Powers of Ten, Units and Standard
Form
Mathematics Skills Guide
This is one of a series of guides designed to help you increase your confidence in
handling mathematics. This guide contains both theory and exercises which cover:1. Units names & conversion
2. Scientific or standard form
3. Degrees of accuracy
There are often different ways of doing things in mathematics and the methods
suggested in the guides may not be the ones you were taught. If you are successful
and happy with the methods you use it may not be necessary for you to change
them. If you have problems or need help in any part of the work then there are a
number of ways you can get help.
For students at the University of Hull
 Ask your lecturers.
 You can contact a math Skills Adviser from the Skills Team on the email shown
below.
 Access more maths Skills Guides and resources at the website below.
 Look at one of the many textbooks in the library.
Web: www.hull.ac.uk/skills
Email: skills@hull.ac.uk
1. UNITS
Metric units are all based on one system of prefixes. This means that just as 1
centimetre is one-hundredth of a metre so 1 centilitre is one-hundredth of a litre and
1 centigram is one-hundredth of a gram. From this it is important to be able to
multiply and divide by 10, 100, 1000 etc.
(i) Multiplying and dividing by 10, 100, 1000 etc.
Multiplying a number by 10 makes it bigger and dividing by 10 makes it smaller which is logical!! If you have £0.40 and multiply that by 10 you get £4.00, but if you
split it between 10 people each gets £0.04.
To multiply
135 by 10 you can split it up as
100 × 10 = 1000, 30 × 10 = 300 and 5 × 10 = 50
giving 135 × 10 = 1350 - every digit becomes 10 times larger
(100 1000, 30300, 550) and so has moved up one place to the left.
The same with 0.246 × 10 - every digit moves up one place to the left
giving 0.246 × 10 = 2.46
To divide, the reverse is true, all the digits move down one place to the right.
795 ÷10 = 79.5; 0.24 ÷ 10 = 0.024
You will see that
× 10
 figures move up
÷ 10
 figures move down
and the above can be shown on a chart as
1000s
100s
10s
135
135 × 10 =
0.246
0.246 × 10 =
795
795 ÷ 10 =
0.24
0.24 ÷ 10 =
1
1
3
7
3
5
9
7
units
5
0
0
2
5
9
0
0
.
.
.
.
.
.
.
.
1 s
10
1 s
100
1 s
1000
2
4
4
6
6
5
2
0
4
2
4
Note the need to put zeros in as ‘place holders’ (shown in bold italic).
100 = 10 × 10 and is written as 102 where the 2 is called the index.
Multiplication by 100 = 102 is multiplication by 10 and then by 10 again, hence the
figures move up two places. 1000 = 103 - move up 3 places etc.
Division by 100 = 102 is division by 10 and then by 10 again, hence the figures move
down two places. 1000 = 103 - move down 3 places etc.
page 1
Examples 1
1 2
0
1 3 5
1 0 0 2
0
.
.
.
.
.
3 4
0 2 3 4
5
0 0 8 9
× 100 =
× 1000 =
÷ 10 000 =
÷ 100 000 =
× 100 000 =
1 2 3 4 .
2 3 .4
0 .0 1 3 5
0 .0 1 0 0 2 5
8 9 0 .
The placeholders (lines 3,4 & 5) have been put in bold italic. In practice it is better to
write the decimal .123 as 0.123 as the 0 ‘protects’ the decimal point.
Long numbers are often split into groups of 3 digits such as
104 000 345, 0.000 040 067 and 0.000 098 76
(some older books use commas to separate the groups of 3 digits).
Exercise 1
Work out the following (without using a calculator)
1) 2.345 × 100
2) 0.456 × 1000
3) 23.076 × 1000
4) 0.003 045 × 1000
5) 234 × 100
6) 0.000 0345 × 1 000 000
7) 972.3 ÷ 100
8) 18.643 ÷ 1000 9) 0.7246 ÷ 100
10) 75 423.045 ÷ 1000 11) 0.0076 ÷ 1000 12) 0.000 0765 ÷ 100 000
(ii) Name those units!
In the metric system the prefixes are given by
Name
Factor
abbreviation
tera giga mega kilo unit centi milli micro nano pico
1012 109
106 103 1 10-2 10-3 10-6 10-9 10-12
Tm Gm Mm km m cm mm m
nm
pm
Reminder on those indices (there will be more later in this worksheet)!
103 means 10 × 10 × 10 = 1000;
109 means 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1 000 000 000;
10-1 means 1 ÷ 10 = 0.1;
10-6 means 1 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 =0.000 001.
