2.6 - Approximate Numbers and Significant Figures

An exact number is one that has no uncertainty. An example is the number of tires on a car (exactly 4) or the number of days in a week (exactly 7). An approximate number is one that does have uncertainty. A number can be approximate for one of two reasons:
  1. The number can be the result of a measurement. For example a certain instrument capable of measuring to the nearest 0.1 cm may measure the length of a certain bolt to be 8.6 cm. A better quality instrument capable of measuring to the nearest 0.001 cm may give the length of the same bolt to be 8.617 cm. This new number is less approximate but is still not exact.

  2. Certain numbers simply cannot be written exactly in decimal form. Many fractions and all irrational numbers fall into this category. For example the fraction 1/3 is approximately but not exactly equal to 0.333 and the irrational number is approximately but not exactly equal to 1.73.

When we state that the measured length of the bolt is 8.6 cm then we actually mean that the value is closer to 8.6 cm than it is to 8.5 cm or 8.7 cm. The true length could be anywhere in the gray area shown here:




And when we state that the more accurate instrument gave the length of the bolt to be approximately 8.617 cm then we mean that the value is closer to 8.617 cm than it is to 8.616 cm or 8.618 cm. The true length could still be anywhere in the gray area shown here:




If someone told us that they used this same instrument and got a reading of 8.61712345 would we believe it? No way! Adding just one atom to the end of the bolt would cause the last digit to change! The extra digits are meaningless and are said to be insignificant. To claim that they are correct is nonsense.



Significant Digits or Figures


Definitions:

In an approximate number the leftmost digit is said to be the most significant digit and the rightmost digit is the least significant digit. All the digits in the number are significant digits (also known as significant figures or sig. figs.) with one exception: if the digit is a zero that is used just to locate the decimal point then it is not significant.

The accuracy of an approximate number is given by the number of significant digits in it.

The precision of an approximate number is given by the position of the rightmost significant digit.



Examples:
  1. The approximate number 8.617 has 4 significant digits. The digit 8 is the most significant digit and the digit 7 is the least significant digit.

  2. The number 1.23, the number 0.000123 and the number 123000000 all have an accuracy of 3 sig. figs. All the zeros are used simply to locate the decimal point.

  3. The number 1.23 has a precision of 0.01, the number 0.000123 has a precision of 0.000001 and the number 123000000 has a precision of 1000000.

  4. The number 1.023, the number 0.01023 and the number 1002000 all have 4 sig. figs. (The zeros shown in red are used simply to locate the decimal point and don't count as sig. figs.)

Notes: The precision of a measuring instrument is the difference between the two closest readings that the instrument can differentiate. For example the above instruments had a precision of 0.1 cm and 0.001 cm.

Precision and accuracy are not the same thing. Accuracy has to do with the quality (and cost!) of the measurement. For example if your instrument has a precision of 1 centimeter then that may not be very accurate if that instrument is designed to measure distances between objects on your desk but it would be very accurate if it was designed to measure the distances between the planets.



Rounding

We saw above that if an instrument capable of measuring to the nearest 0.1 gave a measured value of 8.6 then the true value could be anywhere in the gray area:




Rounding is exactly the same idea but reversed. Rounding to the nearest 0.1 means to replace any number in the gray area (from 8.55 to 8.65) by 8.6. In general, rounding is done like this:


Rounding: When rounding to a certain place value then all digits to the right of that place are dropped. If the first dropped digit is 0, 1, 2, 3, or 4 then the least significant digit kept is not changed. (This is called rounding down.) If the first dropped digit is 5, 6, 7, 8 or 9 then the least significant digit kept is increased by 1. (This is called rounding up.)



You can round to either a given decimal place or to a given number of sig. figs. Here are some examples of rounding to 2 decimal places (the dropped digits are shown in red):
the rounding the rule used
4.384 → 4.38 first dropped digit is a 4 so round down
4.3851 → 4.39 first dropped digit is a 5 so round up
0.00043851 → 0.00 first dropped digit is a 0 so round down

Here are some examples of rounding to 3 sig. figs (the dropped digits are shown in red):
the rounding the rule used
4.384 → 4.38 first dropped digit is a 4 so round down
43851 → 43900 first dropped digit is a 5 so round up
0.00043851 → 0.000439 first dropped digit is a 5 so round up



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