cm > 0.5 mm

Before comparing scientific measurements, make sure the units are the same.

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Real-World Application: Cell Sizes

Suppose:

Cell A = 15 µm

Cell B = 0.025 mm

Convert Cell B:

0.025 mm = 25 µm

Therefore:

25 µm > 15 µm

Cell B is larger.

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Real-World Application: Technology

Suppose two electronic components have widths:

Component A:

0.000004 m

Component B:

0.000007 m

Compare:

0.000004 < 0.000007

Therefore, Component A is narrower.

At microscopic scales, differences of only a few millionths of a metre can be important.

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Why Small Numbers Are Important in Scientific Investigations

Science depends on measurement.

Many important quantities are much smaller than ordinary everyday units.

Scientists may need to measure:

• tiny changes in mass
• microscopic distances
• small concentrations
• short time intervals
• small electrical currents
• small temperature changes
• wavelengths
• particle dimensions

Without accurate ways to represent small numbers, these measurements would be difficult to communicate and compare.

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Small Differences Can Reveal Patterns

Imagine an experiment produces:

Trial 1: 0.0048 g

Trial 2: 0.0051 g

Trial 3: 0.0050 g

Trial 4: 0.0049 g

The differences are small.

However, examining small variations can help scientists identify:

• patterns
• averages
• anomalies
• uncertainty
• experimental error
• relationships between variables

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Small Numbers and Instruments

Different instruments are designed to measure different scales.

Examples include:

• rulers
• vernier calipers
• micrometers
• digital balances
• microscopes
• electronic sensors
• timers
• spectrometers

The instrument must have enough resolution to detect the quantity being measured.

A ruler marked only in centimetres cannot accurately measure a thickness of a few micrometres.

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Common Mistakes

Mistake 1: Thinking more digits means a larger decimal

Incorrect:

0.0045 > 0.02 because 45 > 2

Correct:

0.0045 < 0.0200

Place value must be compared.

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Mistake 2: Ignoring zeros

0.5, 0.05, and 0.005

are very different values.

---

Mistake 3: Thinking trailing zeros change the value

Numerically:

0.4 = 0.40 = 0.400

Trailing zeros do not change the numerical value.

---

Mistake 4: Forgetting units

0.5 mm and 0.5 cm are not equal measurements.

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Mistake 5: Moving the decimal the wrong way with negative powers

10⁻⁶

is:

0.000001

not:

1,000,000

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Mistake 6: Treating every tiny difference as scientifically meaningful

A difference smaller than the uncertainty or resolution of the measuring instrument may not represent a meaningful physical difference.

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Error Analysis

A student says:

0.0008 is smaller than 0.00035 because 8 is smaller than 35.

This is incorrect.

Write both with equal decimal places:

0.00080

0.00035

Now compare.

At the ten-thousandths place:

8 > 3

Therefore:

0.0008 > 0.00035

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Another Error Analysis

A student writes:

6 × 10⁻⁴ = 0.006

This is incorrect.

10⁻⁴ = 0.0001

Therefore:

6 × 10⁻⁴ = 0.0006

A useful check is to count the place values carefully.

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A Reliable Strategy for Small Numbers

When working with very small numbers:

Step 1: Identify the units.

Step 2: Locate the first nonzero digit.

Step 3: Determine its place value.

Step 4: Use trailing zeros if needed for comparison.

Step 5: Compare digits from left to right.

Step 6: Convert units if the measurements use different units.

Step 7: Use scientific notation when many zeros make the number difficult to read.

Step 8: Consider measurement precision and uncertainty in scientific contexts.

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Did You Know?

The difference between everyday and microscopic scales can be enormous.

A useful scale comparison is:

metre → millimetre → micrometre → nanometre

Each step represents a factor of:

1,000

So:

1 metre = 1,000 millimetres

1 millimetre = 1,000 micrometres

1 micrometre = 1,000 nanometres

Therefore:

1 metre = 1,000,000,000 nanometres

Understanding very small numbers allows us to move mathematically from everyday objects to cells, molecules, atoms, and modern nanoscale technology.

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Key Terms

• Decimal: Number containing a decimal point and place values smaller than one.
• Tenths: First decimal place; each unit is 0.1.
• Hundredths: Second decimal place; each unit is 0.01.
• Thousandths: Third decimal place; each unit is 0.001.
• Millionth: 0.000001 or 10⁻⁶.
• Place value: Value of a digit based on its position.
• Leading zero: Zero appearing before a significant digit and helping indicate place value.
• Trailing zero: Zero written after the final nonzero decimal digit.
• Scientific notation: Method for expressing very large or very small numbers using powers of ten.
• Negative exponent: Exponent representing repeated division by 10.
• Order of magnitude: Approximate scale of a quantity represented by a power of ten.
• Millimetre (mm): One thousandth of a metre.
• Micrometre (µm): One millionth of a metre.
• Nanometre (nm): One billionth of a metre.
• Precision: Level of detail represented by a measurement.
• Resolution: Smallest change an instrument can detect.
• Uncertainty: Range associated with limitations in a measurement.
• Scale: Relative size of a quantity.

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Key Relationships

0.1 = 10⁻¹

0.01 = 10⁻²

0.001 = 10⁻³

0.000001 = 10⁻⁶

0.000000001 = 10⁻⁹

For metric measurements:

1 mm = 10⁻³ m

1 µm = 10⁻⁶ m

1 nm = 10⁻⁹ m

Also:

1 mm = 1,000 µm

1 µm = 1,000 nm

1 mm = 1,000,000 nm

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Key Takeaways

• Very small positive numbers are often written as decimals between 0 and 1.
• Decimal place value continues beyond tenths, hundredths, and thousandths.
• Each place to the right is one tenth the value of the previous place.
• Zeros immediately after the decimal point are important because they determine place value.
• Trailing zeros do not change the numerical value of a decimal.
• Small decimals should be compared using place value rather than the apparent size of their digits.
• Adding trailing zeros can make decimal comparisons easier.
• A decimal with more digits is not automatically larger.
• Number lines can be repeatedly divided to represent increasingly small quantities.
• Small decimals can also be represented as fractions.
• Negative powers of ten provide an efficient way to describe very small values.
• Scientific notation makes numbers containing many zeros easier to read, compare, and calculate with.
• Millimetres, micrometres, and nanometres are useful units for increasingly small measurements.
• Very small measurements are common in biology, chemistry, physics, electronics, computing, and materials science.
• Cells are commonly measured in micrometres.
• Molecular and nanoscale structures may be measured in nanometres.
• Light wavelengths are commonly expressed in nanometres.
• Extremely short time intervals are important in computing and electronics.
• Units must be converted before quantities expressed in different units can be compared correctly.
• Precision and resolution become especially important when measuring very small quantities.
• Small differences are not automatically scientifically meaningful; measurement uncertainty must also be considered.
• Scientists use small numbers to detect changes, compare measurements, identify patterns, and describe microscopic phenomena.
• Understanding very small numbers provides an important foundation for scientific notation, significant figures, measurement, microscopy, chemistry, physics, and scientific data analysis.