Significant Figures

What significant figures are, why they show how accurate a number is, and how to count them, with the rule and the reason for every kind of zero.

Before you start

To round a number is to replace it by a simpler number close to it: $3847 \approx 3800$ , correct to the nearest 100. The sign $\approx$ means "is approximately equal to". The two zeros of $3800$ stand where the 4 and the 7 were.

The decimal places (d.p.) of a number are the digits after its decimal point: $6.25$ has 2 decimal places. A zero at the end is written when that place is known: $2.497 \approx 2.50$ , correct to 2 decimal places.

Every digit has a place value. In $0.045$ the 4 is in the hundredths place and the 5 is in the thousandths place.

What Significant Figures Are

No measurement is exact. A ruler, a weighing scale and a stopwatch can each measure only as finely as their markings allow. So when a value is written down, its digits should show how carefully it was measured.

Suppose Mei Ling's height is written as $1.6$ m. It was measured to the nearest $0.1$ m, and two digits were measured. Written as $1.58$ m, it was measured to the nearest $0.01$ m, which is the nearest centimetre, and three digits were measured. In a measurement, the digits that were measured are its significant figures.

Definition

The significant figures of a number are its digits from the first non-zero digit to the last digit that is known. A number that is correct to the nearest $0.01$ is known as far as its hundredths digit. A number that is correct to the nearest 100 is known as far as its hundreds digit.

The first non-zero digit, reading from the left, is the first significant figure.

A zero that is written only to keep the other digits in their places is a placeholder. A placeholder is not a significant figure.

The short form of "significant figures" is s.f.: $1.58$ has 3 s.f.

Significant figures and accuracy

For the same quantity, the more significant figures a value is given to, the higher its degree of accuracy: $1.58$ m (3 s.f.) tells us more about Mei Ling's height than $1.6$ m (2 s.f.).

Changing the unit does not change how carefully something was measured. Mei Ling's height of $1.58$ m is $158$ cm, and it is also $0.00158$ km. All three values have the same three significant figures: 1, 5 and 8. The zeros of $0.00158$ were not measured. They are placeholders, which keep the 1 in the thousandths place.

In the drawings of this lesson, a significant figure is in a solid box with its count under it, and a zero that is not significant is in a dashed box. A zero that may or may not be significant has a question mark under it.

Example 1: Three measurements of one length

Three pupils measure the length of the same whiteboard. Ahmad writes $2$ m, Bala writes $2.4$ m and Chloe writes $2.38$ m.
(a) How many significant figures does each value have?
(b) Was each value measured to the nearest metre, the nearest $0.1$ m or the nearest $0.01$ m?
(c) Which value is given to the highest degree of accuracy?

Solution

(a) No digit is a zero, so every digit was measured. $2$ has 1 significant figure, $2.4$ has 2 and $2.38$ has 3.

(b) The last digit of $2$ is in the ones place, so Ahmad measured to the nearest metre. The last digit of $2.4$ is in the tenths place, so Bala measured to the nearest $0.1$ m. The last digit of $2.38$ is in the hundredths place, so Chloe measured to the nearest $0.01$ m, which is the nearest centimetre.

(c) The three values agree with one another: $2.38$ rounds to $2.4$ and to $2$ . Chloe's value has the most significant figures, so it tells us the most about the length.

(a) $2$ m has 1 s.f., $2.4$ m has 2 s.f. and $2.38$ m has 3 s.f.
(b) Ahmad measured to the nearest metre, Bala to the nearest $0.1$ m and Chloe to the nearest $0.01$ m.
(c) Chloe's value, $2.38$ m, is given to the highest degree of accuracy.

Non-Zero Digits and the Zeros Between Them

Counting significant figures is easy when no digit is a zero. A zero needs thought, because it may be part of the value or it may be only a placeholder. Five rules cover every digit, and one idea lies behind all of them.

