Problem 51 - Master Resource Book in JEE Main Physics

Question

The respective number of significant figures for the number \(23.023,0.0003\) and \(21 \times 10^{-3}\) are [AIEEE 2010] (a) \(5,1,2\) (b) \(5,1,5\) (c) \(5,5,2\) (d) \(4,4,2\)

Step-by-Step Solution

Answer

The correct answer is (a) 5, 1, 2.

Step 1: Understanding Significant Figures

Significant figures are the digits in a number that contribute to its accuracy. This includes all non-zero numbers, any zeros between them, and any trailing zeros in a decimal number.

Step 2: Analyze the Number 23.023

The number 23.023 consists of 5 digits: 2, 3, 0, 2, and 3. Since the zero is between non-zero numbers, it is counted as significant. Therefore, 23.023 has 5 significant figures.

Step 3: Analyze the Number 0.0003

The number 0.0003 consists of 4 zeros and the digit 3. Only the digit 3 is a significant figure, while the leading zeros merely indicate the position of the decimal point. Hence, 0.0003 has 1 significant figure.

Step 4: Analyze the Number 21 × 10^-3

The number 21 is expressed in scientific notation as 21 × 10^-3. The number 21 consists of two non-zero digits, 2 and 1, which are both significant. Therefore, 21 × 10^-3 has 2 significant figures.

Step 5: Compare Choices with Calculated Figures

The given choices are (a) 5,1,2; (b) 5,1,5; (c) 5,5,2; (d) 4,4,2. From steps 2-4, we found the significant figures for the numbers 23.023, 0.0003, and 21 x 10^-3 are 5, 1, and 2, respectively, matching option (a) exactly.

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a concise form. It is particularly useful in mathematical calculations because it can simplify numbers significantly.
In scientific notation, a number is written as the product of two factors:

• A number between 1 and 10
• A power of ten

For example, the number \( 21 \times 10^{-3} \) is in scientific notation. Here, 21 is the coefficient, and it represents a precise value. The exponent \(-3\) on 10 means the decimal point moves three places to the left, converting 21 into 0.021.
Using scientific notation hugs the significant figures concept closely since the coefficient (21 in this case) determines the count of significant figures, which are 2 here.

Leading Zeros

Leading zeros are zeros that appear in front of all other non-zero digits in a number. They serve as placeholders, indicating the position of the decimal point and do not affect the calculation of significant figures.
For example, in the number 0.0003, there are four leading zeros before the 3. These zeros help place the decimal correctly but are not counted as significant figures. Only the 3 is significant in this case.
Being placeholders, leading zeros mainly show lower magnitude values, especially in decimal numbers, without altering the precision or accuracy of the numeric value.

Decimal Point

A decimal point is a period (.) used to separate the whole number part from the fractional part of a number. Its placement affects how we perceive the magnitude and precision of a number.
Numbers with a decimal point can include or exclude zeroes at the end, thus impacting the total count of significant figures.
For example, in 23.023, the decimal point makes it clear that the zero in the middle is significant as it is sandwiched between significant digits. The decimal point controls the position of all digits following it, thus influencing both the number's accuracy and value presentation. Similarly, scientific notation manipulates the position of the decimal point, changing how numbers are expressed without changing their inherent value.