Problem 13 - Technical Mathematics with Calculus

Question

Determine the number of significant digits in each approximate number. $$78.3$$

Step-by-Step Solution

Answer

The number 78.3 has 3 significant digits.

Step 1: Understanding Significant Digits

Significant digits are the numbers that carry meaning contributing to its precision. This includes all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion. Leading zeros are not significant.

Step 2: Identifying Significant Digits in 78.3

In the number 78.3, all digits are significant. The '7' and '8' are non-zero numbers, while '3' is a non-zero digit and a trailing digit in the decimal part. There are no leading zeros or zeros between significant digits to consider.

Step 3: Counting Significant Digits

Count all the significant digits in 78.3. Since 7, 8, and 3 are all significant, there are three significant digits in this number.

Key Concepts

Precision in Measurements

When it comes to the precise sciences, such as chemistry and physics, the accuracy of measurements can be as crucial as the measurements themselves. Precision in measurements refers to the level of detail in a recorded figure, often indicated by the number of significant digits. The more significant digits a number has, the more precise the measurement is considered to be.

This precision indicates the confidence in the reliability of a measurement. For example, a scale that measures weight to the nearest tenth of a gram offers more precision than one that rounds to the nearest gram. Understanding and correctly identifying significant digits is crucial for students, not only to ensure accuracy in their work but also to appropriately convey the reliability of their measurements.

Non-Zero Digits

In the realm of significant digits, non-zero digits are always considered significant because they contribute specific value to a number. A non-zero digit represents a measured or observed quantity and is always included when counting the number of significant digits. For instance, in the number 78.3, the digits '7' and '8' are clearly non-zero and thus inherently significant.

These digits denote the certainty of the quantity measured and exclude any form of guessed or estimated value, which might be the case with zeros. They are the bedrock of significant digits, and their presence is non-negotiable in the accuracy of a measurement.

Trailing Zeros in Decimals

Trailing zeros in decimals have a special place in the concept of significant digits. They are significant only when they come after a decimal point and are preceded by a non-zero digit. The role they play is to indicate the precision of a measurement. For example, in a number like 78.30, the zero after the decimal is significant because it shows the measurement is accurate to the hundredth place.

It’s important for students to note that trailing zeros in a whole number without a decimal point are not considered significant. The rule changes when the decimal point is present, as it signifies that the measurement was precise enough to warrant the inclusion of the zero. In the number 78.3, although there is only one zero, its presence confirms the measurement to a tenth of a unit, indicating a higher precision than a whole number measurement.