The idea behind every rule

A zero is significant when it is part of the value as it is known: it was measured or counted, or its place was kept when the number was rounded. A zero is not significant when it only holds the other digits in their places.

Rule 1

Every non-zero digit is significant.

A digit from 1 to 9 is never a placeholder. It is always part of the value as it is known. So $3681$ has 4 significant figures and $4.6$ has 2.

Rule 2

Every zero between two non-zero digits is significant. The two non-zero digits need not be next to the zero, and the decimal point may lie between them.

The last digit of $1006$ is in the ones place, so the number is known as far as its ones. Its hundreds and its tens are then known too, and each was found to be 0. So $1006$ has 4 significant figures.

Example 2: Zeros between non-zero digits

State the number of significant figures in
(a) $46.215$
(b) $7304$
(c) $20.078$
(d) $90.803$
(e) $500\,002$

Solution

(a) All five digits are non-zero, so all five are significant (Rule 1).

(b) The zero is between 3 and 4, so it is significant (Rule 2). All four digits count.

(c) Both zeros lie between 2 and 7, so both are significant. The decimal point makes no difference. The significant figures are 2, 0, 0, 7 and 8.

(d) One zero is between 9 and 8, and the other is between 8 and 3. The significant figures are 9, 0, 8, 0 and 3.

(e) The four zeros lie between 5 and 2, so all six digits are significant.

(a) $46.215$ has 5 s.f.
(b) $7304$ has 4 s.f.
(c) $20.078$ has 5 s.f.
(d) $90.803$ has 5 s.f.
(e) $500\,002$ has 6 s.f.

Zeros at the Front

A number between 0 and 1, such as $0.0045$ , begins with zeros. They are called leading zeros. The zero before the decimal point shows only that there are no ones. The two zeros after the decimal point keep the 4 in the thousandths place. None of the three was measured.

Rule 3

The zeros before the first non-zero digit are never significant. They are placeholders.

This covers the zero before the decimal point and every zero between the decimal point and the first non-zero digit. So $0.0045$ has 2 significant figures, 4 and 5.

Changing the unit shows why. An ant is $0.0045$ m long, which is $4.5$ mm. It is the same length, measured with the same care, so both values have 2 significant figures. The leading zeros disappear when the unit changes, so they were never part of the measurement.

Only the zeros before the first non-zero digit are leading zeros. In $0.0205$ the first two zeros are leading zeros, but the zero between 2 and 5 is significant by Rule 2. So $0.0205$ has 3 significant figures.

Example 3: Leading zeros

State the number of significant figures in
(a) $0.35$
(b) $0.07$
(c) $0.0008$
(d) $0.0906$
(e) $0.04005$

Solution

(a) The first non-zero digit is 3. The zero before it is a leading zero. The significant figures are 3 and 5.

(b) The first non-zero digit is 7. The two zeros before it are leading zeros. The only significant figure is 7.

(d) The first non-zero digit is 9. The two zeros before it are leading zeros (Rule 3). The zero after it lies between 9 and 6, so it is significant (Rule 2). The significant figures are 9, 0 and 6.

(e) The first non-zero digit is 4. The two zeros before it are leading zeros. The two zeros after it lie between 4 and 5, so they are significant. The significant figures are 4, 0, 0 and 5.

(a) $0.35$ has 2 s.f.
(b) $0.07$ has 1 s.f.
(c) $0.0008$ has 1 s.f.
(d) $0.0906$ has 3 s.f.
(e) $0.04005$ has 4 s.f.

Zeros at the End of a Decimal

$1.6$ and $1.60$ are equal in value, so the last zero of $1.60$ is not needed to keep any digit in its place. There is only one reason to write it: the hundredths place is known, and its digit is 0. A length of $1.60$ m was measured to the nearest $0.01$ m. A length of $1.6$ m was measured only to the nearest $0.1$ m.

Rule 4

In a decimal, the zeros after the last non-zero digit are always significant.

