# CHMB16 Chapter 4

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27 Sep 2011
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Chapter 3: Experimental Error
Significant Figures
Significant Figures: the minimum number of digits needed to write a given
value in scientific notation without loss of precision.
oZeros are significant when they occur in the middle of a number
(101.01) or at the end of a number on the right-hand side of a decimal
point (0.1060)
oThe last significant digit always has some associated uncertainty.
oInterpolation: estimate all readings to the nearest tenth of the distance
between scale divisions (on a 50-mL burette, which is graduated to
0.1-mL, read the level to the nearest 0.01-mL).
Significant Figures in Arithmetic
Addition and Subtraction: express all numbers with the same exponent and
align all numbers with respect to the decimal point; round off the answer
according to the number of decimal places in the number with the fewest
decimal places.
oIf the numbers to be added or subtracted have equal numbers of
digits, the answer goes to the same decimal place (integers are always
exact) as in any of the individual numbers.
Example: 18.9984032 + 18.9984032 + 83.798 = 121.795, since
83.798 only has 3 numbers after the decimal point and
oIf the first insignificant figure is below 5, we round the number down.
oIn the addition or subtraction of numbers expressed in scientific
notation, all numbers should first be expressed with the same
exponent.
Example: 1.632 x 10^5 + 4.107 x 10^3 + 0.984 x 10^6
1.632 x 10^5 + 0.04107 x 10^5 + 9.84 x 10^5 = 11.51307 x
10^5 11.51 x 10^5
Multiplication and Division: we are normally limited to the number of digits
contained in the number with the fewest significant figures.
oExample: (4.3179 x 10^12) x (3.6 x 10^-19) = 1.6 x 10^-6
oThe power of 10 has no influence on the number of figures that should
be retained.
Logarithms and Antilogarithms
oIf n = 10^a, then logn = a.
Example: 2 is the logarithm of 100 because 100 = 10^2 and
the logarithm of 0.001 is -3 because 0.001 = 10^-3.
oIn the equation above, the number n is said to be the antilogarithm of
a.
Example: the antilogarithm of 2 is 100 because 10^2 = 100.
oA logarithm is composed of a characteristic and a mantissa.
Characteristic: the integer part, whole number.
Mantissa: the decimal part.
oNumber of digits in the mantissa of logx = number of significant
figures in x.
oNumber of digits in antilogx (=10^x) = number of significant figures in
mantissa of x.
Example: log(5.403 x 10^-8) = -7.2674
Example: 10^6.142 = 1.39 x 10^6
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