Question

Identify the number of significant figures in each of these measurements of an object’s length: 76.48 cm, 76.47 cm, and 76.59 cm.

Short answer

The number of significant figures in each measurement is as follows: 76.48 cm has \(4\) significant figures, 76.47 cm has \(4\) significant figures, and 76.59 cm has \(4\) significant figures.

Step-by-step solution

Identify the number of significant figures in 76.48 cm

The measurement 76.48 cm has four digits, all of which are non-zero. Therefore, all four digits are significant. The number of significant figures in 76.48 cm is 4.

Identify the number of significant figures in 76.47 cm

The measurement 76.47 cm has four digits, all of which are non-zero. Therefore, all four digits are significant. The number of significant figures in 76.47 cm is 4.

Identify the number of significant figures in 76.59 cm

The measurement 76.59 cm has four digits, all of which are non-zero. Therefore, all four digits are significant. The number of significant figures in 76.59 cm is 4.

Key concepts

Measurement

In the world of science and math, measurements serve as a bridge between abstract concepts and the tangible universe. Measurements allow us to quantify everything, from the length of a table, the temperature of a room, to the concentration of a solution. Key components of a measurement include both a number and a unit. For example, 76.48 cm tells us not just a numerical value, but also that this value refers to centimeters of length.

Whenever you take a measurement, you're combining an observation with a degree of uncertainty inherent to the instruments used. Even the most precise instruments can't eliminate the tiny variations introduced by things like temperature and operator skill. Therefore, understanding the significance of each digit in your measurement assures you that the value you recorded reflects the best possible estimate of the real object or phenomenon.

• Number: Reflects the magnitude of the measurement.
• Unit: Defines what is being measured (length, mass, etc.).
• Significant Figures: Convey the precision of the measurement.

Precision

Precision is the hallmark of good measurement. It's about consistency and repeatability. A measurement with high precision means that repeated measurements will yield results that are close to one another, even if they are not exactly the same. Precision relates to the instrument used for measurement and to the method of measurement.

In the given exercise, each measurement of the object's length has four significant figures, indicating a high level of precision. This tells us that the instrument used was capable of consistently measuring small differences. When you see four significant figures, you're looking at a more precise measurement than if there were only, say, two.

• Sig Fig Importance: More significant figures suggest more precision, not necessarily more accuracy.
• Instrument Quality: Top-quality instruments provide more precise and reliable data.
• Measurement Repetition: Consistent results across attempts highlight precision.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers conveniently. It helps in dealing with calculations involving extremes of scale, both cosmic and microscopic. This notation makes it easier to handle numbers without losing sight of the number of significant figures.

Consider a measurement like Earth's distance from the Sun, which is approximately 149,600,000 kilometers. In scientific notation, we would write this as:\[1.496 \times 10^8 \text{ km}\]. This format makes it clear how precise the measurement is and keeps calculations straightforward, ensuring that the significant figures are preserved throughout calculations.

• Expression: Scientific notation uses a form \(a \times 10^n\), where \(1 \leq a < 10\).
• Conveys Precision: Keeps track of significant figures, maintaining the integrity of the original measurement during calculations.
• Ease of Use: Simplifies complex calculations, especially when dealing with measurements of vastly different magnitudes.