Here "a decimal" means a number written with a decimal point. So $1.60$ has 3 significant figures: 1, 6 and 0.

Rule 4 covers a zero before the decimal point as well. In $40.0$ both zeros come after the last non-zero digit, 4. The number is known as far as its tenths, so its ones are known too. Both zeros are significant, and $40.0$ has 3 significant figures.

Rules 1 to 4 together

In a decimal, every digit from the first non-zero digit to the last digit is significant.

Example 4: Zeros at the end of a decimal

State the number of significant figures in
(a) $0.250$
(b) $7.070$
(c) $1.200$
(d) $0.0700$
(e) $300.0$

Solution

(a) The zero before the decimal point is a leading zero. The zero at the end comes after the last non-zero digit of a decimal, so it is significant (Rule 4). The significant figures are 2, 5 and 0.

(b) The first zero is between the two 7s (Rule 2), and the last zero is at the end of a decimal (Rule 4). All four digits are significant.

(d) The two zeros before the 7 are leading zeros and do not count. The two zeros after the 7 are at the end of a decimal and do count. The significant figures are 7, 0 and 0.

(e) All three zeros come after the last non-zero digit, 3, of a decimal, so all three are significant. The significant figures are 3, 0, 0 and 0.

(a) $0.250$ has 3 s.f.
(b) $7.070$ has 4 s.f.
(c) $1.200$ has 4 s.f.
(d) $0.0700$ has 3 s.f.
(e) $300.0$ has 4 s.f.

Zeros at the End of a Whole Number

The zeros at the end of a whole number are a different matter. They cannot be left out: $48\,000$ without its zeros is 48. So they are written whether they were measured or not.

A newspaper says that $48\,000$ people went to a concert. Suppose the ticket office counted $48\,273$ and the newspaper rounded this to the nearest 1000. Then the three zeros are placeholders, standing where the 2, the 7 and the 3 were, and only the 4 and the 8 are significant. Now suppose exactly $48\,000$ tickets were sold. Then every zero was counted and all five digits are significant. The number on the page is the same both times.

Rule 5

The zeros at the end of a whole number may or may not be significant. The number alone does not say: it depends on how accurately the number is known.

A zero at the end is significant if its place is known. It is a placeholder if its place was rounded away.

One number, four meanings

Where the number came from	Its digits	Significant figures
$4283$ , rounded to the nearest 1000		1
$3962$ , rounded to the nearest 100		2
$3998$ , rounded to the nearest 10		3
Exactly $4000$ things, counted one by one		4

Take the second row. $3962$ lies between $3900$ and $4000$ and is nearer to $4000$ . The rounded number is known as far as its hundreds, and its hundreds digit is 0. That zero is significant. The tens and the ones were rounded away, so the last two zeros are placeholders.

So a question about a whole number that ends in zeros must do one of two things. It may say how accurate the number is, as in " $4000$ , correct to the nearest 100". Or it may ask for every possibility, as in "to how many significant figures could $4000$ be correct?", and then the answer is 1, 2, 3 or 4.

When the question says how accurate the number is

Find the place that the number is correct to. The digit in that place is the last significant figure, even when it is 0. Count from the first non-zero digit to that digit.

Example 5: The question says how accurate the number is

State the number of significant figures in
(a) $76\,000$ , correct to the nearest 1000
(b) $76\,000$ , correct to the nearest 100
(c) $290$ , correct to the nearest 10
(d) $290$ , the number of pupils counted in a school hall
(e) $50\,300$ , correct to the nearest 10

Solution

(a) The number is correct to the nearest 1000, so the thousands digit, 6, is the last significant figure. The significant figures are 7 and 6. The three zeros are placeholders.

(b) The number is correct to the nearest 100, so the hundreds digit, which is the first zero, is the last significant figure. The significant figures are 7, 6 and 0. The last two zeros are placeholders.

