IB Chemistry on Uncertainty, Significant figures and Scientific notation

Tutorial on Uncertainty, Significant figures
and Scientific notation .

Prepared by
Lawrence Kok
http://lawrencekok.blogspot.com

Significant figures

Used in measurements

Degree of precision

Show digits believed to be
correct/certain + 1 estimated/uncertain

All reads 80

80
80.0
80.00
80.000

least precise

Certain
23.00

Uncertain
5

23.005g
more precise

Number sf necessary to express a measurement
• Consistent with precision of measurement
• Precise equipment = Measurement more sf
• Last digit always an estimate/uncertain

measurement
15.831g

(15.831 ± 0.001)g
(5 sig figures)

Deals with precision NOT accuracy!!!!!!!!
Precise measurement doesnt mean, it’s accurate
( instrument may not be accurate)

Significant figures

Used in measurements

Degree of precision

Show digits believed to be
correct/certain + 1 estimated/uncertain

All reads 80

80
80.0
80.00
80.000

least precise

Certain
23.00

Uncertain
5

Zeros bet
(significant)
4.109 = 4sf
902 = 3sf
5002.05 = 6sf

measurement
15.831g

23.005g
more precise

(15.831 ± 0.001)g
(5 sig figures)

Rules for significant figures

All non zero digit
(significant)
31.24 = 4 sf
563 = 3 sf
23 = 2sf

Number sf necessary to express a measurement
• Consistent with precision of measurement
• Precise equipment = Measurement more sf
• Last digit always an estimate/uncertain

Zeros after
decimal point
(significant)
4.580 = 4 sf
9.30 = 3sf
86.90000 = 7sf
3.040 = 4sf
67.030 = 5sf

Zero right of decimal point and
following a non zero digit
(significant)
0.00500 = 3sf
0.02450 = 4sf
0.04050 = 4sf
0.50 = 2sf

Deals with precision NOT accuracy!!!!!!!!
Precise measurement doesnt mean, it’s accurate
( instrument may not be accurate)

Zeros to left of digit
(NOT significant)
0.0023 = 2sf
0.000342 = 3sf
0.00003 = 1sf

Zero without decimal
(ambiguous)
80 = may have 1 or 2 sf
500 = may have 1 or 3 sf

Click here and here for notes on sig figures

Significant figures
1

22

Smallest division = 0.1

22

Max = 21.63
2

Certain
21.6

Uncertainty = 1/10 of smallest division.
= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01

3

Certain = 21.6

4

Uncertain = 21.62 ±0.01

5

(21.62 ±0.01)

Measurement = Certain digits + 1 uncertain digit

Min = 21.61
Answer = 21.62 (4 sf)
21.6
(certain)

2
(uncertain)

Significant figures
1

22


22

Max = 21.63
2

Certain
21.6

= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01

3

Certain = 21.6

4

Uncertain = 21.62 ±0.01

5

(21.62 ±0.01)


Min = 21.61
Answer = 21.62 (4 sf)
21.6
(certain)

1

Smallest division = 1

2

= 1/10 of 1
= 1/10 x 1 = ±0.1

3

Certain = 36

4

Uncertain = 36.5 ±0.1

5


2
(uncertain)

Certain
36

Max = 36.6

(36.5 ±0.1)
Min = 36.4
Answer = 36.5 (3 sf)
36.
5
(certain) (uncertain)

Significant figures
1


Max = 47
2

Certain
40

= 1/10 of 10
= 1/10 x 10 = ±1

3

Certain = 40

4

Uncertain = 46 ±1

5

(46 ±1)


Min = 45
Answer = 46 (2 sf)
4
(certain)

6
(uncertain)

Significant figures
1


Max = 47
2

Certain
40

= 1/10 of 10
= 1/10 x 10 = ±1

3

Certain = 40

4

Uncertain = 46 ±1

5

(46 ±1)


Min = 45
Answer = 46 (2 sf)
4
(certain)

