Measurement Uncertainty

We often deal with experimental data and theoretical calculations. Most of the data is present in an extremely large number of quanta. This uncertainty in measurement is the range of possible values within which the true/real value of the measurement exists. There are practical ways to handle these figures with convenience and present the given data as realistically as possible.

Scientific Notation

• Atoms and molecules have very low masses and are present in exceedingly large numbers.
• We are here dealing with numbers as large as 602,200,000,000,000,000,000,000 for the molecules of just 2 g of H₂ hydrogen gas and as small as 0.00000000000000000000000166 g which is the mass of a hydrogen atom.
• On a similar scale exist the values of constants such as the speed of light, Planck's constant, electric charge of particles, etc.
• Thus, when handling numbers involving such scales of zeroes, it becomes extremely difficult for simple calculations of addition, subtraction, multiplication, and division. To solve these issues, scientists have developed a scientific notation for such numbers.
• Scientific Notation here is the exponential notation in which one can represent a given number in the form of N × 10ⁿ, where N is a digit term that ranges between 1.000... and 9.999..., and n is an exponent of positive or negative values.
• Accordingly, one can write 2702.0109 as 2.7020109 × 10³ in scientific notation.
• It can be observed that once the decimal point was shifted three places to the left, the exponent (3) of 10 was added to the scientific notation.
• Similarly, write 0.00001008565 as 1.008565 ×  10^-5.
• Here, the decimal point is moved to the right by five places, as the (-5) exponent of 10 is in the scientific notation.

Multiplication and Division

Both multiplication and division follow the same rules that exist for exponential numbers.

Example: (4.2342 × 10¹⁹) × (7.32 × 10⁵)

Solution:

(4.2342 × 10¹⁹) × (7.32 × 10⁵)

= (4.2342 × 7.32) × 10⁽¹⁹⁺⁵⁾

= 30.994344 × 10²⁴

= 3.099 × 10²⁵

Example: (6.23 × 10⁶) ÷ (8.33 × 10^-2)

Solution:

(6.23 × 10⁶) ÷ (8.33 × 10^-2)

= (6.23 ÷ 8.33) × 10^6-(-2)

= (0.74789) × 10⁸

= 7.4789 × 10⁷

Addition and Subtraction

For Addition and Subtraction, first, ensure if the numbers are present in the same exponent. After that, the digit terms (coefficients) can be added or subtracted as required.

Example: (5.12 × 10³) + (6.84 × 10⁵)

Solution:

(5.12 × 10³) + (6.84 × 10⁵)

= (5.12 × 10³) + (684 × 10³)

= (5.12 + 684) × 10³

= 689.12 × 10³

= 6.8912 × 10⁵

Example: (2.57 × 10⁵) - (9.46 × 10³)

Solution:

(2.57 × 10⁵) - (9.46 × 10³)

= (2.57 × 10⁵) - (0.0946 × 10⁵)

= (2.57 + 0.0946) × 10⁵

= 2.4757 × 10⁵

Significant Figures

Significant Figures are the digits that are known with certainty plus one which is estimated or uncertain. The uncertainty is thus shown by writing certain digits and the last uncertain digit. For example, if the temperature of the room is 35.2 °C, consider 35 as certain and 2 as uncertain. The uncertainty in this last digit is +1. If it is not stated, consider +1 in the last digit directly.  

Rules for deciding the number of significant figures in a given measurement.

• All Non-zero digits are to be considered significant. For example, 123 has three significant figures, while 0.123 also has three significant figures.
• Zeroes preceding the first non-zero aren't significant. These zeroes indicate the position of the given decimal point. Thus, 0.02 has one significant figure and 0.000027 also has just two significant figures.
• Zeroes between any two non-zero digits are significant. Thus 9.003 has four significant figures.
• Zeroes at the end or on the right of a number (if they are on the right of the decimal point) are significant. For example, 0.500 ml has three significant figures. However, terminal zeroes are not significant if there isn't a decimal point. For example, 9000 has only one significant figure, but 9000. has four significant figures, and 9000.0 has five significant figures. These numbers are better off represented in scientific notation. We would rather express 9000 as 9 × 10³ for one significant figure or 9.00 × 10³ for three significant figures.
• Exact numbers can be represented in infinite significant figures. Like 50 can be written as 50.0000000 or 734 can be written as 743.000000000 and so on. In scientific notation, all digits are significant. For example, 3.545 × 10^-2 has four significant figures while 9.43 × 10⁶ has three significant figures.

Accuracy and Precision

Precision stands for the closeness of different measurements for the same quantity. Accuracy is the consensus of a particular value to the true value of the measurement.

For example

• Suppose three students Alex, Bob, and Carol are measuring the length of the physics textbook.
• The true length that is known of the book is 29.5 cm.
• Alex measures and reports two values 28.0 cm and 28.2 cm.
• Both these values are precise as they are close to each other but not accurate.
• Bob measures and reports two values 28.5 cm and 30.5 cm.
• The average of these values is the true value but this observation though accurate is not precise.
• Carol repeats this experiment and measures 29.4 cm and 29.6 cm.
• Both these values are precise and accurate as they are close to each other and the average is the true value.