Question

The measurement \(30 \mathrm{ft}\) is ambiguous, but the measurement \(30 . \mathrm{ft}\) is not. Explain what the ambiguity is, and how adding the decimal point eliminates the ambiguity.

Short answer

The ambiguity in the measurement \(30 \mathrm{ft}\) arises from the last digit "0", which could either be significant or a placeholder, making it unclear how precise the measurement is. By adding a decimal point, as in \(30 . \mathrm{ft}\), it shows that the "0" is significant rather than just a placeholder, indicating that the measurement is accurate to the nearest foot. This eliminates the ambiguity, allowing for proper interpretation of the significant figures and more confident use in calculations.

Step-by-step solution

In the measurement \(30 \mathrm{ft}\), there are two digits: 3 and 0. The number has 2 significant figures - the digit "3" and the digit "0" can be considered as a placeholder digit or as a significant measurement digit. The ambiguity arises because the "0" can be either significant or not depending on how the measurement was taken, which we don't know in this case. #Step 2: Explain the ambiguity in the given measurement#

The measurement \(30 \mathrm{ft}\) is ambiguous because the last digit, "0", can be significant or not. It could be representing an exact 30-foot measurement that has three significant digits (30.0) or just be a placeholder and indicate that the measurement is only accurate to the nearest ten feet (approximately 30 feet). This ambiguity makes it difficult to determine how precise the measurement is and how it should be used in calculations involving other measurements. #Step 3: Describe the effect of adding the decimal point to the measurement#

When the decimal point is added to the measurement, as in \(30 . \mathrm{ft}\), it clarifies the ambiguity by showing that the "0" is a significant figure and not just a placeholder. It indicates that the measurement is accurate to the nearest foot, rather than only to the nearest 10 feet. #Step 4: Explain how the ambiguity is eliminated by adding the decimal point#

By including the decimal point in the measurement, like \(30 . \mathrm{ft}\), it indicates that there is no more precision beyond this point. This means that the final "0" is a significant measurement digit and not just a placeholder. Therefore, we know that the measurement is accurate to the nearest foot and can be used more confidently in calculations involving other measurements. It also signals to others using this measurement that it has a certain level of precision and ensures the proper interpretation of the significant figures.

Key concepts

Measurement Precision

Measurement precision refers to how close the repeated measurements of a quantity are to each other. It is an indication of the fineness of measurement and provides insight into the smallest interval that can be reliably indicated by the measurement instrument used.

For example, if one were to repeatedly measure a length with a standard ruler, one might obtain measurements such as 30.2 cm, 30.3 cm, and 30.1 cm. These measurements are precise if the ruler is capable of reliably showing the difference up to 0.1 cm. However, the precision of a tool can be compromised by several factors, including the skill of the person doing the measurement and the quality of the instrument.

Significant Digits

Significant digits, also known as significant figures, are the digits in a number that carry meaning contributing to its precision. This includes all non-zero numbers, any zeroes between significant digits, and any trailing zeroes in the decimal portion of the number.

For example, in the measurement 30 ft, if the 'zero' is intended as a significant digit, it means the measurement is accurate to the foot. If it is not significant, then the measurement might only be accurate to the nearest ten feet. The significant figures tell us how precisely something is measured and indicate the quality of the measurement.

Decimal Point in Measurements

The positioning of a decimal point in measurements is crucial for understanding the level of precision. It tells us how far to the right we can trust the numbers to be accurate.

Let's say we have the measurement 30 ft. Without any additional context, it's not clear whether the measurement is to the nearest ten feet or to the exact foot. However, when this measurement is written as 30. ft, the decimal point indicates that the 'zero' is indeed a significant digit. The measurement is thus precise to the foot, and not just an approximation to the nearest ten feet. It's a subtle but powerful way to express the exactitude of measurements.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in measurements to clearly indicate the precision and to make it easier to handle very large or very small numbers.

The general form of scientific notation is \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer. The number \(a\) is the \(significant\) part, which includes all the significant figures of the number. For instance, the Avogadro's number \(6.02214076 \times 10^{23}\) contains all significant digits that define the number of entities in one mole of substance, written compactly and precisely in scientific notation.