APPROXIMATE
NUMBERS
Precision & Accuracy
EXACT VS
APPROXIMATE NUMBERS
Exact Number
Approximate Number
No uncertainty
Has a degree of uncertainty
Counting Process
• Computer counts number of
students majoring in science = 768
Definition
• 1 𝑚𝑖𝑛 = 60 𝑠𝑒𝑐
• 1 𝑖𝑛 = 2.54 𝑐𝑚
Any type of measurement
A number that cannot be
written as an exact decimal
• 2/3 = 0.66666666
• √3 = ~ 1.73
APPROXIMATE
NUMBERS
• When performing calculations on approximate
numbers accuracy & precision must be considered
• Affects the accuracy of the results obtained
• Ex. Measured bolt length
PRECISION VS ACCURACY
PRECISION
& ACCURACY
Precision
• Refers directly to the place value of the last significant
digit
• Or the value that the rightmost significant digit measures
Accuracy
• Refers to the number of significant digits in the number
• Example:
• 321.04
• Number precise to the nearest hundredths
• Number is accurate to 5 significant digits
REVIEW
SIGNIFICANT DIGITS
• All nonzero digits are significant digits
• Zeros are significant when they
a) Are between two significant digits:
b) Follow the decimal point and a significant digit
• Zeros used as placeholders are not significant digits
PRECISION VS ACCURACY
• Electrical current measured with two ammeters
a)
0.031 𝐴
• Precise to the thousandths
• Accurate to two significant digits
b) 0.0312 𝐴
• Precise to the ten thousandths
• Accurate to three significant digits
• More precise & more accurate
PRECISION VS ACCURACY
• A machine part is measured to be 2.5 𝑐𝑚 long. It is
coated with a film 0.025 𝑐𝑚 thick.
• Thickness of the film measured to a greater precision
• Measured to the nearest thousandths vs machine part
measured to nearest tenths
• Two measurements have the same accuracy
• 2 significant digits
ROUNDING OF
SIGNIFICANT DIGITS
• Round 52.35471 𝑚 to 4 significant digits
1. Locate the fourth significant digit, counting left to right
52.35471
2. Examine the digit immediately to the right of the
significant digit that is being rounded
52.35471
3. If digit is less than 5, indicated SD remains the same
52.35 𝑚
4. If digit is 5 or greater, indicated SD increases by 1
5. If necessary use zeros to replace other digits in order to
locate decimal properly
ROUNDING EXAMPLES
• Round the following measurements to the indicated
number of significant digits.
• It may be necessary to use scientific notation to show
appropriate number of significant digits.
a)
b)
c)
d)
e)
f)
g)
h)
70 430 𝑐𝑚 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 3 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 70400 𝑐𝑚
1 572 𝑚 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 2 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 1600 𝑚
187.35 𝑔 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 4 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 187.4 𝑔
39.72 𝑠 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 2 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 4.0×10A
203 𝐿 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 2 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 2.0×10C 𝐿
501.46 𝑁 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 3 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 501 𝑁
32 768 𝑚𝑚 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 1 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡 = 30 000 𝑚𝑚
0.021505 𝑘𝑚 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 2 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑑𝑖𝑔𝑖𝑡𝑠 = 0.22 𝑘𝑚
ADDING & SUBTRACTING
• Answer expressed with the precision of the least precise
number
a) 16.3 𝑓𝑡 + 14.92 𝑓𝑡 = 31.22 𝑓𝑡 → 31.2 𝑓𝑡
b) 10.821 𝑠 + 10.82 𝑠 + 11.0 𝑠 = 32.641 𝑠 → 32.6 𝑠
c) 23 𝑚 − 2.5 𝑚 − 4.65 𝑚 = 15.85 𝑚 → 16 𝑚
MULTIPLYING & DIVIDING
• Answer is expressed with the same accuracy of the
least accurate number
a)
b)
207.54 𝑚 81.4 𝑚 = 16 893.756 𝑚 → 16 900 𝑚
IJ K
L.MN O
= 8.896797153 𝑚⁄𝑠 → 8.9 𝑚⁄𝑠
ROOTS
• Answer is expressed with the same as accuracy of the
original number
1.
2.
3.
54 = 7.34846 → 7.3
216.7
T
R
S
=
S
216.7 = 6.006474 → 6.006
101 = 3.17015 → 3.17
CALCULATIONS
• Before and during the calculation, all numbers except
the least precise or least accurate may be rounded
off to one decimal place beyond that of the least
precise or least accurate
• Procedure helpful when calculator not used
PRACTICE PROBLEMS
• Perform the indicated operations on the following
approximate numbers:
a) 26.3 𝑐𝑚 + 412.07 𝑐𝑚 + 0.349 𝑐𝑚
b) 19.8062 𝑔 − 8.92 𝑔
c) (3.96 𝑓𝑡)(0.030 𝑓𝑡)
d) 4.924 𝐿 ÷ 86 𝐿
e) 6.80 𝑖𝑛 ÷ 0.0327 𝑖𝑛
f)
g)
48 𝑐𝑚C
87.4
R
T
𝑠