1

Certain
3.4


2

= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01

3

Certain = 3.4

4

Uncertain = 3.41±0.01

5


6
(uncertain)

Max = 3.42

(3.41 ±0.01)
Min = 3.40
Answer = 3.41 (3sf)
3.4
(certain)

1
(uncertain)

Significant figures
1


Max = 0.48

0.1
2

0.2
0.3
0.4
0.5

Certain
0.45

= 1/10 of 0.05
= 1/10 x 0.05 = ±0.005 (±0.01)

3

Certain = 0.45

4

Uncertain = 0.47 ± 0.01

5

(0.47 ±0.01)


Min = 0.46
0.4
(certain)

7
(uncertain)

Significant figures
1


Max = 0.48

0.1
2

0.2
0.3

Certain
0.45

= 1/10 of 0.05
= 1/10 x 0.05 = ±0.005 (±0.01)

3

Certain = 0.45

4

Uncertain = 0.47 ± 0.01

5

0.4

(0.47 ±0.01)


Min = 0.46

0.5

0.4
(certain)

7
(uncertain)

Measurement
1


2

= 1/10 of 0.1
= 1/10 x 0.1 = ±0.01

3

Certain = 5.7

4

Uncertain = 5.72 ± 0.01

(5.72 ±0.01)
Answer = 5.72 (3sf)
5.7
(certain)

2
(uncertain)

1


2

= 1/10 of 1
= 1/10 x 1 = ±0.1

3

Certain = 3

4

Uncertain = 3.0 ± 0.1

(3.0 ±0.1)
Answer =3.0 (2 sf)
3
0
(certain) (uncertain)

Scientific notation
How many significant figures

Written as
a = 1 to 9

Scientific - notation = a ´10b

b = integer

Number too big/small
3 sf

6,720,000,000

Size sand

= 6.72 ´109
4 sf

0.0000000001254

=1.254 ´10-10
3 sf

Speed of light

300000000

= 3.00 ´108

Scientific notation

Written as
a = 1 to 9


b = integer

3 sf


6,720,000,000

Size sand

= 6.72 ´109
4 sf

0.0000000001254

=1.254 ´10-10
3 sf

Speed of light

300000000


4.66 x 10 6
4.660 x 10 6

4 sf

4.6600 x 10 6

4660000

3 sf

5 sf

Click here practice scientific notation


= 3.00 ´108

Scientific notation

Written as
a = 1 to 9


b = integer

3 sf


6,720,000,000

Size sand

= 6.72 ´109
4 sf

0.0000000001254

=1.254 ´10-10
3 sf

Speed of light

= 3.00 ´108

300000000

Scientific notation

80
3 ways to write 80


4.66 x

4660000

10 6

3 sf

4.660 x 10 6

5 sf

80 – 8 x 101 – (1sf)
Digit 8 uncertain
It can be 70 to 90

80.
80. – 8.0 x 101 – (2sf)
Digit 8 is certain
It can be 79 to 81

80.0
80.0 – 8.00 x 101 – (3sf)
Digit 80 is certain
It can be 79.9 or 80.1

4 sf

4.6600 x 10 6

80

90 or 9 x 101
80 or 8 x 101
70 or 7 x 101

81 or 8.1 x 101
80 or 8.0 x 101
79 or 7.9 x 101

80.1 or 8.01 x 101
80.0 or 8.00 x 101
79.9 or 7.99 x 101
More precise



✔

Rules for sig figures addition /subtraction:
• Last digit retained is set by the first doubtful digit.
• Number decimal places be the same as least number of decimal places in any numbers being added/subtracted