Examples 2
1. State which of the following are equivalent:
a)
1.2 m; 12 cm; 1200 mm
Convert all measurements to mm 1.2 m = 1.2 × 103 mm = 1200 mm;
12 cm = 12 × 10 mm = 120 mm;
hence 1.2 m and 1200 mm are equivalent
b)
0.0065 Mm; 65000 cm; 6.5 km
Convert all measurements to metres
0.0065 Mm = 0.0065 × 106 m = 6500 m;
65000cm = 65000 × 10-2 m = 650 m;
6.5 km = 6.5 × 103 m = 6500 m
hence 0.0065 Mm and 6.5 km are equivalent.
page 2
c)
123 400 mg; 1234 cg; 0.000 012 34 Mg
Convert all measurements to grams
123 400 mg = 123 400 × 10-3 g = 123.4 g;
1234cg = 1234 × 10-2 g = 12.34 g;
0.000 0123 4 Mg = 0.000 012 34 × 106 g = 12.34 g
hence 1234 cg and 0.000 012 34 Mg are equivalent
2. Add 37 cm, 1600 mm and 0.034 km, giving your answer in metres.
Convert the measurements to metres: 37 cm = 0.37 m, 1600 mm = 1.6 m and
0.034 km = 34 m giving a total of 0.37 + 1.6 + 34 m = 35.97 m.
3. Find the total length of 30 rods of length 145 cm; give your answer in metres.
Total length is 145 × 30 cm = 4350 cm = 43.5 m.
Exercise 2
1. Find which of the following are equivalent:
(a) 123 mm, 0.0123 m, 0.000 123 km, 12.3 cm
(b) 456 000 g, 456 cg, 0.456 g, 456 000 000 ng
(c) 0.7643 Tm, 764 300 Mm, 764 300 000 km, 764 300 000 000 cm
2. Add the following measurements
(a) 1.45cm, 234 mm, 1.2045 m (answer in metres)
(b) 945g, 23.6 mg, 19763 ng (answer in g)
3. Giving your answer in kg, find the total weight of 240 hamsters
each weighing 34 g.
4. Find the average of the following lengths 23.78 m, 0.842 km and
0.000 087 29 Mm, giving your answer in km.
5. A plank of length 3.567 m is cut into 34 pieces. If the thickness of the cut is 3 mm
what is the exact length of each piece in mm?
(iii) Areas and Volumes
We know that there are 10 mm in 1 cm, so how many square millimetres (mm2) are
there in a square centimetre (cm2)?
The diagrams show a
square of side 1 cm
(=10 mm). The areas
are respectively 1 cm2
and 100 mm2. Hence
1cm
1cm
Area
1 cm2
10mm
10mm
Area
100 mm2
1 cm2 = 100 mm2
Note that cm2 is read as square centimetres and NOT as centimetres squared.
page 3
Examples 3
(a) Express an area of 3.4m2 in cm2
1 m = 100 cm
2
1 m2 = 1002 cm2 = 10000 cm2
3.4 m2 = 3.4  10000 = 34000 cm2
(b) Express a volume of 24560 mm3 in cm3
10 mm = 1 cm
103 mm3 = 1000 mm3 = 13 cm3
24560 mm3 =
24560
cm3 = 24.56 cm3
1000
(c) Express a speed of 44 feet per second in km h -1 (use 1 foot as equivalent to
0.305 m)
44 feet per second = 44  0.305 m per second
= 44  0.305  60  60 m per hour
44  0.305  60  60
=
km per hour
1000
= 48.3 km h-1
Exercise 3
1. Express areas of 2.35 km2 and 0.0045 km2 in m2.
2. Express an area of 34.98 m2 in mm2 and in km2.
3. Express a volume of 0.34 m3 in cm3 and mm3.
4. Express an area of 2.74 km2 in cm2 and m2.
5. Express a volume of 345.6 cm3 in mm3 and m3.
6. Express a speed of 10cm s-1 in km h-1
7. Express a pressure of 30 pounds per square inch in kg cm -2 (kg/square cm)
(1lb is approx 454g)
2. SCIENTIFIC NOTATION OR STANDARD FORM
a) A way of representing large and small numbers
The mass of an atom is 0.000 000 000 000 000 000 000 000 001 67 kg.
The speed of light is 300 000 000 metres per second.
In Scientific Notation these are written as 1.67 × 10-27 kg and 3 × 108 ms-1. In this
form they are easier to multiply or divide, though not to add or subtract! The notation
is also used to show large and small numbers on a calculator, though it’s not a
modern notation having been in use for some 400 years!
Sometimes when large numbers are printed, a gap is left between each group of 3
digits to aid reading such as 2345678 being written as 2 345 678 (this was used
above for the mass of an atom and the speed of light.) Your calculator does not do
this!
