significant figures in a product

Question

If I take a digit like $5$ (say in m) and multiply it by a digit say $5$ (in m) then I will get $25$ ($\mathrm{m}^2$---an area). Now, $5$ can be expressed in Scientific notation as $5 \times 10^0$ to show that it has $1$ significant digit. But then what would the product result $25$ be represented as when expressed to $1$ significant figure using the Scientific notation? $3 \times 10^1$ means the product answer is $30$ and is not $25$. Now $30$ m is far from the correct $25 \; \mathrm{m}^2$.

I have seen the rule that a product of two numbers cannot have more significant digits than any of the multiplicand or the multiplier. And I have read that it is good to use the Scientific notation when working with significant figures to avoid spurious significant digits.

Answer 1

$30\ \mathrm{m}^2$ is correct to one significant figure. The fact that it's far from $25$ is simply because one significant figure isn't very precise.

Scientific notation isn't particularly helpful with numbers such as $25$ or $30$. It becomes more useful when we want to express very small or large numbers succinctly. As an example, $3\times 10^{21}$ is easier to read than $3000000000000000000000$.

Answer 2

Suppose for the moment that measuring $5$ m could somehow contain a possible error whereby a more precise measuring tool would yield $5.8$ m. Suppose as well that this is multiplied: $5.8\cdot 5.8= 33$. Clearly $25 < 33$, but we want to ensure our data it consistently read and therefore a rule is adhered to for some uniformity with which we can rely on. $30$ satisfies the supposition that only one significant figure is being used and $25$ does not.