23.112233
1.3324
+ 0.25
24.694633

uncertain
least number
decimal places

round down

4.7832
1.234
+ 2.02
8.0372

uncertain
least number
decimal places

round down

1247
134.5
450
+ 78
1909.5

uncertain

least number
decimal places

1.0236
- 0.97268
0.05092

4.2
2.32
+ 0.6157
7.1357

8.04
least number
decimal places
uncertain

round down

round up

0.03

uncertain
least number
decimal places

68.7
- 68.42
0.28

0.0509
least number
decimal places
uncertain

7.987
- 0.54
7.447

uncertain
least number
decimal places

round up

round down

round up

0.3

16.96

7.1
1.367
- 1.34
0.027

1910

12.587
4.25
+ 0.12
16.957

uncertain

round down

round up

24.69

least number
decimal places

uncertain
least number
decimal places

2.300 x 103
+ 4.59 x 103
6.890 x 103

least number
decimal places

7.45
Convert to same exponent

47.68 x 104
+ 23.2 x 103

476.8 x 103
+ 23.2 x 103
500.0 x 103

round up

6.89 x 103

500.0 x 103
5.000 x 105

least number
decimal places

Rules for sig figures addition /subtraction:
• Last digit retained is set by the first doubtful digit.
• Number decimal places be the same as least number of decimal places in any numbers being added/subtracted

23.112233
1.3324
+ 0.25
24.694633

uncertain
least number
decimal places

round down

4.7832
1.234
+ 2.02
8.0372

uncertain
least number
decimal places

round down

1247
134.5
450
+ 78
1909.5

uncertain

least number
decimal places

1.0236
- 0.97268
0.05092

8.04

4.2
2.32
+ 0.6157
7.1357

least number
decimal places
uncertain

round down

0.03
3 x 10-2 (1sf)

1.910 x 103 (4sf)
uncertain
least number
decimal places

5.09 x 10-2 (3sf)

least number
decimal places

68.7
- 68.42
0.28

uncertain

7.987
- 0.54
7.447

uncertain
least number
decimal places

round up

round down

0.3

16.96

7.1 x 100 (2sf)

round up

12.587
4.25
+ 0.12
16.957

0.0509

round up

7.1
1.367
- 1.34
0.027

1910

8.04 x 100 (3sf)

2.469 x 101 (4sf)

uncertain

round down

round up

24.69

least number
decimal places

1.696 x 101 (4sf)
uncertain
least number
decimal places

2.300 x 103
+ 4.59 x 103
6.890 x 103

7.45

3 x 10-1 (1sf)
least number
decimal places

7.45 x 100 (3sf)

Convert to same exponent

47.68 x 104
+ 23.2 x 103

476.8 x 103
+ 23.2 x 103
500.0 x 103

round up

6.89 x 103
6.89 x 103 (3sf)

500.0 x 103
5.000 x 105

least number
decimal places

Rules for sig figures - multiplication/division
• Answer contains no more significant figures than the least accurately known number.

12.34
3.22
x 1.8
71.52264

least sf (2sf)

round up

23.123123
x
1.3344
30.855495

least sf (5sf)

21.45
x 0.023
0.49335

round down

round down

30.855

72
16.235
0.217
x
5
17.614975

least sf (1sf)

round up

4.52
÷ 6.3578
7.1093775

least sf (3sf)

923
÷ 20312
0.045441

least sf (3sf)

1300
x 57240
round down
74412000

4.6

0.00435
x
4.6
0.02001

least sf (2sf)

round down

7.11

0.020

least sf (2sf)
Scientific notation

least sf (2sf)

round up

0.49

round up

20

2.8723
x
I.6
4.59568

least sf (2sf)

6305
÷ 0.010
630500

least sf (2sf)

round down

630000

I.3*103
x 5.724*104
7.4412 x 107

round down

0.0454

74000000

7.4 x 107
Click here for practice notes on sig figures

Rules for sig figures - multiplication/division
• Answer contains no more significant figures than the least accurately known number.

