# Reasonable rounding

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?