The first time I ever corrected a teacher ((You would be right to conclude that it was far from the only time)) was when Mr Hawkins – an absolute legend of a teacher, don’t get me wrong – tried to explain that 3.45 rounded up to 4 because you’d round up to 3.5 and then up to 4.

Nine-year-old me was having none of it. I can’t remember if I was articulate or even made any sense, but I protested in the absolute certainty I was right ((A certainly only ever matched during the debacle involving Mrs Hannent and a rounders ball that was very clearly below my knees.))

So, despite rounding not really being maths so much as an arbitrary convention, I have a soft spot for such problems.

On reddit, someone asked: “What is the correct answer to $1.472 \times 10^{-7} + 4.32 \times 10^{-9}$?”

The expected answer

I see a certain amount of logic behind the answer the student was meant to give of $1.52\times 10^{-7}$. The thinking, I assume, is that since the less-accurately stated number is correct to three significant figures, that’s how precisely we should state the other.

While I see the logic, I simply flat-out disagree with it. $1.23\times 10^7 + 4.567\times 10^{-7}$ isn’t $1.230\times 10^7$, there’s absolutely no justification for a fourth sig fig there.

Smartarse answer #1

First, change the standard form so you have the same exponent: I get $147.2 \times 10^{-9} + 4.32 \times10^{-9}$. (I multiplied the number bit by 100 and divided the exponent part by 100, leaving me with the same number written differently.)

Adding these gives $151.52 \times 10^{-9}$, or $1.5152\times 10^{-7}$. This is the unambigiously correct answer, since there is nothing in the question to suggest there has been any rounding at all. However, I guess there is some context I’m missing and we’re actually meant to make rounding decisions.

Sensible answer

I think the “conventional” approach is to round using the smaller number of decimal places once you’re using the same exponent. 147.2 has only one decimal place, so that’s as far as we should take it – giving an answer of $1.515\times 10^{-7}$.

Smartarse answer #2

Given that the numbers we have are rounded, we can say that the larger number is between $147.15\times 10^{-9}$ and $147.25\times10^{-9}$. The smaller is between $4.315\times10^{-9}$ and $4.325\times 10^{-9}$. Their sum is between $151.465\times10^{-9}$ and $151.575\times10^{-9}$.

The most precise number whose bounds contain both of those possibilities is $1.5\times 10^{-9}$, and I don’t think you can give a more accurate answer than that.

Any other suggestions or approaches?