Download SIGNIFICANT DIGITS EXAMPLE SHEET #2.1.0

SIGNIFICANT DIGITS EXAMPLE SHEET #2.1.0
I. RULES FOR DETERMINING SIGNIFICANT DIGITS
There are several rules on determining how many significant digits are in a number. You MUST
memorize each of them!
1.
Non-zero digits (1, 2, 3, 4, 5, 6, 7, 8 and 9) in numbers are ALWAYS SIGNIFICANT.
For example: 47,381 has 5 significant digits and
59 has 2 significant digits
2.
“Inside zeros” or zero(s) between two significant digits are ALWAYS SIGNIFICANT. The
zeros in the numbers: 3,809; 54,000,027 and 106,003 are ALWAYS SIGNIFICANT.
For example: 3,809 has 4 significant digits;
54,000,027 has 8 significant digits and
106,003 has 6 significant digits
3.
“Beginning” or “leading” zero(s) in numbers are NEVER SIGNIFICANT --- REGARDLESS
OF WHETHER THERE IS A DECIMAL POINT IN THE NUMBER OR NOT!
For example:
00493 has 3 significant digits; and
00067.812 has 5 significant digits
4.
“Ending” or “trailing” zero(s) in NUMBERS HAVING A DECIMAL POINT ARE ALWAYS
SIGNIFICANT.
For example:
81.7000 has 6 significant digits; and
45,000.00 has 7 significant digits
5.
“Ending” or “trailing” zero(s) in NUMBERS NOT HAVING A DECIMAL POINT ARE
NEVER SIGNIFICANT!
For example:
680,000 has 2 significant digits; and
2,150 has 3 significant digits
Focus on these rules and learn them well. They will be used extensively throughout the remainder of
this course. Please remember that, in science, all numbers are based upon measurements (except
for a very few that are defined). Since all measurements are uncertain, we must only use those
numbers that are meaningful. A common metric ruler cannot measure something to be 22.4072643
cm long. Not all of the digits have meaning (significance) and, therefore, should not be written
down. In science, only the numbers that have significance (derived from measurement) are written.
(Continued on the back of this page)
- 2EXACT NUMBERS
Exact numbers, such as the number of people in a room, have an infinite number of significant digits.
Exact numbers are counting up how many of something are present, they are not measurements
made with instruments. Another example of this are defined numbers, such as 1 foot = 12 inches
or 1 meter = 1000 millimeter. There are exactly 12 inches in 1 foot and exactly 1000 mm in 1 m.
Therefore, if a number is exact, it DOES NOT affect the accuracy of a calculation nor the precision
of the expression. Some more examples of exact numbers are: there are 100 years in a century and
there are 12 donuts in 1 dozen donuts.
You may recall from your math classes that exact numbers can be written with the “repeating digit
bar” written over the repeating digit(s) to the right of the decimal point:
1. 0 foot = 12. 0 inches and 1. 0 m
= 1000. 0 mm
II. ADDITION and SUBTRACTION with SIGNIFICANT DIGITS
In mathematical operations involving significant digits, the answer is reported in such a way that it
reflects the reliability of the least precise operation. Let's state that another way: a chain is no
stronger than its weakest link. An answer is no more precise that the least precise number used to get
the answer. Let's do it one more time: imagine a team race where you and your team must finish
together. Who dictates the speed of the team? Of course, the slowest member of the team. Your
answer cannot be MORE precise than the least precise measurement.
For addition and subtraction of significant digits, look at the decimal portion (for example, look to the
right of the decimal point) of the numbers ONLY. Here is what to do:
1)
Count the number of significant digits in the decimal portion of each number in the problem.
2)
Add or subtract the numbers in the usual manner.
3)
Round the answer to the LEAST number of decimal places (NOT DIGITS) in the decimal
portion of any number in the problem.
Examples:
12.52
349.0
+ 8.24
369.76 (Calculator Ans.)
