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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

rounding of the final answer.

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