How do significant numbers maintain accuracy?

There is a certain uncertainty associated with every measurement. The uncertainty arises from the measuring device and the ability of the person taking the measurement. Scientists report measurements with significant numbers to reflect this uncertainty.

Let's take volume measurement as an example. Let's say you are in a chemistry lab and you need 7 ml of water. You could take an unlabeled coffee mug and add water until you think you have about 7 milliliters. In this case, most of the measurement error is related to the ability of the person taking the measurement. You can use a beaker marked in 5 ml increments. With the beaker you can easily get a volume between 5 and 10 ml, probably close to 7 ml, 1 ml give or take. If you use a pipette labeled 0.1 ml, you can pretty reliably get a volume between 6.99 and 7.01 ml. It would be wrong to report that you measured 7,000 ml with one of these devices because you did not measure the volume to the nearest microliter. You would report your measurement with significant numbers. This includes all the digits you know for sure, plus the last digit, which contains some uncertainties.

Important figure rules

  • Digits other than zero are always important.
  • All zeros between other significant digits are significant.
  • The number of significant digits is determined by starting with the leftmost digit other than zero. The leftmost non-zero digit is sometimes called the most significant digit or the denotes the most significant number . For example, in the number 0.004205, the '4' is the most significant number. The zeros on the left are not significant. The zero between the '2' and the '5' is significant.
  • The rightmost digit of a decimal number is the least significant digit or number. Another way to look at the least significant number is to think of it as the rightmost digit when the number is written in scientific notation. The least significant numbers are still significant! In the number 0.004205 (which can be written as 4.205 x 10, -3 ), the '5' is the least significant figure. In the number 43,120 (which is called 4.3210 x 10 1 can be written ) the '0' is the least significant number.
  • If there is no decimal point, the rightmost non-zero digit is the least significant number. In the number 5800, the least significant number is '8'.

Calculation uncertainty

Measurements are often used in calculations. The accuracy of the calculation is limited by the accuracy of the measurements on which it is based.

  • Addition and subtraction
    When adding or subtracting measures, the uncertainty is determined by the absolute uncertainty of the least accurate measurement (not the number of significant numbers). Sometimes this is viewed as the number of digits after the decimal point.
    32.01 m
    5.325 m
    12 m
    Together they add up to 49.335 meters, but the total should be reported as 49 meters.
  • Multiplication and division
    When experimental quantities are multiplied or divided, the number of significant numbers in the result is the same as that in the set with the smallest number of significant numbers. For example, when performing a density calculation that divides 25.624 grams by 25 ml, the density should be reported as 1.0 g / ml, not 1.0000 g / ml or 1000 g / ml.

Losing significant numbers

Sometimes significant numbers are lost in the calculation. For example, if you find the mass of a cup to be 53.110 g, add water to the cup and find the mass of the cup plus water to be 53.987 g, the mass of the water from 53.987 to 53.110 g = 0.877 g
The final value only has three significant numbers, although each mass measurement contained 5 significant numbers.

Round and cut numbers

There are several methods of rounding numbers. The usual method is to round numbers with digits less than 5 down and numbers with digits greater than 5 up (some people round exactly 5 up and others down).

Example:
If you subtract 7.799 g - 6.25 g, your calculation is 1.549 g. This number would be rounded to 1.55 g because the digit '9' is greater than '5'.

In some cases, numbers are truncated or truncated, rather than rounded, to give suitable significant numbers. In the example above, 1.549 g could have been cut off to 1.54 g.

Exact numbers

Sometimes the numbers used in a calculation are more accurate than approximate. This applies when using defined quantities, including many conversion factors, and when using pure numbers. Pure or defined numbers do not affect the accuracy of a calculation. You can imagine that they have an infinite number of significant numbers. Pure numbers are easy to spot because they have no units. Like measured values, defined values ​​or conversion factors can have units. Practice identifying them!

Example:
You want to calculate the average height of three plants and measure the following heights: 30.1 cm, 25.2 cm, 31.3 cm; with an average height of (30.1 + 25.2 + 31.3) / 3 = 86.6 / 3 = 28.87 = 28.9 cm. There are three significant figures in the heights. Even though you divide the total by a single digit, the three significant numbers should be kept in the calculation.

Accuracy and Precision

Accuracy and precision are two separate concepts. The classic illustration that distinguishes the two is looking at a target or a porthole. Arrows surrounding a porthole indicate a high degree of accuracy; Arrows that are very close together (may not be near the porthole) indicate a high level of precision. To be precise, there must be an arrow near the target. To be precise, consecutive arrows must be close together. Consistently hitting the center of the porthole indicates both accuracy and precision.

Consider a digital scale. If you weigh the same empty beaker repeatedly, the balance will provide values ​​with high precision (e.g. 135.776 g, 135.775 g, 135.776 g). The actual mass of the cup can vary widely. Scales (and other instruments) need to be calibrated! Instruments usually give very accurate readings, but accuracy requires calibration. Thermometers are notoriously inaccurate and often need to be recalibrated several times over the life of the instrument. Scales also need to be recalibrated, especially if they are moved or mistreated.

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  • de Oliveira Sannibale, Virgínio (2001). "Measurements and Significant Numbers". Freshman Physics Laboratory . Department of Technology, Physics, Mathematics, and Astronomy at the California Institute.
  • Myers, R. Thomas; Oldham, Keith B .; Tocci, Salvatore (2000). chemistry . Austin, Texas: Get Rinehart Winston. ISBN 0-03-052002-9.