If you multiply 2540 and 165 480 your calculator it may give you 420319200 or
4.203192 E8 depending on the capacity of the display on your calculator.
4.203 192 E8 is shorthand for 4.203 192 × 108
Check 4.203 192 × 108
= 4.203 192 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 420 319 200
page 4
In Scientific Notation ( a  10n ) there are two parts to each number:
the decimal part a where 1  a  10
and the power of 10, n , which is always an integer, positive or negative. The
value of n tells you how many places you move the digits from one form to the
other.
Examples 4
a) 12 345 = 1.2345 × 104
In 12345 the 1 is ‘worth’ 1 × 104
Or the 1, in 1.2345, moves 4 places to the left
b) 2 567.895 = 2.567895 × 103 In 2567.895 the 2 is ‘worth’ 2 × 103
Or the 2, in 2.567895, moves 3 places to the left
c) 0.567 = 5.67 × 10-1
In 0.567 the 5 is ‘worth’ 5 × 10-1
Or the 5, in 5.67, moves 1 place to the right
d) 0.000 345 6 = 3.456 × 10-4
In 0.0003456 the 3 is ‘worth’ 3 × 10-4
Or the 3, in 3.456, moves 4 places to the right
Exercise 4
A. Write as ordinary numbers
1) 3.124  104
2) 1.9306  103
4) 8.345  10 2
5) 1.0546  103
B. Write in scientific notation
7) 348.22
8) 18 589.02
10)
0.987 23
11) 0.000 34
3) 9.8256  105
6) 5.34  101
9) 14 000
12) 0.000 003 082
b) Using numbers in Scientific Notation (Standard Form)
You can deal with numbers in Scientific Notation by putting them into your calculator.
3.24 × 104 is typed in by using the EE (or EXP) button on the calculator as
3.24 EE 4. The number 3.24  105 would be typed in as 3.24EE-5.
However, it is important that you have some estimate of the size of your answer as a
check on the answer given by your calculator.
Examples 5
a) Multiply
2.34 × 105 and 1.456 × 102
2  105  1 102  2  107
Estimate
Calculator 2.34 E5 × 1.456 E2 = 3.40704 E7 which is of the same order of
magnitude and is close to the estimate.
b) Division is more tricky but still worth trying to estimate!
Divide
1.9876 × 10-5 by 7.84987 × 106
2  105
 0.25  1011  2.5  1012
Estimate
2 × 10-5 ÷ (8 × 106) =
6
8  10
Calculation 1.9876 E-5 ÷ 7.84987 E6 = 2.5320 E-12 which is of the same order of
magnitude and remarkably close!
page 5
Exercise 5
A. Multiply the following pairs of numbers together, working out an estimate first.
Write your answers both in Standard Form and as ordinary numbers.
  5102 
4) 3.2  101  3  103 
6) 7  1087  3  1088 

 2 102 
3) 6 102  1.2  104 
5) 3.5  103  3  103 
1) 1.3  104
2) 9  103
B. Estimate the answer in each of the following:
7)
9)
11)
12)
1.356 102   2.753 102 
8) 9.712  103   5.6  102 
2
6.74102   1.284 104 
10) 3.2  101 
3.075 103   3.98 103 
7.8343 104   3.863  102   2.3  103 
3. DEGREES OF ACCURACY
Most calculators give answers with up to 10 digits in the display. In most cases you
need to approximate the answer to something more sensible. For instance if you
want to share £1000 between 11 people, the calculator shows that each should get
£90.90909091! In practice they’d get £90.90 each (and argue over the 10p left!).
To approximate or round off 36.7274002 to, say, 2 decimal places (dec pl or d.p.) we
can see that it lies between 36.72 and 36.73. This means that the approximate
value to 2 decimal places will be 36.72 or 36.73.
36.73
36.729
36.728
36.727
36.7274002
36.726
36.725
36.724
36.723
36.722
36.721
36.72
Putting these onto a section of the number line we have:
As 36.7274002 is closer to 36.73 than 36.72 we write:
36.7274002 = 36.73 (to 2 d.p.).
In practice when rounding to 2 d.p.
if the digit in the 3rd d.p. is 5 or greater then increase the 2nd digit by 1,
if the digit in the 3rd d.p. is less than 5 leave the 2nd digit unchanged.
It is possible to have what appear to be ‘awkward’ numbers as shown in the last
example below but approximating to 2 decimal places means having 2 figures after
the decimal point even if they are both zero!
page 6
Examples 6a
Number
Degree of accuracy 2 d.p.
