# Uncertainty and Significant Figures

Uncertainty of significant figures

Kinds of numbers in scientific work:
• Exact numbers: those whose values are known exactly. For Example, there is exactly 1000 g in kilogram.
• Inexact numbers: those whose values have some uncertainty like all numbers that obtained by measurement.

Precision and Accuracy:
Precision is the measure of how closely individual measurements agree with one another.
Accuracy is how closely individual measurements with the correct value.

Significant Figures:
• Significant figures is all digits of a measured quantity, including the uncertain one.
• The greater the number of significant figures, the greater is the certainty implied for the measurement.

Rules for identifying significant figures when writing numbers:
1. Zeros between nonzero digits are always significant. Example: 1008 has four significant figures.
2. Zeros at the beginning of a number are never significant. They only indicate the position of the decimal point. Example: 0.0038 has only two significant figures.
3. Trailing zeros in a number containing a decimal point are significant. Example: 0.0050 and 6.0 have only two significant figures.
4. When a number ends in zeros but contains no decimal point, the zeros may or may not be significant. Example: 160 has two or three significant figures. If a decimal is not shown the end zero(es) cannot be counted as significant.
5. If you can/must get rid of the zeroes, then they are NOT significant.
Scientific notation eliminates the potential ambiguity about the significance of trailing zeros. For example 1600 can be written in scientific notation showing four, three or two significant figures:
• 1.600 x 10^3 (four significant figures)
• 1.60 x 10^3 (three significant figures)
• 1.6 x 10^3 (two significant figures)

Addition and Subtraction
When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures). Sometimes this is considered to be the number of digits after the decimal point.

Multiplication and Division
When experimental quantities are mutiplied or divided, the number of significant figures in the result is the same as that in the quantity with the smallest number of significant figures.

Rounding Significant Figures
If you are rounding a number to a certain degree of significant digits if the number following that degree is less than five the last significant figure is not rounded up, if it is greater than 5 it is rounded up.
ex. 10.5660 rounded to four significant figures is
10.57

Resources: