Using Numbers in Science
Significant Digits
Significant digits are all those digits that occupy places for which an
actual measurement was taken. These include the last digit which
should always be an estimated value
Determining the Number of Significant Digits
You start by counting the first non-zero digit, and then all those that
follow.
Exceptions
1. If there is no decimal in the number, any zero(s) that the number
ends with are not significant.
2. Numbers that are arrived at when counting and exactly defined
quantities have an unlimited number of significant digits
Examples
1. 5.3451
5 sig. dig.
2. 0.00845
3 sig. Dig.
3. 0.0431000
6 sig. dig.
4. 0.003200
4 sig. dig.
5. 70 000
1 sig. dig.
6. 4.821 x 105
4 sig. dig.
Try These
1.
2.
3.
4.
5.
6.
7.
1.98
0.02850
6.509
900.00
8.02 x 10-9
0.00016
505







3
4
4
5
3
2
3
SD
SD
SD
SD
SD
SD
SD
Scientific Notation
When writing a number in scientific notation, it is composed of two parts.
The mantissa, and the exponent.
2.45 x 106
mantissa
exponent
The number is always written so that there is one non-zero digit in front of
the decimal point.
Converting from Scientific to Standard Form
If the exponent becomes larger the mantissa must become smaller and if
the exponent become smaller the mantissa must become larger to
compensate for the change to the exponent. The overall value of the
number cannot change.
ie.
3.45 x 105
4.285 x 10-3
6.9 x 107
345 000
.004285
69 000 000
* Keep the same number of SD
Converting Standard Form to Scientific Form
If the mantissa becomes larger the exponent must become smaller and
if the mantissa become smaller the exponent must become larger to
compensate for the change to the mantissa. The overall value of the
number cannot change.
ie.
0.000 663
36 297.6
0.000 000 000439
6.63 x 10-4
3.62976 x 104
4.39 x 10-10
Try These
1.
2.
3.
4.
5.
6.
7.
8.
25 387
6.92 x 107
0.000 03659
1.00 x 10-5
88 543 756
5.21 x 10-7
3 974 000
0.000 000 241
 2.538 7 x 104
 69 200 000
 3.659 x 10-5
 0.0000100
 8.854 375 6 x 107
 0.000 000 521
 3.974 x 106
 2.41 x 10-7
Rounding Off
Rules
1.
If the number is larger than 5, round up.
ie. 52548 = 52550
2.
If the number is smaller than 5, round down.
ie. 0.047325 = .047
3.
If the number is a 5 followed by all zeroes, or no digits, round to
the nearest even number.
ie. 8 775 = 8 780
4.
If the number is a 5 followed by any non zero digits, round up.
ie. 0.00445002 = 0.0045
Try These
1)
2)
3)
4)
5)
6)
7)
34 783
.0021754
750 043
68 500
29 384
5 550.1
0.9735
 34 800
 .002175
 800 000
 68 000
 29 000
 5 600
 0.974
Conversions
Multiply
terra
T
giga
G
mega M
kilo
k
hecta h
deca
da
deci
d
centi
c
milli
m
micro µ
nano
n
pico
p
femto f
1012
109
106
103
102
101
100
10-1
10-2
10-3
10-6
10-9
10-12
10-15
Divide
When doing conversions, if the unit is getting larger, the number must get
smaller and if the unit is getting smaller, the number must get larger.
However, you must maintain the same number of significant digits.
ie.
1000 mm = 1 m
The meter is 1 000 times larger than the millimeter,
therefore the number must become 1 000 times smaller.
.00462 km = 46.2 dm
The dm is 10 000 times smaller than the km,
therefore the number is 10 000 times bigger.
ie.
.193 g = 19.3 cg
923 ps = 9.23 x 10-10 s
0.0446 Mm = 4.46 x 105 dm
Try These
1)
2)
3)
4)
5)
.905 hg =
mg
822 cm =
dm
81.4 pm =
m
.00775 g =
µg
3.76 x 106 kg =
dg





90500
82.2
8.14 x 10-11
7.75 x 103
3.76 x 1010
Multiplying and Dividing
When multiplying or dividing, your answer must be rounded off so that it
contains the same number of SD as the value with the least number of SD.
ie.
1. 17 / 42 = .404761904 (must contain 2 SD)
= .40
2. 3.125 x .11 = .34375 (must contain 2 SD)
= .34
3. (7.58 x 104) (8.32 x 10-8) / (4.18 x 10-5) = 150.8746411 (3 SD)
= 151
Try These
1. (35.72) (0.00590)
9. (7.79 x 104) (6.45 x 104)
(5.44 x 106)
2. (707000) (3.1)
3. (0.05432) (62000)
4. (6.090 x 10-1) (9.08 x 105)
5. (1.101 x 109) (4.75 x 109)
6. (6810.12) / (2.4)
7. (.4832) / (5.12)
8. (1.18 x 10-2) / (2.2 x 103)
10. (4.87 x 106) (9.69 x 101)
( 9.765 x 10-3)
11. (2.1 x 103) (2.593 x 10-2)
(5.23 x 10-3) (6 x 10-5)
Answers
1)
2)
3)
4)
5)
6)
.211
2.2 x 106
3400
5.53 x 105
5.23 x 1018
2800
7)
.0944
8)
5.4 x 10-6
9)
924
10) 4.83 x 1010
11) 2 x 108
Adding and Subtracting
When you add or subtract, the answer must be rounded off to the
place value of the least accurate value.
ie. 677
39.2
6.23
722.43
Must be rounded off to the ones place
722
Correct rounded off answer
2) 201
3) 5.32 - 0.75938 = 4.56062
3.57
= 4.56 (hundredths)
98.493
303.063 = 303 (ones)
Try These
Answers
1) 35.6 + 4.987 + 0.09135
40.7
2) 2798 + 33.00 + 45.991
2877
3) .98 - 629.9 + 5300
4700
Rearranging Equations
To rearrange an equation for a new variable it must be
isolated in the numerator.
Rules
To move a number or variable to the opposite side of an
equation, you must perform the opposite operation to that
number or variable, and then do the same thing to both sides
of the equation.
In order to invert both sides of an equation, you must have a
single denominator on both sides of the equation.
𝐴 = 𝜋𝑟 2
Rearrange for r
1. Divide both sides by 𝜋
𝐴
= 𝑟2
𝜋
2. Take the square root of both sides.
𝐴
= 𝑟
𝜋
Using and Expressing Measurements
Measurement – is a quantity that has both a number and a unit.
- it is fundamental to experimental science.
- it is important to be able to make measurements
and decide if they are correct
Accuracy, Precision and Error
Accuracy – a measure of how close a measurement is to the actual.
Precision – is a measure of how close a series of measurements are to
one another.
Determining Error
Error = experimental value – accepted value
Percent Error =
|error|_____
accepted value
x 100%