Question

How many significant figures are understood for the numbers in the following definition: 1 in. \(=2.54 \mathrm{~cm} ?\)

Short answer

In the given definition, 1 in. = 2.54 cm, there are 5 significant figures: 1, 2, 5, 4, and the zero between 5 and 4.

Step-by-step solution

Identify non-zero digits

Non-zero digits are any of the digits from 1 to 9. In the given definition, the non-zero digits are 1, 2, 5, and 4.

Count the zeroes between non-zero digits

In 2.54, there is a zero between the two non-zero digits (5 and 4). We count this zero as a significant figure.

Count the significant figures

We now have the non-zero digits (1, 2, 5, and 4) and the zero between them (0). Count all these significant figures, which is 5 in total. The number of significant figures in the given definition (1 in. = 2.54 cm) is 5.

Key concepts

Measurement Precision

Understanding measurement precision is key when dealing with significant figures. Measurement precision refers to how closely repeated measurements of a quantity agree with each other. It indicates the level of detail in a measurement. The more significant figures a number has, the more precise it is. Significant figures consist of:

• All non-zero digits (e.g., 1, 2, 5, 4 in the given example).
• Any zeroes between non-zero digits (e.g., zero between 5 and 4 in 2.54).
• Trailing zeroes in the decimal portion (e.g., 0.300 has three significant figures).

These rules help in maintaining accuracy, ensuring that the degree of precision is sustained throughout calculations, particularly in scientific and engineering work. Remember, precision is not just about accuracy; it's about consistency between similar measurements.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers conveniently. It simplifies the handling and calculation of such numbers by expressing them in terms of powers of ten. This method is ideal when working with measurements that have many significant figures, as it helps in maintaining precision without cumbersome notations. For instance, the number 2500, if precise to three significant figures, would be written as \(2.50 \times 10^3\). By converting numbers into scientific notation, it becomes easier to perform arithmetic operations such as multiplication and division, since you primarily work with the coefficients and simply adjust the exponents. Therefore, scientific notation is a crucial tool in scientific calculations, ensuring both precision and simplicity.

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another, requiring careful consideration of significant figures to preserve measurement precision. In the given example, 1 inch is defined as 2.54 centimeters. When converting units, you must maintain the number of significant figures to convey the measurement's original precision.

• Start with identifying the conversion factor, such as 1 in = 2.54 cm.
• Apply the conversion by multiplying by the appropriate factor ensuring significant figures are kept.
• For example, converting 5 inches to centimeters: \(5 \times 2.54 = 12.70\). Here, the converted value also has three significant figures, consistent with the conversion factor's significant figures.

Whether you're converting inches to centimeters or meters to feet, the principles of unit conversion centered around significant figures ensure that measurements remain precise and true to their defined values.