1.5 Scientific and Engineering Notation
As the names indicate, scientific notations are typically used by science and engineering notation
is utilized primarily by engineering and engineering technology. The usefulness of engineering
notation will become apparent in the next section in determining unit prefixes.
Scientific Notation
Scientific notation expresses a value as a power of 10 such that the numeric part is greater than or
equal to I but less than 10. That is, the numeric part always has one and only one digit left of the
decimal point. The form of scientific notation is
N x 10n
where
•
N is a number between 1 and 10, that is 1 ≤ N < 10
•
N must have one and only one digit left of the decimal point.
•
The number of significant figures of N. corresponds to its original value's number of
significant figures.
•
The exponent n is an appropriate number to force the number N to be a value between 1
and 10.
•
The exponent n can be a positive or a negative number.
Example 1-14
Express the number 123 in scientific notation.
Solution
You need to move the decimal point 2 places to the left to get only one digit left of the
decimal point, that is, from 123 to 1.23. To compensate, add 2 to the 10's exponent, that
is, from 100 to 102.
1.23 x 102
Scientific notation
Practice
Express 12,300,000 in scientific notation.
Answer: 1.23 x 107
Example 1-15
Express the number 0.00012 in scientific notation.
Solution
You need to move the decimal point 4 places to the right to get only one digit left of the
decimal point, that is, from 0.00012 to 1.2. To compensate, subtract 4 from the 10's
exponent, that is, from 100 to 10-4.
1.2 x 10-4
Scientific notation
Practice
Express 0.00000012 in scientific notation.
Answer:
1.2 x 10-7
Engineering Notation
Engineering notation expresses a number as a power of 10 such that the numeric value is greater
than or equal to 1 but less than 1000, and the 10’s exponent is a multiple of three. The form of
engineering notation is
N x 10n, where
N is a number between 1 and 1000, that is 1 ≤ N < 1000
The number of significant figures of N corresponds to its original value's number of significant
figures.
The exponent n must be a multiple of 3 and is an appropriate value to force the number N to be a
value between 1 and 1000.
The exponent n can be a positive or negative number.
Example 1-16
Express the number 12,300 in engineering notation.
Solution
You need to move the decimal point 3 (a multiple of 3) places to the left to get a number
between 1 and 1000, that is, from 12,300 to 12.300.
To compensate, add 3 to the 110’s exponent, that is, from 100 to 103.
12.3 x 103
Engineering notation
Practice
Express 12,300,000 in engineering notation.
Answer:
12.3 x 106
Example 1-17
Express the number 0.00012 in engineering notation.
Solution
You need to move the decimal point 6 (a multiple of 3) places to the right to get a number
between 1 and 1000, that is, from 0.000 12 to 120. Note: you had to pad with a trailing 0
to create the appropriate number.
To compensate, subtract 6 from the 10's exponent, that is, from 100 to 104.
120 x 104
Engineering notation
Practice
Express 0.00000012 in engineering notation.
Answer: 120 x 10-9
1.6 Unit Prefixes
Unit prefixes are a simple way to communicate very large and very small unit values. As humans
we like to communicate in words. Expressing simple values like 10 seconds is very comfortable
and meaningful for us; but, how about 0.000001 seconds? We realize this is a very small amount
of time. But how small is it? It is difficult to visualize.
Expressing extreme numbers in scientific notation or engineering notation is more helpful. For
example, the above value would be expressed as 1 x 10-6 seconds. It is also very helpful to
establish names that represent large and small unit values. These names help us more effectively
visualize and communicate values. The name "micro" stands for the value 10-6. Thus 1 x 10-6
second can be written as 1.0 microsecond. The term 10' is replaced directly with the word
"micro".
The unit prefix naming convention adopted for the SI system of units is based upon engineering
notation. Look at Table 1-5. Use uppercase symbols for mega and above; use lowercase letters for
kilo and below. The symbol g for the prefix micro is a Greek letter. Be careful to clearly print the
prefix symbols g (long leading line) for micro (10-6), lowercase m for milli (10-3), and uppercase
M for mega (106). Otherwise you will convey incredibly inaccurate information
Table 1-5 SI unit prefixes
Prefix Name Prefix
Value
Symbol
tera
T
10 12
giga
G
109
mega
M
106
kilo
K
103
100
unit
milli
M
10-3
micro
U
10-6
nano
N
10-9
Pico
femto
P
f
10-12
10-15
Example 1-18
Express 1 kilogram in the basic unit of grams.
Solution
• 1 kilogram = 1 kg
• Directly replace "k” with its equivalent value of 103.
• 1 kg =1 (103) g = 1X103 g = 1000 g
• Note: "k" is written in the lowercase. Do not use uppercase.
Practice
Express 2 megagrams in the basic unit of grams.
Answer: 2 x 106g
Example 1-19
Express 2 milliseconds in the basic unit of seconds.
Solution
2 milliseconds = 2 ms
Directly replace "m" with its equivalent value of 10-3.
