Learning objectives
 Define accuracy and precision and
distinguish between them
 Make measurements to correct precision
 Round numbers to correct number of
decimal places.
 Determine number of SIGNIFICANT
FIGURES in a number
 Round numbers to correct number of
significant figures
RECALL( on
the board)
Challenge
Corner
Class Activity: Measuring the
length of their table
TWO TYPES OF NUMBERS: EXACT AND INEXACT.
EXACT NUMBERS ARE OBTAINED BY COUNTING
OR BY DEFINITIONS – A DOZEN OF WINE,
HUNDRED CENTS IN A DOLLAR
ALL MEASURED NUMBERS ARE INEXACT.
All analog measurements involve
a scale and a pointer

Errors arise from:
 Quality of scale
 Quality of pointer
 Calibration
 Ability of reader
• In reading the number the last digit quoted is a best estimate.
Conventionally, the last figure is estimated to a tenth of the smallest
division
2.3 6
2.0
2.1
2.2
2.3
2.4
2.5
The last figure written is always an estimate
• In this example we recorded the measurement to be 2.36
• The last figure “6” is our best estimate
• It is really saying 2.36 ± .01
2.0
2.1
2.2
2.3
2.4
2.5
Accuracy and Precision
• A number can only be as accurate and
precise as the instrument that has made
the measurement to produce that
number.
• Accuracy refers to how close a
measured value is to an accepted value.
• Precision refers to how close a series of
measurements are to one another.
What is the difference?
Let’s take a look
at an archery
contest to
determine what
accuracy and
precision really
are!
Accuracy vs. Precision
Accurate
• The arrow in the
center of the target
demonstrates high
accuracy.
Accuracy vs. Precision
Precise…NOT accurate
• The arrows have all
hit the same location
but not the center.
Accuracy vs. Precision
Accurate and Precise
• The arrows are both
precise (in the same
place) and accurate
(in the bull’s-eye).
Accuracy vs. Precision
Neither accurate or precise.
• The arrows fail to hit
the same location nor
do they hit the
intended location
(center).
Rounding and Approximations
A Question…
Imagine you are walking along the street and someone stops you and asks you to do this
nice little sum in your head in 30 seconds…
2
6.0602
3.1092  5.95
With a little knowledge about rounding and approximations, you should be able
to tell that person that the answer is about 2, and then ask them to kindly leave
you alone
1. Rounding
Now, there are lots of degrees of accuracy you will need to know how to round to, but
the way to tackle any question you could ever possibly be asked is always the same:
1. Circle the last digit you need – what I will call the Key Digit
2. Look at the unwanted digit to the right of it – if it is 5 or above add one
on to your Key Digit, if it is less than five, leave your Key Digit alone.
3. Be very careful of the dreaded number 9…
decimal places
The most common degree of accuracy you are asked to round to is a number of decimal places.
Because mathematicians are lazy, this is normally shortened down to dp.
e.g. 5.96 (2dp) means that the answer was probably really long, but when
rounded to two decimal places, it was 5.96
The thing you need to remember, and the thing that sounds really obvious, is that if
the question asks for two decimal places, you must give two, no more, no less!
Example 1 Round 5.639 to 1dp
5 . 6 3 9
1. We start by putting a ring around our
Key Digit. Now the question has asked for 1
decimal place, so our key digit is the 6, as
it occupies the 1st decimal place
2. Next we look at the digit to the right of
it – the unwanted number 3. It is less than 5,
so we leave the key digit alone.
3. So, to one decimal place, our answer is:
5.6 to 1dp
Example 2
Round 12.0482 to 2dp
1 2 . 0 4 8 2
1. This time the Key Digit is in the 2nd
decimal place, which makes it the 4
2. The unwanted digit to the right of it is
an 8, which is definitely 5 or above, so we
must add one onto our Key Digit
3. So, to two decimal places, our answer is:
12.05 to 2dp
Example 3
Round 25.72037 to 3dp
2 5 . 7 2 0 3 7
Example 4 Round 3.7952 to 2dp
1. This time the Key Digit is in the 3rd decimal
place, which makes it the 0
1. This time the Key Digit is in the 2nd
decimal place, which makes it the 9
2. The unwanted digit to the right of it is 3,
which is definitely less than 5, so just leave our
Key Digit alone
2. The unwanted digit to the right of it is a
5, which is 5 or above, so we must add one
onto our Key Digit
3. So, to three decimal places, our answer is:
But: if we add one to our key digit, we get 10!
