SI Unit Conversions, Scientific Notation, and Significant Figures

Hello! This is the first lesson in your density test sprint. Before calculating density, you need to be able to trust the numbers and units you put into the formula. This lesson builds the measurement skills behind nearly every density question: SI prefixes, scientific notation, and significant figures.

By the end, you should be able to convert values such as milligrams to kilograms or cubic centimetres to cubic metres, write very large or small values compactly, and round a final calculated answer without accidentally claiming more precision than the measurement supports.

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1. SI units and prefixes: the language of measurements

Physics uses the International System of Units, abbreviated SI. Its base units include:

• length: metre, $\mathrm{m}$
• mass: kilogram, $\mathrm{kg}$
• time: second, $\mathrm{s}$
• temperature: kelvin, $\mathrm{K}$

Density is a derived quantity because it combines mass and volume. You will commonly see density written in $kg/{\mathrm{m}}^3$ or $g/c{\mathrm{m}}^3$.

A prefix changes a unit by a power of ten. The prefix is part of the unit, so $\mathrm{mg}$ is not “metres times grams”; it means milligram.

Prefix	Symbol	Meaning	Relative to the base unit
mega	$\mathrm{M}$	million	$10^6$
kilo	$\mathrm{k}$	thousand	$10^3$
centi	$\mathrm{c}$	hundredth	$10^{-2}$
milli	$\mathrm{m}$	thousandth	$10^{-3}$
micro	$\mu$	millionth	$10^{-6}$
nano	$\mathrm{n}$	billionth	$10^{-9}$

The essential translations are:

$$1\mathrm{kg}=10^3\mathrm{g}$$ $$1\mathrm{cm}=10^{-2}\mathrm{m}$$ $$1\mathrm{mL}=10^{-3}\mathrm{L}$$

Pay close attention to capital letters. $\mathrm{M}$ is mega, while $\mathrm{m}$ is milli. Therefore, $\mathrm{Mg}$ and $\mathrm{mg}$ differ by a factor of $10^9$. That is a billionfold difference.

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2. Convert units by making the unwanted unit cancel

A quick “move the decimal point” shortcut can work for simple metric conversions, but a more reliable test method is to write a conversion factor: a fraction equal to 1.

For example, because $1\mathrm{g}=1000\mathrm{mg}$,

$$\frac{1000\mathrm{mg}}{1\mathrm{g}}=1$$

Multiplying by this fraction changes the unit but not the physical amount.

Suppose a sample has mass $0.0425\mathrm{g}$, and you need milligrams:

$$0.0425\mathrm{g}\left(\frac{1000\mathrm{mg}}{1\mathrm{g}}\right)=42.5\mathrm{mg}$$

The $\mathrm{g}$ units cancel, leaving $\mathrm{mg}$. This cancellation is your built-in error check.

A useful size check

Before calculating, decide whether the numerical value should get bigger or smaller:

• Converting to a smaller unit gives a larger numerical value.
  For instance, $0.0425\mathrm{g}$ becomes $42.5\mathrm{mg}$.

• Converting to a larger unit gives a smaller numerical value.
  For instance, $4250\mathrm{mg}$ becomes $4.250\mathrm{g}$.

If your result contradicts this expectation, recheck the conversion factor.

Density-unit conversion: grams per cubic centimetre to kilograms per cubic metre

This conversion appears often because $g/c{\mathrm{m}}^3$ is convenient in school labs, while $kg/{\mathrm{m}}^3$ is the SI form.

Start with the mass conversion:

$$1\mathrm{g}=10^{-3}\mathrm{kg}$$

For volume, remember that volume has three dimensions:

$$1\mathrm{c}{\mathrm{m}}^3={\left(10^{-2}\mathrm{m}\right)}^3=10^{-6}{\mathrm{m}}^3$$

So, convert $0.750g/c{\mathrm{m}}^3$:

$$0.750\frac{\mathrm{g}}{\mathrm{c}{\mathrm{m}}^3}\left(\frac{10^{-3}\mathrm{kg}}{1\mathrm{g}}\right)\left(\frac{1\mathrm{c}{\mathrm{m}}^3}{10^{-6}{\mathrm{m}}^3}\right)=7.50\times 10^2\frac{\mathrm{kg}}{{\mathrm{m}}^3}$$

The result has the same physical density, just expressed in different units.

A shortcut worth memorising is:

$$1\frac{\mathrm{g}}{\mathrm{c}{\mathrm{m}}^3}=1000\frac{\mathrm{kg}}{{\mathrm{m}}^3}$$

This means a density of $2.7g/c{\mathrm{m}}^3$ is $2700kg/{\mathrm{m}}^3$.

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3. The cubic-unit trap: volume conversions must be cubed

A common density-test error is correctly converting a length but forgetting that a volume is cubed.