(c) The number is correct to the nearest 10, so the tens digit, 9, is the last significant figure. The significant figures are 2 and 9. The zero is a placeholder.

(d) The pupils were counted one by one, so the ones digit is known to be 0. The significant figures are 2, 9 and 0.

(e) The zero between 5 and 3 is significant (Rule 2). The number is correct to the nearest 10, so the tens digit, which is the first of the two zeros at the end, is the last significant figure. The significant figures are 5, 0, 3 and 0. The last zero is a placeholder.

(a) $76\,000$ , correct to the nearest 1000, has 2 s.f.
(b) $76\,000$ , correct to the nearest 100, has 3 s.f.
(c) $290$ , correct to the nearest 10, has 2 s.f.
(d) $290$ , counted exactly, has 3 s.f.
(e) $50\,300$ , correct to the nearest 10, has 4 s.f.

Example 6: The question does not say

Nothing is known about how these numbers were rounded. To how many significant figures could each be correct?
(a) $5000$
(b) $60\,200$
(c) $830$

Solution

(a) The 5 is certainly significant. Each of the three zeros may or may not be. If the number is correct to the nearest 1000 it has 1 significant figure; to the nearest 100, 2; to the nearest 10, 3; to the nearest whole number, 4.

(b) The 6, the zero between 6 and 2, and the 2 are certainly significant: that is 3. The two zeros at the end may or may not be. If the number is correct to the nearest 100 it has 3 significant figures; to the nearest 10, 4; to the nearest whole number, 5.

(c) The 8 and the 3 are certainly significant. If the number is correct to the nearest 10 it has 2 significant figures; to the nearest whole number, 3.

(a) $5000$ could be correct to 1, 2, 3 or 4 s.f.
(b) $60\,200$ could be correct to 3, 4 or 5 s.f.
(c) $830$ could be correct to 2 or 3 s.f.

A writer can remove the doubt by saying how accurate the number is, or by changing the unit. A distance written as $4.0$ km has 2 significant figures by Rule 4, where $4000$ m would leave the reader guessing.

Decimal Places and Significant Figures

Decimal places and significant figures are counted in different ways. To count decimal places, count every digit after the decimal point, zeros included. This tells you the place value of the last digit. To count significant figures, start at the first non-zero digit, wherever it is, and count to the last digit that is known. This tells you how many digits of the number are known.

Number	Decimal places	Significant figures
$54.332$	3	5
$0.0045$	4	2
$7.0$	1	2
$120.50$	2	5
$0.070$	3	2

Two values can have the same number of decimal places and very different numbers of significant figures. A mass of $0.0045$ kg and a mass of $12.3456$ kg are both given to the nearest $0.0001$ kg and both have 4 decimal places. But the first has 2 significant figures and the second has 6.

Example 7: Decimal places and significant figures

Write down the number of decimal places and the number of significant figures in
(a) $0.0308$
(b) $25.0$
(c) $0.900$
(d) $1002.5$

Solution

(a) There are four digits after the decimal point: 0, 3, 0 and 8. The first non-zero digit is 3, so the significant figures are 3, 0 and 8.

(b) There is one digit after the decimal point. The zero is at the end of a decimal (Rule 4), so the significant figures are 2, 5 and 0.

(c) There are three digits after the decimal point. The first non-zero digit is 9, and the two zeros after it are at the end of a decimal. The significant figures are 9, 0 and 0.

(d) There is one digit after the decimal point. The two zeros lie between 1 and 2 (Rule 2). The significant figures are 1, 0, 0, 2 and 5.

(a) $0.0308$ has 4 d.p. and 3 s.f.
(b) $25.0$ has 1 d.p. and 3 s.f.
(c) $0.900$ has 3 d.p. and 3 s.f.
(d) $1002.5$ has 1 d.p. and 5 s.f.

Counting Significant Figures in Any Number

The five rules come down to three steps.

