Animated Solution for Physics - Physics and Measurement: The respective number of significant figures for the numbers $23.023$, $0.0003$ and $2.1\times 10^{-3}$ are

Understanding Significant Figures

In the realm of physics and measurement, numbers are not just abstract mathematical entities; they represent physical realities. When we measure a quantity, the precision of our measuring instrument dictates how many digits we can confidently report. This brings us to the concept of significant figures. Significant figures are the reliable digits in a measurement plus the first uncertain digit. They tell us exactly how precise our measurement is.

In this problem, we are tasked with determining the number of significant figures for three distinct numbers: $23.023$, $0.0003$, and $2.1\times 10^{-3}$. Each of these numbers tests a different rule of significant figures. Let us embark on a journey to decode these rules one by one.

The Trapped Zero

Our first candidate is the number $23.023$. To find its significant figures, we must recall the most fundamental rules.

First, all non-zero digits are always considered significant. This immediately tells us that the digits $2$, $3$, $2$, and $3$ are significant. But what about the zero sitting right in the middle?

This brings us to the second rule: any zero that is "trapped" between two non-zero digits is also significant. Why? Because if the digits on either side of the zero are reliably measured, the zero in between must also be a reliable part of the measurement. Therefore, the zero in $23.023$ is significant.

Counting them all up, we have five digits in total. Thus, the number $23.023$ has $5$ significant figures.

The Leading Zeros

Next, we turn our attention to the number $0.0003$. This number often trips students up. It is a number less than $1$, and it is packed with zeros.

The rule for numbers less than $1$ is strict: all zeros to the right of the decimal point but to the left of the first non-zero digit are merely placeholders. They indicate the position of the decimal point and the magnitude of the number, but they do not represent measured precision. These are known as leading zeros, and they are never significant.

In the number $0.0003$, the first three zeros after the decimal point are leading zeros. We must ignore them when counting significant figures. The only digit that represents a true measurement here is the final $3$.

Consequently, the number $0.0003$ possesses only $1$ significant figure.

Scientific Notation

Our final number is written in scientific notation: $2.1\times 10^{-3}$. Scientific notation is a brilliant tool used by physicists to handle extremely large or extremely small numbers without writing out endless strings of zeros.

The rule for scientific notation is beautifully simple. When a number is expressed in the standard form $A\times 10^B$, the number of significant figures is determined entirely by the coefficient $A$. The exponential term, $10^B$, only dictates the scale or order of magnitude and has absolutely no bearing on the significant figures.

In our case, the coefficient is $2.1$. This coefficient consists of two non-zero digits, both of which are significant. We completely ignore the $10^{-3}$ part.

Therefore, the number $2.1\times 10^{-3}$ has exactly $2$ significant figures.

We have meticulously applied the rules of significant figures to all three numbers.

For $23.023$, we found $5$ significant figures. For $0.0003$, we found $1$ significant figure. For $2.1\times 10^{-3}$, we found $2$ significant figures.

Putting it all together, the respective number of significant figures is $5$, $1$, and $2$. Looking at our options, this perfectly matches option (a).

Mastering significant figures is crucial for any aspiring physicist or engineer. It ensures that we communicate our measurements with honesty and precision, never claiming more accuracy than our instruments can provide. Keep practicing these rules, especially the tricky cases involving zeros, and you will never make a silly mistake on these fundamental concepts again!