Significant Figures

Significant figures are the digits in a measurement that determine its sensitivity and express the reliability of the measurement result. They are determined based on the precision of the measuring instrument and must be taken into account in calculations. Significant figures are of great importance in scientific calculations, engineering, physics and chemistry like.

Determining Significant Figures

Specific rules are followed to determine the significant figures of a number. These rules explain how the measurement result should be interpreted and how it should be handled during calculations.

Non-Zero Digits

All non-zero digits are significant. For example:

4829 (This number has four significant figures.)

7.31 (It contains three significant figures.)

Zeros Between Digits

Zeros located between other digits are significant. For example:

505 (It contains three significant figures.)

30.06 (It contains four significant figures.)

Leading Zeros

Leading zeros are not significant. For example:

0.0045 (It contains two significant figures: 4 and 5.)

0.000306 (It contains three significant figures: 3, 0 and 6.)

Trailing Zeros

If there is a decimal point: All zeros after the decimal point are significant.

10.00 (It contains four significant figures.)

0.500 (It contains three significant figures.)

If there is no decimal point: Trailing zeros may not be significant.

45000 (It contains two significant figures.)

3.200 × 10³ (It contains four significant figures; in scientific notation all digits are significant.)

Determining the Number of Significant Figures-KARADENİZ TECHNICAL UNIVERSITY

Exponential Notation (Scientific Notation)

Exponential notation is a method used to write very large or very small numbers in a more short and understandable form. Numbers are written in the form a × 10ⁿ, where a is a decimal number between 1 and 10 and n is an complete integer.

Significant Figures in Exponential Notation

1. In exponential notation, only the digits in the a value are significant.
2. The power of 10 (the exponent) does not change the number of significant figures.
3. When decimal places are preserved in writing, no loss of precision occurs.

Examples

The number 450000 is written in exponential notation as 4.5 × 10⁵ and contains two significant figures.

The number 0.0006789 is written in exponential notation as 6.789 × 10⁻⁴ and contains four significant figures.

The number 3.0200 × 10³ contains five significant figures because both the zeros and the decimal places are explicitly shown.

Rounding in Exponential Notation

If a number must be expressed with a specific number of significant figures, rounding rules must be applied:

Expressing 4.56789 × 10⁵ with three significant figures: 4.57 × 10⁵.

Expressing 0.0034567 with two significant figures: 3.5 × 10⁻³.

Scientific notation allows numbers to be expressed more practically and better preserves the level of precision of measurements.

Exponential Notation-KARADENİZ TECHNICAL UNIVERSITY

Arithmetic Operations with Significant Figures

Addition and Subtraction

In addition and subtraction, the result must have the same number of decimal places as the number with the least number of decimal places.

23.56 + 0.8 = 24.4 (The result matches the number with the least decimal places, 0.8.)

45.678 - 3.2 = 42.5 (Since 3.2 has the fewest decimal places, the result must be expressed with one decimal place.)

Multiplication and Division

In multiplication and division, the result must have the same number of significant figures as the number with the least number of significant figures.

6.24 × 1.3 = 8.1 (Since 1.3 has two significant figures, the result must also have two significant figures.)

15.678 ÷ 2.1 = 7.5 (Since 2.1 has two significant figures, the result must also have two significant figures.)

Rules for Rounding Numbers

Long calculation results must be rounded according to the rules for significant figures. When rounding, follow these rules:

If the digit to be dropped is 5 or greater, increase the preceding digit by one.

12.5678 → 12.57 (Rounded.)

6.789 → 6.79 (Rounded.)

If the digit to be dropped is 4 or less, the preceding digit remains unchanged.

12.344 → 12.34 (Rounded.)

9.8321 → 9.83 (Rounded.)

Rounding Significant Figures-KARADENİZ TECHNICAL UNIVERSITY

Difference Between Accuracy and Precision

When evaluating measurement results, the concepts of accuracy and precision are distinct:

1. Precision: The repeatability of a measurement, meaning obtaining values close to each other. For example, measurements on a scale yielding values such as 1.21, 1.22, and 1.215 indicate high precision.

2. Accuracy: How close the measurement results are to the true value. For example, measuring a substance whose true value is 12.350 grams as 12.3495 grams indicates high accuracy.

A measurement can be precise but not accurate. For example, an object that actually weighs 9.85 grams may be measured as 10.00 grams. In this case, the measurement is precise but not accurate.

Applications of Significant Figures

Significant figures are important in many scientific and technical fields:

1. Chemistry and Physics: Used to minimize measurement errors.

2. Engineering: Used in precise calculations and tolerance analysis.

3. Statistics: Used to ensure correct analysis of data.

4. Finance and Economics: Used in calculations requiring high precision.