Method

Find the first non-zero digit. It is the first significant figure. Zeros before it do not count.
If the number is a decimal, count every digit from there to the last digit.
If the number is a whole number, count from there to the last non-zero digit. Then count the zeros at the end as far as the place the number is correct to. If you are not told, give every possible count.

A negative sign makes no difference: $-0.0620$ has the same significant figures as $0.0620$ .

Example 8: A mixed list

State the number of significant figures in
(a) $0.0620$
(b) $305.0$
(c) $8009$
(d) $0.5$
(e) $12\,000$ , correct to the nearest 100
(f) $100.20$

Solution

(a) The first non-zero digit is 6. The number is a decimal, so count from the 6 to the last digit: 6, 2 and 0.

(b) The first non-zero digit is 3. The number is a decimal, so count every digit from the 3: 3, 0, 5 and 0.

(d) The first non-zero digit, 5, is also the last digit.

(e) This whole number ends in zeros. It is correct to the nearest 100, so the hundreds digit, which is the first zero, is the last significant figure: 1, 2 and 0.

(f) The first non-zero digit is 1. The number is a decimal, so count every digit: 1, 0, 0, 2 and 0.

(a) $0.0620$ has 3 s.f.
(b) $305.0$ has 4 s.f.
(c) $8009$ has 4 s.f.
(d) $0.5$ has 1 s.f.
(e) $12\,000$ , correct to the nearest 100, has 3 s.f.
(f) $100.20$ has 5 s.f.

Example 9: A measurement

A nurse weighs a newborn baby and records its mass as $3.50$ kg.
(a) How many significant figures does this value have?
(b) Was the baby weighed to the nearest kilogram, the nearest $0.1$ kg or the nearest $0.01$ kg?
(c) The baby's father writes the mass as $3.5$ kg. What has been lost?
(d) Write $3.50$ kg in grams. How many digits of your answer are significant?

Solution

(a) $3.50$ is a decimal, so every digit from the first non-zero digit counts: 3, 5 and 0.

(b) The last digit is in the hundredths place, so the baby was weighed to the nearest $0.01$ kg, which is the nearest 10 g.

(c) $3.5$ and $3.50$ are equal in value. But $3.5$ kg has 2 significant figures and says only that the mass is correct to the nearest $0.1$ kg, which is the nearest 100 g. A reader can no longer tell that the hundredths digit was measured and found to be 0.

(d) $3.50 \times 1000 = 3500$ , so the mass is $3500$ g. Changing the unit does not change the accuracy: the mass is still known to the nearest 10 g. So the tens digit, which is the first zero, is the last significant figure, and the last zero is a placeholder.

(a) $3.50$ kg has 3 s.f.
(b) The baby was weighed to the nearest $0.01$ kg.
(c) $3.5$ kg has only 2 s.f.: it no longer shows that the hundredths digit is known.
(d) The mass is $3500$ g, and 3 of its 4 digits are significant.

Common Mistakes

Counting the zeros at the front. $0.0052$ has 2 significant figures, not 4 or 5. Start counting at the first non-zero digit.
Leaving out a zero at the end of a decimal. $2.50$ has 3 significant figures, not 2. $2.5$ and $2.50$ are equal in value, but $2.50$ is given to a higher degree of accuracy.
Stopping at the decimal point. In $40.0$ and $305.0$ the zeros on both sides of the decimal point count. These numbers have 3 and 4 significant figures.
Saying that the zeros at the end of a whole number never count, or that they always count. $4000$ may have 1, 2, 3 or 4 significant figures. Look for the words that say how accurate it is.
Mixing up decimal places and significant figures. $0.0450$ has 4 decimal places but 3 significant figures.

Practice

Work each question on paper before you open its solution.

Question 1

State the number of significant figures in
(a) $728$
(b) $5.914$
(c) $3007$
(d) $60.05$
(e) $0.62$
(f) $0.00091$

ShowHide solution

(a) All three digits are non-zero.

(b) All four digits are non-zero.