12.34
3.22
x 1.8
71.52264

23.123123
x
1.3344
30.855495

least sf (2sf)

round up

21.45
x 0.023
0.49335

least sf (5sf)

round down

round down

30.855

72
7.2 x 101 (2sf)

3.0855 x 101 (5sf)

16.235
0.217
x
5
17.614975

least sf (1sf)

round up

4.52
÷ 6.3578
7.1093775

0.00435
x
4.6
0.02001

923
÷ 20312
0.045441

1300
x 57240
round down
74412000

least sf (2sf)

0.020

7.11 x 100 (3sf)
least sf (3sf)

4.6 x 100 (2sf)

round down

7.11

2 x 101 (1sf)

4.6

4.9 x 10-1 (2sf)

least sf (3sf)

2.0 x 10-2 (2sf)
least sf (2sf)
Scientific notation

least sf (2sf)

round up

0.49

round up

20

2.8723
x
I.6
4.59568

least sf (2sf)

6305
÷ 0.010
630500

least sf (2sf)

round down

630000
6.3 x 105 (2sf)

I.3*103
x 5.724*104
7.4412 x 107

round down

0.0454

74000000

7.4 x 107

4.54 x 10-2 (3sf)
Click here for practice notes on sig figures

Rules for sig figures – Multiplication/Division/Addition/Subtraction
Answer contains no more significant figures than the least accurately known number.
Avoid rounding off error.

•
•

0.0000673 x 291 ÷ 0.125 = ?
least sf (3sf)

0.0000673
x
291
0.0195843

0.0000673
x
291
0.0195843

Intermediate step
– leave more sf
DO NOT ROUND UP/DOWN
intermediate steps
Leave extra sf to avoid
rounding off error

0.0195843
÷
0.125
0.15632

ROUND UP/DOWN
intermediate step

0.0196
÷ 0.125
0.1568
round up

round down

0.156
1.56 x

10-1

✔
(3sf)

Rounding off error

0.157

✗

10-1

(3sf)

1.57 x

Rules for sig figures – Multiplication/Division/Addition/Subtraction
Answer contains no more significant figures than the least accurately known number.
Avoid rounding off error.

•
•

0.0000673 x 291 ÷ 0.125 = ?
least sf (3sf)

0.0000673
x
291
0.0195843

0.0000673
x
291
0.0195843

Intermediate step
– leave more sf
intermediate steps
rounding off error

0.0195843
÷
0.125
0.15632

ROUND UP/DOWN
intermediate step

0.0196
÷ 0.125
0.1568
round up

round down

1.56 x

10-1

Rounding off error

✔

0.156

(3sf)

0.157

✗

10-1

(3sf)

1.57 x

(21.5 + 21.53 + 22.548 ) x 8.45 = ?
least number
decimal places

21.5
21.53
+ 22.548
65.578

Intermediate step
– leave more sf

65.578
x 8.45
554.1341
round down

554

least number
decimal places

✔

5.54 x 102 (3sf)

rounding off error
least sf (3sf)

ROUND UP/DOWN
intermediate step
least sf (3sf)

21.5
21.53
22.548
65.578
65.5
x 8.45
553.47
round down

553 ✗

5.53 x 102 (3sf)

Significant figures in measurement
Recording measurement
using significant figures

Volume, V = 4/3πr3

Radius, r = 2.15 cm


Circumference, C = 2πr

Radius, r = 3.0 cm


Displacement, s = ½ gt2

Time, t = 2.25 s

G = (20 )
H = (16 )
Z = (106)

Speed, s = (G + H)
Z


Radius, r = 2.15 cm

Volume, V = 4/3πr3

V = 4/3 x π x (2.15)3
= 4/3 x 3.14 x 2.15 x 2.15 x 2.15
= 41.60


C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495


Time, t = 2.25 s

G = (20 )
H = (16 )
Z = (106)

least sf (2sf)

g and ½ – constant
sf is not taken
(not a measurement)


s = 1/2 x 9.8 x (2.25)2
= 24.80625

least sf (3sf)

Speed, s = (G + H)
Z
20 + 16 = 36
÷ 106
0.339

least sf (3sf)

2 and π – constant
sf is not taken
(not a measurement)


Radius, r = 3.0 cm

4/3 – constant
π – constant
sf is not taken
(not a measurement)

least sf (2sf)