74.626
-28.34
46.286 (Calculator Ans.)
46.29 (Correct Ans.)
369.8 (Correct Ans.)
127.30
+340
467.30 (Calculator Ans.)
470 (Correct Ans.)
15 600
-2 731.49
12 868.51 (Calculator Ans.)
12 900 (Correct Ans.)
(Continued on the next page)
- 3WARNING: the rules for add/subtract are different from multiply/divide. A very common student
error is to swap the two sets of rules. Another common error is to use just one rule for both types of
operations.
III. MULTIPLICATION and DIVISION with SIGNIFICANT DIGITS
In mathematical operations involving significant digits, the answer is reported in such a way that it
reflects the reliability of the least precise operation. Let's state that another way: a chain is no
stronger than its weakest link. An answer is no more precise that the least precise number used to
get the answer. Let's do it one more time: imagine a team race where you and your team must
finish together. Who dictates the speed of the team? Of course, the slowest member of the team.
Your answer cannot be MORE precise than the least precise measurement.
The following rule applies for MULTIPLICATION and DIVISION:
The LEAST number of significant digits in any number of the problem determines the number of
significant digits in the answer.
This means you MUST know how to recognize significant
digits in order to use this rule.
Example #1:
2.5 H 3.42 = 8.55 (Calculator Answer) =>
8.6 (Significant Digit Answer)
Why is the answer to this problem 8.6 (which was rounded
from the calculator answer of 8.55)? The answer is 8.6 because 2.5 has two significant digits while
3.42 has three. Two significant digits is less precise than three, so the answer has two significant
digits.
Example #2: How many significant digits will the answer to 3.10 H 4.520 have? You may have said 2
significant digits. This is too few. A common error is for the student to look at a number like 3.10
and think it has two significant digits. The zero in the hundredth's place is not recognized as
significant when, in fact, it is. The number 3.10 has three significant digits --- 3 significant digits is
the correct answer. (REMEMBER: All trailing zeros in a number having a decimal point are
ALWAYS SIGNIFICANT).
3.10 H 4.520 = 14.012 (Calculator Answer) =>
14.0 (Significant Digit Answer)
Another common error is for the student to think
that 14 and 14.0 are the same thing. THEY ARE NOT! The number 14.0 is ten times more precise
than 14. The two numbers have the same value, but they convey different meanings about how
accurate they are. Four significant digits is also an incorrect answer given by some students. It is
too many significant digits. One possible reason for this answer lies in the number 4.520. This
number has four significant digits while 3.10 has three. Somehow, the student (YOU!) maybe got
the idea that it is the GREATEST number of significant digits in the problem that dictates the
answer. It is the LEAST. Sometimes students will answer this with 5 significant digits. Most likely
you responded with this answer because it says 14.012 on your calculator. This answer would have
been correct in your math class because mathematics does not have the significant figure rules.
(Continued on the next page)
- 4-
Example #3:
2.25 H 6.085 ) 5.0 = 2.73825 (Calc. Ans.) =>
2.7 (Signif. Digit Ans.)
How many significant digits are in the answer? There are
2 significant digits in the answer.
Which number decides this? The number 5.0.
Why? Because the number 5.0 has the least number of significant digits in the problem. It
is, therefore, the least precise measurement.
Example #4:
(4.52 H 104) ) (3.980 H 106 ) = (4.52 ) 3.980) H (104 ) 106 )
= 1.135678392 H 104 - 6 = 1.135678392 H 10 - 2 (Calc. Ans.)
= 1.14 H 10 -2 (Significant Digit Answer)
How many significant digits are in the answer? There are 3 significant digits in the answer.
Which number decides this? The number 4.52 H 104.
Why? It has the least number of significant digits in the problem. It is, therefore, the least
precise measurement. Notice it is the 4.52 portion that plays the role of determining
significant digits; the exponential portions (the 104 and 106) of the numbers do not
determine the number of significant digits.