(d) The two zeros lie between 6 and 5, so all four digits are significant.

(e) The zero before the decimal point is a leading zero. The significant figures are 6 and 2.

(f) The four zeros before the 9 are leading zeros. The significant figures are 9 and 1.

(a) $728$ has 3 s.f.
(b) $5.914$ has 4 s.f.
(c) $3007$ has 4 s.f.
(d) $60.05$ has 4 s.f.
(e) $0.62$ has 2 s.f.
(f) $0.00091$ has 2 s.f.

Question 2

State the number of significant figures in
(a) $8.0$
(b) $0.50$
(c) $12.300$
(d) $0.00900$
(e) $600.0$
(f) $0.0407$

ShowHide solution

(a) The zero is at the end of a decimal, so it is significant: 8 and 0.

(b) The zero before the decimal point is a leading zero. The zero at the end of the decimal is significant: 5 and 0.

(d) The three zeros before the 9 are leading zeros. The two zeros after it are at the end of a decimal: 9, 0 and 0.

(e) All three zeros come after the last non-zero digit of a decimal, so all of them are significant: 6, 0, 0 and 0.

(f) The two zeros before the 4 are leading zeros. The zero between 4 and 7 is significant: 4, 0 and 7.

(a) $8.0$ has 2 s.f.
(b) $0.50$ has 2 s.f.
(c) $12.300$ has 5 s.f.
(d) $0.00900$ has 3 s.f.
(e) $600.0$ has 4 s.f.
(f) $0.0407$ has 3 s.f.

Question 3

State the number of significant figures in
(a) $8000$ , correct to the nearest 1000
(b) $8000$ , correct to the nearest 10
(c) $35\,000$ , correct to the nearest 100
(d) $150$ , correct to the nearest whole number
(e) $700\,000$ , correct to the nearest $10\,000$

ShowHide solution

(a) The thousands digit, 8, is the last significant figure. The three zeros are placeholders.

(b) The tens digit, which is the second zero, is the last significant figure: 8, 0 and 0. The last zero is a placeholder.

(c) The hundreds digit, which is the first zero, is the last significant figure: 3, 5 and 0. The last two zeros are placeholders.

(d) The ones digit is known, so every digit is significant: 1, 5 and 0.

(e) The ten-thousands digit, which is the first zero, is the last significant figure: 7 and 0. The other four zeros are placeholders.

(a) 1 s.f.
(b) 3 s.f.
(c) 3 s.f.
(d) 3 s.f.
(e) 2 s.f.

Question 4

Nothing is known about how these numbers were rounded. To how many significant figures could each be correct?
(a) $900$
(b) $2500$
(c) $10\,040$

ShowHide solution

(a) The 9 is certainly significant. Each of the two zeros may or may not be: 1 significant figure if the number is correct to the nearest 100, 2 if to the nearest 10, and 3 if to the nearest whole number.

(b) The 2 and the 5 are certainly significant: 2 significant figures if the number is correct to the nearest 100, 3 if to the nearest 10, and 4 if to the nearest whole number.

(c) The 1, the two zeros between 1 and 4, and the 4 are certainly significant: that is 4. Only the last zero is in doubt: 4 significant figures if the number is correct to the nearest 10, and 5 if to the nearest whole number.

(a) $900$ could be correct to 1, 2 or 3 s.f.
(b) $2500$ could be correct to 2, 3 or 4 s.f.
(c) $10\,040$ could be correct to 4 or 5 s.f.

Question 5

Write down the number of decimal places and the number of significant figures in
(a) $6.08$
(b) $0.0072$
(c) $30.150$
(d) $0.400$
(e) $205.0$

ShowHide solution

(a) There are two digits after the decimal point. The zero is between 6 and 8, so the significant figures are 6, 0 and 8.

(b) There are four digits after the decimal point. The first non-zero digit is 7, so the significant figures are 7 and 2.