Radius, r = 2.15 cm

Volume, V = 4/3πr3

V = 4/3 x π x (2.15)3
= 4/3 x 3.14 x 2.15 x 2.15 x 2.15
= 41.60

4/3 – constant
π – constant
sf is not taken
(not a measurement)
least sf (3sf)

round down

41.6

Radius, r = 3.0 cm

2 and π – constant
sf is not taken
(not a measurement)


C = 2 x π x (3.0)
= 2 x 3.14 x 3.0
= 18.8495

least sf (2sf)

round up

19

Time, t = 2.25 s

g and ½ – constant
sf is not taken
(not a measurement)


s = 1/2 x 9.8 x (2.25)2
= 24.80625

least sf (3sf)

round down

24.8
G = (20 )
H = (16 )
Z = (106)

Speed, s = (G + H)
Z
20 + 16 = 36
÷ 106
0.339

least sf (2sf)

round down

0.34


Period, T = 2π √L
√g

Length, I = 1.25 m


Area, A = I x h

Length, I = 4.52 cm
Height, h = 2.0 cm


Conc, c
= (2.00) M
Volume, v = (2.0 )dm3

Moles, n = Conc x Vol


Length, I = 1.25 m


Length, I = 4.52 cm
Height, h = 2.0 cm


Conc, c
= (2.00) M

√g

least sf (3sf)

T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399

Area, A = I x h

4.52
x 2.0
9.04

least sf (2sf)


2.00
x 2.0
4.00

least sf (2sf)

2, π and g – constant
sf is not taken
(not a measurement)


Length, I = 1.25 m

√g

least sf (3sf)

T = 2 x π x √(1.25/9.8)
= 2 x 3.14 x 0.35714
= 2.24399
round down

2.24

Length, I = 4.52 cm
Height, h = 2.0 cm

Area, A = I x h

4.52
x 2.0
9.04

least sf (2sf)

round down

9.0

Conc, c
= (2.00) M


2.00
x 2.0
4.00

least sf (2sf)

round down

4.0

2, π and g – constant
sf is not taken
(not a measurement)


Density = Mass
Volume

Mass, m = 482.63g
Volume, v = 258 cm3


Enthalpy, H = m x c x ΔT

Mass water, m = 2.00 g
ΔTemp, ΔT
= 2.0 C


Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s

t ´ I2
Energy = 1/2
v


Density = Mass
Volume

Mass, m = 482.63g
Volume, v = 258 cm3


482.63
÷
258
1.870658


ΔTemp, ΔT
= 2.0 C

x


Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s

least sf (3sf)

2.00
4.18
2.0
16.72

t ´ I2
Energy = 1/2
v
4.52
3.0
x 3.0
40.68
÷ 1.414
28.769

c (4.18) - constant
sf is not taken
(not a measurement)

least sf (2sf)

least sf (2sf)


Density = Mass
Volume

Mass, m = 482.63g
Volume, v = 258 cm3

482.63
÷
258
1.870658

least sf (3sf)

round down

1.87


ΔTemp, ΔT
= 2.0 C

x

2.00
4.18
2.0
16.72

c (4.18) - constant
sf is not taken
(not a measurement)

least sf (2sf)

round up


Volt, v = 2.0 V
Current, I = 3.0A
Time, t = 4.52s

17
t ´ I2
Energy = 1/2
v
4.52
3.0
x 3.0
40.68
÷ 1.414
28.769
round up

29

least sf (2sf)

Acknowledgements
Thanks to source of pictures and video used in this presentation
http://crescentok.com/staff/jaskew/isr/tigerchem/econfig/electron4.htm
http://pureinfotech.com/wp-content/uploads/2012/09/periodicTable_20120926101018.png
http://www.wikihow.com/Find-the-Circumference-and-Area-of-a-Circle

Thanks to Creative Commons for excellent contribution on licenses
http://creativecommons.org/licenses/

Prepared by Lawrence Kok

Check out more video tutorials from my site and hope you enjoy this tutorial
http://lawrencekok.blogspot.com

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