Problem 15 - Technical Mathematics with Calculus

Question

Determine the number of significant digits in each approximate number. $$4.008$$

Step-by-Step Solution

Answer

The number 4.008 has 4 significant digits.

Step 1: Identify Significant Digits

Significant digits in a number are those that carry meaning contributing to its measurement resolution. This includes all non-zero digits, zeroes between non-zero digits, and trailing zeroes in a decimal number.

Step 2: Analyze the Number

In the number 4.008, all of the digits are significant. The digit 4 is a non-zero digit, therefore it is significant. The zero digits are between non-zero digits, making them significant. And the last digit, 8, is a non-zero digit, therefore it is significant.

Step 3: Count the Significant Digits

Counting each significant digit in the number 4.008, we find that there are a total of four significant digits.

Key Concepts

Measurement Resolution

When we encounter the term 'measurement resolution,' we're referring to the smallest change in a quantity being measured that a device or method can detect. Consider the tick marks on a ruler: the spacing between these marks dictates how precisely we can measure the length of an object. In the context of significant digits, measurement resolution is closely related to the precision of a number. The more significant digits a number has, the higher its measurement resolution and thereby its precision. So, for instance, the number 4.008 is more precise than 4.0 because it has more digits that are significant, allowing us to detect even smaller changes in measurement.

Non-Zero Digits

Non-zero digits are always considered significant because they contribute key information about a number's accuracy. Any digit from 1 to 9 falls into this category. Let’s take our number 4.008; the digits 4 and 8 are non-zero digits and are unquestionably significant. They provide the most straightforward indication of precision and cannot be omitted without altering the value of the number. These digits matter for every measurement or calculation, as they directly impact the result's accuracy and reliability.

Trailing Zeroes

The topic of 'trailing zeroes' can be somewhat tricky in the realm of significant digits. The rule is that trailing zeroes are significant only if they come after a decimal point in a number. For example, in 50.0, the trailing zero is significant because it's after a non-zero digit and a decimal point, implying a precision to the tenths place. Conversely, in the number 100, the zeroes are not considered significant because they merely represent the magnitude of the number rather than its precision, unless specified otherwise by a decimal point as in 100.