(d) There are three digits after the decimal point. The first non-zero digit is 4, and the two zeros after it are at the end of a decimal: 4, 0 and 0.

(e) There is one digit after the decimal point. Every digit from the 2 counts: 2, 0, 5 and 0.

(a) $6.08$ has 2 d.p. and 3 s.f.
(b) $0.0072$ has 4 d.p. and 2 s.f.
(c) $30.150$ has 3 d.p. and 5 s.f.
(d) $0.400$ has 3 d.p. and 3 s.f.
(e) $205.0$ has 1 d.p. and 4 s.f.

Question 6

Which of these six numbers have exactly 3 significant figures?
$0.408$ , $4.80$ , $0.048$ , $480.0$ , $4008$ , $0.00480$

ShowHide solution

$0.408$ : the significant figures are 4, 0 and 8. It has 3.

$4.80$ : the significant figures are 4, 8 and 0. It has 3.

$0.048$ : the significant figures are 4 and 8. It has 2.

$480.0$ : the significant figures are 4, 8, 0 and 0. It has 4.

$4008$ : the significant figures are 4, 0, 0 and 8. It has 4.

$0.00480$ : the three zeros before the 4 are leading zeros, and the significant figures are 4, 8 and 0. It has 3.

$0.408$ , $4.80$ and $0.00480$ have exactly 3 significant figures.

Question 7

Say whether each statement is true or false, and give a reason.
(a) $0.050$ has 2 significant figures.
(b) $2.0$ cm and $2$ cm show the same degree of accuracy.
(c) A number with 3 decimal places always has at least 3 significant figures.
(d) $101$ has more significant figures than $0.00101$ .

ShowHide solution

(a) True. The two zeros before the 5 are leading zeros. The significant figures are the 5 and the zero at the end.

(b) False. $2.0$ cm has 2 significant figures and is correct to the nearest $0.1$ cm. $2$ cm has 1 significant figure and is correct to the nearest centimetre.

(d) False. Both numbers have the significant figures 1, 0 and 1. The three zeros at the front of $0.00101$ are leading zeros.

(a) True
(b) False: $2.0$ cm is given to a higher degree of accuracy.
(c) False: $0.004$ has 3 d.p. and 1 s.f.
(d) False: both have 3 s.f.

Question 8

Ravi finds the mass of the same coin on three balances in the science laboratory. The readings are $7.6$ g, $7.62$ g and $7.620$ g.
(a) How many significant figures does each reading have?
(b) Which reading is given to the highest degree of accuracy? Is it correct to the nearest $0.1$ g, the nearest $0.01$ g or the nearest $0.001$ g?
(c) Ravi says that $7.62$ g and $7.620$ g are the same reading. Is he right?

ShowHide solution

(a) $7.6$ has the significant figures 7 and 6. $7.62$ has 7, 6 and 2. $7.620$ is a decimal, so the zero at its end counts: 7, 6, 2 and 0.

(b) $7.620$ g has the most significant figures. Its last digit is in the thousandths place, so it is correct to the nearest $0.001$ g.

(c) The two readings are equal in value, but they do not say the same thing. $7.62$ g is correct to the nearest $0.01$ g: its thousandths digit was not measured. $7.620$ g says that the thousandths digit was measured and found to be 0.

(a) $7.6$ g has 2 s.f., $7.62$ g has 3 s.f. and $7.620$ g has 4 s.f.
(b) $7.620$ g, which is correct to the nearest $0.001$ g.
(c) He is right about the value but not about the accuracy: $7.620$ g is given to a higher degree of accuracy than $7.62$ g.

Question 9

The distance from Aisha's home to her school is $2.40$ km, correct to the nearest $0.01$ km.
(a) How many significant figures does $2.40$ km have?
(b) Write the distance in metres.
(c) How many digits of your answer to (b) are significant? Which digit is a placeholder?
(d) A reader sees only " $2400$ m" and is not told how accurate it is. To how many significant figures could it be correct?

