Significant Figures

Every time you measure something, there is a limit to how precisely your instrument can read. A regular ruler marked in millimetres lets you be confident about the centimetre and millimetre digits, but the next digit is just your best guess. So how do you honestly report what you measured? That is where significant figures come in.

Why Significant Figures Matter

Think about it this way: if you measure the period of a pendulum and get 1.62 s, the digits 1 and 6 are solidly reliable, while the 2 is your best estimate from the last readable division of the stopwatch. That gives you three significant figures. Similarly, if you measure the length of an object and record it as 287.5 cm, the digits 2, 8, and 7 are certain while 5 is uncertain, giving four significant figures. Writing the pendulum result as 1.6200 s would be misleading because it implies a precision your instrument never had.

Significant figures (also called significant digits) are all the digits in a measured value that are known reliably, plus the very first digit that carries some uncertainty. They tell anyone reading your result exactly how precise the measurement actually was.

A few important things follow from this:

• Reporting extra digits beyond your significant figures is misleading. It gives a false impression of higher precision.
• The number of significant figures depends on the instrument, specifically on its least count (the smallest division it can resolve). A vernier calliper reading to 0.01 mm naturally gives more significant figures than a metre stick marked in centimetres.
• Switching units does not change the number of significant figures. If you measure a length as 2.308 cm (four significant figures), the same measurement is 0.02308 m, 23.08 mm, or 23080 $\mu$m. All four representations have the same four significant digits: 2, 3, 0, and 8. The position of the decimal point shifts, but the precision of the measurement remains unchanged.

This last point is crucial and makes most of the counting rules intuitive once you keep it in mind.

Rules for Counting Significant Figures

Here are the rules, with examples for each:

Rule 1: Every non-zero digit is significant

This one is straightforward. In 2.308, all four digits (2, 3, 0, 8) are significant. In 735, all three digits are significant.

Rule 2: Zeros between non-zero digits are always significant

A zero that sits between two non-zero digits is doing real work. It carries information about the measurement, not just about decimal placement. In 2.308, the 0 between the 3 and the 8 is significant. In 20508, all five digits are significant. This holds regardless of whether the number has a decimal point or not.

Rule 3: Leading zeros are not significant

When a number is less than 1, the zeros that appear after the decimal point but before the first non-zero digit are not significant. They are just placeholders that show where the decimal point falls.

For example, in 0.002308, the underlined zeros (0.00…) are not significant. The significant digits are 2, 3, 0, and 8, giving four significant figures. If you convert 2.308 cm to metres, you get 0.02308 m, but the number of significant figures stays at four.

Rule 4: Trailing zeros without a decimal point are not significant

If a number greater than 1 has trailing zeros and no decimal point is written, those trailing zeros are not considered significant. For instance, a measurement of 123 m can also be expressed as 12300 cm or 123000 mm. In all three forms (none of which has a decimal point), only the digits 1, 2, and 3 are significant, giving three significant figures. The trailing zeros simply reflect the change in units, not any additional precision from the instrument.

This is where confusion often creeps in, as you will see shortly.

Rule 5: Trailing zeros with a decimal point are significant

If the number does include a decimal point, trailing zeros at the end are significant. They are deliberately written to show that the measurement is precise to that digit.

For example:

• 3.500 has four significant figures (3, 5, 0, 0)
• 0.06900 has four significant figures (6, 9, 0, 0)
• 4.700 has four significant figures (4, 7, 0, 0)

The trailing zeros in these numbers are not decoration. If the measurement were only precise to one decimal place, you would write 3.5 or 4.7 instead.

Rule 6: The placeholder zero before the decimal is never significant

In a number like 0.1250, the single zero before the decimal point is a conventional placeholder. It is never counted as significant. The significant digits here are 1, 2, 5, and the trailing 0, giving four significant figures.

The Trailing-Zero Problem and How Scientific Notation Solves It

Here is a real scenario that shows why Rules 4 and 5 can cause trouble.

Suppose you measure a length to be 4.700 m. The trailing zeros clearly convey precision: your instrument could read to the nearest millimetre, and the result has four significant figures. Now, convert that to millimetres:

$4.700 \text{ m} = 4700 \text{ mm}$

The number 4700 has no decimal point, so by Rule 4, it looks like it has only two significant figures. But you know the original measurement had four. A simple change of units appears to have destroyed information about precision.

Scientific notation eliminates this ambiguity entirely. In scientific notation, every number is written as $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is any integer (positive or negative). When you use this format:

$4.700 \text{ m} = 4.700 \times 10^2 \text{ cm} = 4.700 \times 10^3 \text{ mm} = 4.700 \times 10^{-3} \text{ km}$

No matter what unit you pick, the base number is always 4.700, and it clearly has four significant figures. The power of 10 only sets the scale; it plays no role in counting significance. Every zero that appears in $a$ is significant, period.

Practical guideline: If you are not using scientific notation, follow these summary rules:

• For a number greater than 1, without a decimal point, trailing zeros are not significant.
• For a number with a decimal point, trailing zeros are significant.

When there is any room for doubt, scientific notation is the safest choice.

Order of Magnitude

Scientific notation also gives you a quick way to estimate the scale of a quantity. When you express a number as $a \times 10^b$, you can round $a$ to get a rough sense of size:

• If $a \leq 5$, round $a$ down to 1, so the number is approximately $10^b$
• If $a > 5$, round $a$ up to 10, so the number is approximately $10^{b+1}$

The exponent you end up with is called the order of magnitude of the quantity.

For example:

• The diameter of the Earth is $1.28 \times 10^7$ m. Since 1.28 $\leq$ 5, the order of magnitude is 7.
• The diameter of a hydrogen atom is $1.06 \times 10^{-10}$ m. Since 1.06 $\leq$ 5, the order of magnitude is $-10$.

This means the Earth’s diameter is about $10^{17}$ times the diameter of a hydrogen atom, or 17 orders of magnitude larger. Order-of-magnitude estimates are extremely useful in physics for quick sanity checks and rough comparisons.

Exact Numbers Have Infinite Significant Figures

Not every number in a calculation comes from a measurement. Some numbers are exact by definition, and they carry no uncertainty at all.

For instance, in the formula $r = \dfrac{d}{2}$, the factor 2 is not a measured value. It is an exact mathematical relationship: the radius is always precisely half the diameter. You can write it as 2.0, 2.00, or 2.0000000 with as many digits as you want, because it is perfectly precise.

Similarly, in $T = \dfrac{t}{n}$, the number $n$ (counting how many oscillations you timed) is an exact integer, not a measurement.

The same applies to constants like $2\pi$ in $s = 2\pi r$. The factor $2$ and $\pi$ are mathematical constants with infinite precision.

The key distinction: numbers that come from counting or from mathematical definitions are exact. Numbers that come from measurement with instruments are not. Only measured values have a finite number of significant figures.