Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of significant figures in \(306.45\) and 40440 are respectively (a) 4,5 (b) 5,5 (c) 5,4 (d) 4,6

Step-by-Step Solution

Answer

The answer is (c) 5,4.

Step 1: Understanding Significant Figures

Significant figures in a number are the digits that carry meaning contributing to its precision. This includes all the non-zero digits, zeros between significant digits, and trailing zeros in a decimal portion.

Step 2: Counting Significant Figures in 306.45

Write down the digits of 306.45: 3, 0, 6, 4, 5. All these digits are significant because: 3, 6, 4, and 5 are non-zero, and 0 is between 3 and 6. This makes a total of 5 significant figures.

Step 3: Counting Significant Figures in 40440

Write down the digits of 40440: 4, 0, 4, 4, 0. Non-zero digits 4, 4, and 4 are significant. The zeros are between and after other significant digits, indicating a precision in measurement. With no decimal point, trailing zeros are not counted as significant. Thus, there are 4 significant figures.

Step 4: Comparing to Answer Choices

(b) 5,5 indicates both numbers have 5 significant figures. (c) 5,4 correctly states 5 for 306.45 and 4 for 40440, which aligns with our findings.

Key Concepts

Precision in Measurement

Precision in measurement is all about how detailed and exact the measurement is when recorded. The number of significant figures in a measurement can reveal its precision. They tell us how sure we can be about certain values and how much uncertainty we have to account for.

• More significant figures generally mean higher precision.
• Less significant figures might mean greater uncertainty.
• Each digit plays a specific role, including zeroes.

Identifying significant figures helps determine how precise a measurement is, guiding scientific calculations and results.

Non-zero digits

Non-zero digits are inherently significant. Every digit from 1 to 9 is considered significant because they are not place-holders and always count towards the precision of a measurement. Here's why they matter:

• They give immediate indications of scale and magnitude.
• They are the digits that contribute directly to the measurement's precision.

In both numbers, 306.45 and 40440, the non-zero digits form the core of what we count as significant figures. Thus, capturing their presence is crucial in accuracy for reporting a value.

Trailing zeros

Trailing zeros can sometimes be tricky because their significance can change depending on the context they appear in. Trailing zeros are the zeros that come after all the non-zero digits. Here’s how they work:

• In a decimal number, trailing zeros are always significant because they indicate the precision of the measurement, such as in 3.00.
• In a whole number without a decimal point, trailing zeros are not considered significant, like in 40440.

This is significant when measuring data since it dictates how you interpret the data's precision.

Decimal portion

The decimal portion of a number contains digits after the decimal point. These play a vital role in determining the number's precision in measurement. Here's why they are essential:

• The digits after the decimal point provide more detailed precision.
• Zeros in the decimal portion are significant as they define the measurement accuracy, like in 306.450.

The precise placement of numbers after the decimal shows how detailed the measurement is, adding depth to the data it's representing.