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Introduction:
Introduction:
Significant figures, also known as significant digits, play a crucial role in science and mathematics. They are used to express the precision or accuracy of a measurement or calculation. By using significant figures, scientists can convey the level of confidence they have in their data and avoid misleading others with unnecessary precision. In this guide, we will explore the reasons for using significant figures, distinguish between significant and non-significant numbers, understand how to handle numbers with infinite significant figures, and learn the rules for performing arithmetic operations with significant figures.
Why Use Significant Figures:
Why Use Significant Figures:
In science, measurements and calculations are subject to uncertainties due to limitations in the precision of instruments or the inherent variability of natural phenomena. For example, when measuring the length of an object using a ruler, the smallest division on the ruler determines the precision of the measurement. If the smallest division is 1 mm, it means that the measurement can be accurate to the nearest millimeter.
Significant figures are used to communicate the precision of measurements and calculations. They enable scientists to represent the uncertainty associated with a value. By adhering to the rules of significant figures, scientists can avoid overestimating the precision of their results or making false claims about the accuracy of their data. This ensures that scientific communication remains transparent and consistent.
Rules of Significant Figures: Significant vs Non-Significant Numbers
Rules of Significant Figures: Significant vs Non-Significant Numbers
Significant figures are the digits in a number that carry meaning regarding its precision. They include all the certain digits plus one uncertain digit. Non-significant numbers, on the other hand, are placeholders that indicate the position of the significant digits but do not contribute to the precision of the value. To determine whether a digit is significant or non-significant, we follow these rules:
Rule #1: All non-zero digits are significant.
Rule #1: All non-zero digits are significant.
Example:
Example:
978 has 3 significant figures
4574 has 4 significant figures
Rule #2: Zeros between nonzero digits are significant.
Rule #2: Zeros between nonzero digits are significant.
Example:
Example:
2005 has 4 significant figures, both zeros are significant
9070 has 3 significant figures, the last zero is not significant
Rule #3: Leading zeros (zeros at the beginning of a number) are not significant
Rule #3: Leading zeros (zeros at the beginning of a number) are not significant
Example:
Example:
0.0005 has one significant figure
0.0098 has two significant figures
Rule #4: Trailing zeros to the right of the decimal ARE significant.
Rule #4: Trailing zeros to the right of the decimal ARE significant.
Trailing zeros to the right of the decimal ARE significant. There are FOUR significant figures in 92.00. 92.00 is different from 92: a scientist who measures 92.00 milliliters knows his value to the nearest 1/100th milliliter; meanwhile their colleague who measured 92 milliliters only knows his value to the nearest 1 milliliter. It's important to understand that "zero" does not mean "nothing." Zero denotes actual information, just like any other number. You cannot tag on zeros that aren't certain to belong there
Example:
Example:
9.0000 has 5 significant figures
0.0000090 has 2 significant figures, the trailing zero is significant
Rule #5: Trailing zeros in a whole number with no decimal shown are NOT significant.
Rule #5: Trailing zeros in a whole number with no decimal shown are NOT significant.
Writing just "540" indicates that the zero is NOT significant, and there are only TWO significant figures in this value.
Example:
Example:
984600 has four significant figures
4000 has one significant figure
Rule #6: Trailing zeros in a whole number with the decimal shown ARE significant.
Rule #6: Trailing zeros in a whole number with the decimal shown ARE significant.
Placing a decimal at the end of a number is usually not done. By convention, however, this decimal indicates a significant zero.
Example:
Example:
540 has two significant figures
540. has 3 significant figures because the decimal point communitates that all numbers before the decimal are significant)
5400. has 4 significant figures
Numbers with Infinite Significant Digits
Numbers with Infinite Significant Digits
In certain cases, numbers or values can have infinite significant figures. This occurs when the value is defined precisely without any uncertainty. Let's explore a few examples to understand this concept.
Exact numbers: Some numbers are defined precisely and do not involve any measurement or uncertainty. For instance, when counting objects, the number of objects is exact. For example, if you have 5 apples, the number 5 is an exact value with infinite significant figures.
Defined constants: Certain physical and mathematical constants are defined precisely and have infinite significant figures. For example, the value of π (pi) is approximately 3.14159, but it is defined precisely and has infinite significant figures.
Conversion factors: Conversion factors, such as those used in unit conversions, are defined precisely and have infinite significant figures. For example, 1 inch is precisely equal to 2.54 centimeters. Although the conversion factor has an approximate value, when used in calculations, it is considered to have infinite significant figures.
In these cases, the numbers or values are not subject to measurement uncertainties, and therefore, they are treated as having infinite significant figures. When performing calculations involving infinite significant figures, it is important to ensure that the final result is rounded to an appropriate number of significant figures based on the least precise value involved in the calculation.
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