The Significance of Numbers:
TYPES OF NUMBERS:
Exact Numbers: Only two types of numbers may be known exactly.
Counted numbers (a whole number of objects): 14 apples and 4 eggs are exact, 2.5 pies is not exact.
Defined numbers (relationships between two measures that are true by definition): There are exactly 12 inches in 1 foot.
There are exactly 10 decimeters in 1 meter.
Measured Numbers: Numbers which are determined using a measuring device (scale, ruler, etc.). These numbers are never
known exactly. The accuracy of the measuring device limits the number of digits in a measurement. (a bathroom scale nearest pound vs an analytical balance - nearest millionth of a pound)
Because these numbers are not exact, we must recognize which digits are measured (known) and which are not.
Examples: The patient was given 2 aspirin tablets each containing 325mg of aspirin. Aspirin are counted, so the 2 is an exact
number. Milligrams are measured on a balance, so the 325 is not exact and has only 3 significant figures (see below).
Generally whole number relationships are exact but decimal relationships are measured.
exact
exact
exact
measured
3 feet = 1 yard
1 quart = 0.947 liters
SIGNIFICANT FIGURES:
Exact Numbers: Have an infinite (unlimited) number of significant digits.
Measured Numbers: Unless instructed otherwise, you should estimate one digit beyond the smallest increment (marking) on
the measuring scale. For example, on our meter sticks the smallest increment is a millimeter (tenths of a centimeter).
Therefore you should estimate one more digit (read between the lines; hundredths of a centimeter).
Reported Values: When you read a number recorded by someone else, it is the significance of zeros which is in question.
Non-zero digits are significant.
Leading Zeros are never significant. These are zeros to the left of the non-zero digits. (0.006090). Captive Zeros are always
significant. These are found between nonzero digits. (0.006004090). Trailing Zeros are significant if the decimal is
shown These are zeros to the right of other significant digits. (sinnificant 0.006090; significant 6090. ; not significant
6090)
Assigning:
1. Find the first and last nonzero digits in the number. These and all the digits in between are significant.
2. Trailing zeros are significant if decimal is shown.
Calculated Values: Don’t believe your calculator when it comes to significant figures. It sometimes displays zeros which are
not significant or doesn’t display zeros that are significant. You must determine how many digits are significant based
on the numbers used and the type of calculation.
Assigning:
Multiplication/Division: The number with the fewest significant digits determines the number of significant digits in the
answer.
Examples: 0.25 x 8.00 = (calculator 2) 2.0
8.39 x 160.950 = (calculator
1350.3705) 1.35 x 103
Addition/Subtraction: The number with the fewest decimal places (..., hundreds, tens, ones, tenths, hundredths, ...)
determines the number of decimal places in the answer.
Examples: 160.95 - 8.3 = (calculator 152.65) 152.7
0.83056 + 0.0037 = (calculator 0.83426) 0.8343
Rounding: Rounding is done after the calculation if the result contains digits that are not significant. If the first insignificant
digit is
5-9 round up (add one to the last significant digit and delete the insignificant digits). If the first insignificant digit is
0-5 round down (delete the insignificant digits)
Rounding must never cause the decimal place to move.
SCIENTIFIC NOTATION:
Scientific notation is an alternative to normal decimal notation as a way of writing numbers. Usually the numbers have exactly
the same meaning when expressed either way (823 is the same as 8.23X102.) Sometimes, however, it is necessary to
express the answer to a calculation in scientific notation to clarify the significant figures. Example: In the calculation
50. x 10. = 500, the answer appears to have three significant digits (or one depending on whether the decimal is shown)
though it should have two. This is clarified by writing the number as 5.0 x 102 (worked out below.)
Assigning:
1. Write the significant digits. (example: 50 from calculation above) Put the decimal after the first digit. (example: 5.0)
2. Write times ten ( X10). (example: 5.0 X 10)
3. Count the number of places the decimal has been moved. (example: the decimal was moved 2 places - from 500. to
5.00) Write this number as the exponent of 10 (10exponent)
4. If the value of the original number is greater than 10 the exponent is positive. If the value of the original number is
less than 1 the exponent is negative. (example: 5.0 X 102)
UNITS
Remember that numbers have little meaning without a unit (dimension) to identify the numbers type.
Examples: “A person weighs 13 more after the Christmas holiday.” By itself the number means nothing. It makes a big
difference whether this is 13 grams, or 13 ounces, or 13 pounds. “There are five remaining.” Is this 5 gallons of gas, 5
apples, or 5 people?
Conversions
A conversions can be used to express the equivalent value in a different unit (ex. 18 in can be expressed as 1.5 ft). To do a
conversion requires a relationship between the size of the units. (ex. 12 in = 1 ft). The two numbers in the relationship can be
expressed as a fraction used as a conversion factor.
 1ft 
18in 
 = 1.5ft
 12in 
 12in 
1.5ft
 = 18in
 1ft 
Many metric conversion can be done by moving the decimal the correct number of places to the right or to the left.
G
M
k
←
bigger units
UNIT
d
c
meter,liter,gram
smaller numbers
m
µ
smaller units
→
n
bigger numbers
Conversion Practice
Use the conversion factors listed below and those in your book to solve the following problems. Some problems may require
more than one step while others may be solved by the shortcut method of moving the decimal point.
(1g = 15grains; 1in3 = 16.39mL; 1cal = 4.184J; 1mole of NaOH = 40g of NaOH)
4 megabytes =
bytes
2361cm =
km
3.7kg =
dg =
mg
65mL =
µm
287cc =
mL =
L
13kJ =
J
3.6L
(
)=
____________
5.0grains
16.2gal
4.7ins
(
(
(
in3
)=
____________
)=
____________
)=
____________
83g NaOH
(
g=
L=
cm =
)=
____________
(
4.6qts (
53lbs (
31cal (
55km
mg
dL
m
moles NaOH
)=
)=
)=
)=
____________
feet
____________
L=
____________
kg =
____________
J=
mL
g
kJ