For length:

$$1\mathrm{cm}=10^{-2}\mathrm{m}$$

For volume:

$$1\mathrm{c}{\mathrm{m}}^3=10^{-6}{\mathrm{m}}^3$$

The exponent is multiplied by three because there are three length dimensions:

$${\left(10^{-2}\right)}^3=10^{-6}$$

Likewise:

$$1{\mathrm{m}}^3=10^6\mathrm{c}{\mathrm{m}}^3$$

Two volume facts are especially helpful in density questions:

$$1\mathrm{c}{\mathrm{m}}^3=1\mathrm{mL}$$ $$1\mathrm{L}=1000\mathrm{mL}=1000\mathrm{c}{\mathrm{m}}^3$$

And in SI cubic metres:

$$1\mathrm{L}=10^{-3}{\mathrm{m}}^3$$

Do not treat $\mathrm{c}{\mathrm{m}}^3$ and ${\mathrm{m}}^3$ as though they differed by only $100$. They differ by $1,000,000$.

Test habit: whenever you see a squared or cubed unit, put parentheses around the conversion before applying the power. For example,

$${\left(100\mathrm{cm}\right)}^3=10^6\mathrm{c}{\mathrm{m}}^3$$

That written structure makes it much harder to lose the exponent.

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4. Scientific notation: a clean way to handle extreme values

Scientific notation, also called standard form, has the format

$$A\times 10^n$$

where

$$1\le A<10$$

and $n$ is an integer.

The first number, $A$, contains the meaningful digits. The exponent tells you the scale.

For a large value:

$$45000=4.5\times 10^4$$

For a small value:

$$0.0045=4.5\times 10^{-3}$$

A positive exponent makes the ordinary number large; a negative exponent makes it smaller than 1.

Writing a number in scientific notation

1. Place the decimal after the first non-zero digit.
2. Count how many places it moved.
3. If the original number was large, the exponent is positive.
4. If the original number was less than 1, the exponent is negative.

For example:

$$0.0000720=7.20\times 10^{-5}$$

The decimal moves five places to form $7.20$, so the exponent is $-5$.

Returning to ordinary notation

To interpret a scientific-notation result, use the exponent as the number of decimal places:

$$3.61\times 10^3=3610$$ $$6.2\times 10^{-4}=0.00062$$

Scientific notation is very useful in density conversions because it avoids long strings of zeros:

$$0.000001{\mathrm{m}}^3=1\times 10^{-6}{\mathrm{m}}^3$$

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5. Significant figures: report only the precision your data supports

Significant figures, often shortened to sig figs, show the precision of a measured value. They matter because a calculator can display many digits that your measurements did not justify.

Counting significant figures

Use these rules.

1. Every non-zero digit is significant.

$$42.7$$

has 3 significant figures.

2. Zeros between non-zero digits are significant.

$$1002$$

has 4 significant figures.

3. Leading zeros are not significant. They only locate the decimal point.

$$0.00470$$

has 3 significant figures: $4$, $7$, and the final zero.

4. Trailing zeros after a decimal point are significant.

$$2.300$$

has 4 significant figures.

5. Trailing zeros in a whole number without a decimal point may be unclear. Scientific notation removes the ambiguity.

$$1500$$

could mean different levels of precision. But:

$$1.5\times 10^3$$

has 2 significant figures, while

$$1.500\times 10^3$$

has 4 significant figures.

Rounding in calculations

The operation determines the rounding rule.

Calculation type	Rule
Multiplication or division	Round to the fewest significant figures in the measured inputs.
Addition or subtraction	Round to the fewest decimal places in the measured inputs.

Density is found from a mass divided by a volume, so it follows the multiplication/division rule.

For a density-style calculation, suppose measurements are $12.34\mathrm{g}$ and $4.5\mathrm{c}{\mathrm{m}}^3$. The calculator gives:

$$\frac{12.34}{4.5}=2.742\dots$$

The volume has only 2 significant figures, so the final result must have 2 significant figures:

$$2.7\frac{\mathrm{g}}{\mathrm{c}{\mathrm{m}}^3}$$

Do not round intermediate steps early. Keep a few guard digits on your calculator, then round only the final answer.

Exact numbers do not limit significant figures

Some values are exact, not measured. Examples include:

$$1\mathrm{m}=100\mathrm{cm}$$

and

$$1\mathrm{L}=1000\mathrm{mL}$$

These conversion definitions are exact, so they do not reduce the precision of a measurement. If $45.0\mathrm{cm}$ is measured to 3 significant figures, then:

$$45.0\mathrm{cm}=0.450\mathrm{m}$$

The converted result should still show 3 significant figures.

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A test-ready workflow

For any question involving measurements, use this sequence:

1. Write the given value with its unit.
2. Identify the required unit.
3. Use a conversion factor so unwanted units cancel.
4. Treat squared and cubed units with their corresponding powers.
5. Use scientific notation when zeros make the calculation hard to read.
6. Keep full calculator precision until the last line.
7. Round the final answer using the correct significant-figure rule.
8. Write the unit in the final answer. A number without a unit is incomplete physics.

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You now have the numerical foundation for density questions: prefixes express powers of ten, conversion factors protect your units, cubic units require cubic conversions, scientific notation keeps extremes manageable, and significant figures keep answers physically honest.

Next, you will focus on the quality of the measurements themselves: choosing instruments, distinguishing accuracy from precision, identifying anomalous data, and improving an investigation.