ShowHide solution

(a) $2.40$ is a decimal, so the zero at its end counts: 2, 4 and 0.

(b) $2.40 \times 1000 = 2400$ , so the distance is $2400$ m.

(c) $0.01$ km is 10 m, so the distance is correct to the nearest 10 m. The tens digit, which is the first zero, is the last significant figure: 2, 4 and 0. The ones digit, which is the last zero, is a placeholder.

(d) The 2 and the 4 are certainly significant. The reader cannot tell about the two zeros: 2 significant figures if the distance is correct to the nearest 100 m, 3 if to the nearest 10 m, and 4 if to the nearest metre.

(a) $2.40$ km has 3 s.f.
(b) The distance is $2400$ m.
(c) 3 of the 4 digits are significant. The last zero is a placeholder.
(d) It could be correct to 2, 3 or 4 s.f.

Question 10

At the last count, $239\,951$ people lived in a town.
(a) Round $239\,951$ to the nearest $10\,000$ , to the nearest 1000 and to the nearest 100.
(b) How many significant figures does each of your three answers have?
(c) What does this show about a number such as $240\,000$ ?

ShowHide solution

(a) To the nearest $10\,000$ : the thousands digit is 9, so round up from $230\,000$ to $240\,000$ .

To the nearest 1000: the hundreds digit is 9, so round up from $239\,000$ to $240\,000$ .

To the nearest 100: the tens digit is 5, so round up from $239\,900$ to $240\,000$ .

(b) Correct to the nearest $10\,000$ , the ten-thousands digit, 4, is the last significant figure: 2 and 4. Correct to the nearest 1000, the thousands digit is the last significant figure: 2, 4 and 0. Correct to the nearest 100, the hundreds digit is the last significant figure: 2, 4, 0 and 0.

(a) $240\,000$ each time.
(b) 2 s.f., 3 s.f. and 4 s.f.
(c) The zeros at the end of a whole number may or may not be significant. The number alone does not say: it depends on how the number was rounded.

Question 11

Each part of this question has many correct answers. Write down one number that
(a) is between 0 and 1 and has 3 significant figures, one of which is a zero
(b) has 2 decimal places and 4 significant figures
(c) has 4 decimal places and 1 significant figure
(d) is a whole number greater than 1000 with 2 significant figures. Say how accurate your number is.

ShowHide solution

(a) Begin with leading zeros, then write three digits that have a zero after the first of them, such as $0.205$ or $0.0640$ .

(b) Two of the four significant figures are after the decimal point, so the other two are before it, as in $37.41$ or $60.50$ .

(d) Write two digits and then placeholders, and say which place the number is correct to: $5300$ , correct to the nearest 100, or $46\,000$ , correct to the nearest 1000.

One set of answers:
(a) $0.205$
(b) $37.41$
(c) $0.0007$
(d) $5300$ , correct to the nearest 100
Check your own numbers by counting their decimal places and their significant figures.

Key Points

The significant figures of a number are its digits from the first non-zero digit to the last digit that is known. For the same quantity, more significant figures show a higher degree of accuracy.
A zero is significant when it is part of the value as it is known: it was measured or counted, or its place was kept when the number was rounded. It is not significant when it only holds the other digits in their places.
Rule 1: every non-zero digit is significant.
Rule 2: zeros between non-zero digits are significant: $1006$ has 4 s.f.
Rule 3: zeros before the first non-zero digit are never significant: $0.0045$ has 2 s.f.
Rule 4: zeros at the end of a decimal are always significant: $1.60$ has 3 s.f. and $40.0$ has 3 s.f.
Rule 5: zeros at the end of a whole number may or may not be significant. $4000$ has 1, 2, 3 or 4 s.f. according to how it was rounded, so the question must say.
Decimal places count the digits after the decimal point. Significant figures count from the first non-zero digit: $0.0450$ has 4 d.p. and 3 s.f.