Writing A Number In Floating Point With $5$ Significant Digits

Question

Write the number $496354.1$ with $5$ significant digits by chopping and rounding

Now for chopping we can write it as $0.49635*10^5$ or $4.9635*10^4$ which is the correct way?

Answer 1

Beware! Your initial attempts are off by one power of 10.

Let $x = 496354.1$. The normalized scientific representation is $x = 4.963541 \times 10^5$. Rounding to 5 significant figures yields $\hat{x} = 4.9635 \times 10^5$, as the tail, i.e., $0.41$ is strictly less than $\frac{1}{2}$. In this case, chopping yields the same approximation, $\bar{x} = 4.9635 \times 10^5$.

The difference between rounding and chopping is better illustrated in the case of $4$ significant figures. Here $\hat{x} = 4.964 \times 10^5$ as the relevant tail, i.e., $0.541$ is greater than $\frac{1}{2}$. Chopping produces the approximation $\bar{x} = 4.963 \times 10^5$.

Let $y = -4.35218$. This number is already written in normalized scientific notation. There is no rule stating that we must explicitly write the exponent, i.e., $y = -4.35218 \times 10^0$ and it is not an error to omit it. Rounding $y$ to $4$ significant figures produces the approximation $\hat{y} = -4.352$. Chopping produces the same result $\bar{y} = -4.352$.