23.3486
12.45055
0.025467
1298.4549999
223.996
Answer to 2 d.p.
23.35
12.45
0.03
23.34 | 86
12.45 | 055
0.02 | 5467
1298.45 | 49999
223.99 | 6
1298.45
224.00
The rules for significant figures (sig figs or s.f.) are similar but you need to be
careful when there are zeros at the beginning of a decimal. The zeros in 0.000 0234
are not considered to be significant figures but must be included in the final result to
ensure the answer is of the same magnitude as the original number. The zeros in
200 046 and in 2.000 034 are significant as they have non-zero digits to the left.
Examples 7
Number
Degree of accuracy 3 s.f.
Answer to 3 s.f.
23.348 6
124 290
0.002 546 7
1 298.454 9
23.3 | 486
124 | 290
0.002 54 | 67
1 29 | 8.454 99
23.3
124 000
0.002 55
1 300
One way of checking whether there are enough zeros in the answer is to check the
‘order of magnitude’ of each number and its approximation.
In the number 23.3486 the value of the 2 is 2 × 10 1. Hence the order of magnitude
of 23.3486 is 101; the order of magnitude of 23.3 is the same.
The order of magnitude of 124 290 is 105; it is the same for 124 000
The order of magnitude of 0.002 546 7 is 10-3; it is the same for 0.002 55
The order of magnitude of 1 298.454 9 is 10 3; it is the same for 1300.
Exercise 6
Write each of the following to the degree of accuracy specified.
1) 9.739
(3s.f.)
2) 0.362 18 (2d.p.) 3) 147.49
(1d.p.)
4) 28.63
(2s.f.)
5) 0.525 2
(2s.f.)
6) 4.198 3
(2d.p.)
7) 1243.4
(3s.f.)
8) 0.004 25
(3d.p.)
9) 0.004 24
(2s.f.)
10) 145 560
(2s.f.)
11) 0.002 876 (2d.p.) 12) 0.999 67
(3s.f.)
13) 1300.002 (1d.p.) 14) 0.999 34
(3d.p.) 15) 10 099.09 (3s.f.)
page 7
Answers
Exercise 1
1) 234.5
4) 3.045
7) 9.723
10) 75.423 045
2) 456
5) 23 400
8) 0.018 643
11) 0.000 0076
3) 23 076
6) 34.5
9) 0.007 246
12) 0.000 000 000 765
Exercise 2
1) (a) equivalent: 123 mm, 0.000123 km and 12.3 cm
(b) equivalent: 456 000 g, 0.456 g, 456 000 000 ng
(c) equivalent: 0.7643 Tm, 764 300 Mm and 764 300 000 km
2) (a) 1.453 m
(b) 24564.763 g
3) 8.16 kg
4) 0.31769 km
5) 102 mm exactly (if your answer is not exact did you realise there are
only 33 cuts?)
Exercise 3
1) 2350000 m2, 4500 m2
3) 340000 cm2, 340000000 mm3
4) 2740000 m2, 27400000000 cm2
Exercise 4
1) 31240
5) 0.0010546
9) 1.4 × 104
2) 1930.6
6) 0.534
10) 9.8723 × 10-1
Exercise 5
1) 2.6 × 106 = 2 600 000
4) 9.6 × 10-4 = 0.000 96
2) 34980000 mm2, 0.00003498 km2
5) 345600 mm3, 0.0003456 m3
6) 0.36 km h-1
7) 2.1 kg cm-2
3) 982560
7) 3.4822 × 102
11) 3.4 × 10-4
2) 4.5 × 106 = 4 500 000
5) 1.05 × 101 = 10.5
4) 0.08345
8) 1.858902 × 104
12) 3.082 × 10-6
3) 7.2 × 102 =720
6) 2.1 × 100 = 2.1
(The following are estimates and your answers may differ slightly from these!)
7) 4 × 104 = 40 000 8) 5 × 106 = 5 000 000 9) 6 × 10-6 = 0.000 006
10) 9 × 10-2 = 0.09
11) 1.2 × 101 = 12
12) 1 × 106 = 1 000 000
Exercise 6
1) 9.74
2) 0.36
3) 147.5
4) 29
7) 1240 8) 0.004 9) 0.0042 10) 150 000
13) 1300.0
14) 0.999
15) 10100
5) 0.53
11) 0.00
6) 4.20
12) 1.00
We would appreciate your comments on this worksheet, especially if
you’ve found any errors, so that we can improve it for future use. Please
contact the Maths Skills Adviser by email at skills@hull.ac.uk
The information in this leaflet can be made available in an alternative format on
request using the email above.
page 8