2 ms = 2 (10-3) s = 2x 10-3 s = 0.002 s
Practice
Express 2 nanoseconds in the basic unit of seconds.
Answer: 2 x 10-9s
Example 1-20
Express 120,000,000 seconds with the proper prefix.
Solution
Express in engineering notation: 120 x 106 S
Replace 106 with the mega "M" prefix: 120 Ms
Example 1-21
Express 0.00034 seconds with proper prefix.
Solution
• Express in engineering notation:
• Replace 10-6 with the micro "u" prefix:
340 x 104 s
340 us
1.7 Unit Prefix Conversion
Sometimes it is necessary to convert from one unit prefix to another. Remember a prefix is just a
name for an engineering notation power of 10 whose exponent is a multiple of 3.
Example 1-22
Convert 0.00003 kilograms to milligrams.
Solution
•
•
•
•
•
Move decimal point to the right in groups of 3 digits and convert to the next
prefix until the number is between 1 and 1000.
Start with original value.
0.00003 kg
Move decimal point 3 places right.
0.03 g
Move decimal point 3 places right.
30 mg
Note as the number gets larger, the prefix gets smaller.
Practice
Convert 0.00000003 kilograms into micrograms.
Answer:
30 ug
Example 1-23
Convert 4,000,000 picoseconds to milliseconds.
Solution
• Move decimal point to the left in groups of 3 digits and convert to the next prefix until
the number is between 1 and 1000.
• Start with original value.
4,000,000 ps
• Move decimal point 3 places left:
4,000 ns
• Move decimal point 3 places left:
4 us
• Move decimal point 3 places left:
0.004 ms
• Note as number gets smaller, the prefix gets larger.
Practice
Convert 50,000 milligrams into kilograms.
Answer: 0.05 kg
1.8 Unit Consistency and Conversions
You must take units very seriously as noted earlier in this chapter. It is sometimes necessary to
convert to other units within the same system of units or to convert a value to a different system
of units. You must carry units throughout your calculations to ensure accurate units in the
resulting answer. Calculations without units are not meaningful.
To convert to a new unit, you must multiply by an equivalent value of 1 or a unity multiplier, for
example, 60 seconds per minute. The following converts 5 minutes (min) to seconds (s):
 60s 
 = 300s
 min 
time = 5 min x 
Notice the unity multiplier of 60 seconds per minute does not change the value of the answer. It
only changes the value's units from "minutes" to "seconds". In the unity term, "minutes" is
strategically placed in the denominator to cancel the original "Minutes" in the numerator. And the
equivalent number of "seconds" per minute is placed in the numerator. The "minutes" cancel, and
the answer is left in units of "seconds".
Example 1-24
You are offered a major contract at 2 cents per millisecond for 2 hours. The actual job
cost is $100,000. Will you make money? Prove your answer. Use unit conversion,
multiplying by a conversion factor that is an identity. Minutes are abbreviated "min" and
hour is abbreviated "hr".
Solution
Note: units are set up in the conversion factor so as to cancel out the unit you want to
eliminate. For example, if you want to cancel out a unit in the numerator then you must
multiply by a conversion factor with that unit in the denominator.
2cents  1$   100ms   60s   60 min 
x
 x
 x
 x
 x 2hr = $144,000
1ms  100cents   s   min   hr 
Looks like an excellent deal, a profit of $44,000 in 2 hours.
Practice
Convert 3 cents per us to dollars per minute.
Answer: 1.8 X 106 dollars per minute
Example 1-25
Convert 230,000 microseconds to seconds.
Solution
Use a unity multiplier to convert from us to s. 10-6 s
 10 −6 s 
 = 0.23s
230,000usx 
 us 
Practice
Convert 30,000,000 nanoseconds to seconds.
Answer: 0.03 s
Example 1-26
Convert 0.00005 grams to micrograms.
Solution
Use a unity multiplier to convert ftom g to gg.
 10 6 ug 
 = 50ug
0.00005gx 
 g 
Practice
Convert 0.0000007 seconds to nanoseconds.
Answer: 700 ns
Example 1-27
Convert 670,000,000 milligrams to kilograms.
Solution
Use a unity multiplier to convert from mg to g, then another unity multiplier to convert
from g to kg.
 10 −3 g   kg 
 x  3  = 670kg
670,000,000mgx 
 mg   10 g 
Practice
Convert 70,000 nanoseconds to microseconds using two unity multipliers: (from ns to s)
and (from s to gs).
Answer: 70 us
The next example demonstrates converting between systems of units, from metric to
English.
Example 1-28
The Arecibo National Astronomy and Ionosphere Center in Puerto Rico has the most sensitive
astronomical telescope in the world. The diameter of its spherical shaped reflector is 305 meters,
the largest antenna dish in the world. Convert the diameter expressed in SI units of meters to the
English units of feet.
 100cm   1inch   1 foot 
305mx 
 = 1000 feet
 x
 x
 1m   2.54cm   12inch 
Note that each fraction is a unity and its denominator is set up to eliminate (convert) a unit.