So, we must add one to the next digit as well,
which is the 7
25.720 to 3dp
Be careful: Some silly people will put 25.72 down as
the answer thinking that the 0 makes no
difference. But it does! The question has asked for
3dp, so give them 3dp!
3 . 7 9 5 2
3. So, to two decimal places, our answer is:
3.80 to 2dp
Test Your Understanding
A
B
1dp
2dp
3dp
4dp
1dp
2dp
3dp
4dp
7.7 ?
7.74 ?
7.740?
?
7.7403
N
1dp
10.0
?
2dp
9.99
?
3dp
9.990
?
4dp
9.9901
?
13.5 ?
13.50?
?
13.496
?
13.4958
Significant Figures
• Each of the digits of a number that are
used to express it to the required
degree of accuracy, starting from the
first nonzero digit.
Rules for Significant Figures
1. All digits 1-9 are significant.
a. Example: 213 has 3 significant digits)
2. Zeros between significant digits are always significant.
a. Example: 3,013 has 4 significant digits.
3. Trailing zeroes in a number are significant only if the number contains
decimal point.
a. Example: 200.0 has 4 significant digits and 200 has 1 significant
digit.
Rules for Significant Figures
4. Zeros in the beginning of a number whose only function is to
place the decimal point are not significant.
a. Example: 0.0043 has 2 significant digits.
5. Zeros following a decimal significant figure are significant.
a. Example: 0.000342 has 3 significant digits and 0.43000 has 5
significant digits.
Remember: the size of your rounded number should be a similar size to the number in
the question, and you must use zeros to help you with this
Example 1
Round 28.53 to 1 sig fig
2 8 . 5 3
1. The Key Digit has the be the first
significant figure, which must be the 2, as it
is the first non-zero number
2. Now we carry on as normal looking to the number
to the right, which is an 8, so we add one on.
3. So, keeping the size of the answer the
same as the question with a zero, to 1 sig fig
the answer must be:
30
Example 2
Round 5,322 to 2 sig figs
5 3 2 2
1. The Key Digit is in the place of the 2nd
significant figure, which is the 3
2. The unwanted digit to the right of it is 2,
which is definitely less than 5, so we leave
our Key Digit alone
3. So, again using zeros to help us, to two sig
figs, our answer is:
5300
Example 3
Example 4
Round 0.027 to 1 sig fig
Round 305,216 to 3 sig figs
0 . 0 2 7
3 0 5 2 1 6
1. Our first significant figure is the first
non-zero number, which means it’s the 2
2. The unwanted digit to the right of it is 7,
so we add one to our Key Digit.
1. The 1st sig fig is the 3, the 2nd is the 0 (it
is after the 3, so it’s significant), so the Key
Digit is the 5
2. The unwanted digit to the right of it is a
2, so we leave our Key Digit alone.
3. No need for extra zeros here, so to the 1
significant figure our answer is:
3. We need some zeros to make our answer
the correct size, so to 3 sig figs::
0.03
305,000
Example 5
Example 6
Round 4.0004 to 2 sig figs
Round 0.089722 to 2 sig figs
4 . 0 0 0 4
0 . 0 8 9 7 2 2
1. The 1st sig fig is the 4, and so the 2nd is
the 0 (it is after the 4, so it’s significant).
1. Our 1st non zero number is the 8, so the
Key Digit must be the 9.
2. The unwanted digit to the right of it is 0,
which is definitely less than 5, so we leave
our Key Digit alone
2. The unwanted digit to the right of it is a
7, so we add one on, but that gives us 10, so
we must add one to our 8 as well.
3. So, to 2 sig figs, our answer is:
3. Keeping our answer the right size, we have:
4.0
0.090 to 2 sig figs
Examples
Round 17.4864 to:
Round 49 329 to:
• 1 sf: 20 ?
• 2 sf: 17 ?
• 3 sf: 17.5 ?
?
• 4 sf:
17.49
• 1 sf: 50 000?
• 2 sf: 49 000?
• 3 sf: 49 300?
• 4 sf:
49 ?330
Round 0.0429028 to:
• 1 sf: 0.04 ?
• 2 sf: 0.043 ?
• 3 sf: 0.0429
?
• 4 sf:
0.04290
?
Test Your
Understanding
Vote with the coloured cards in your diaries (use
the front for blue)
Round 7494.4924 to 2 sf.
Round 540 693 to 3 sf.
Round 0.04046 to 2 sf.
Round 69311 to 1 sf.
70000