Subtraction with a negative result

My son was doing a math problem and getting the wrong result, so he asked me to help out. The problem was:

$4.5-8.2 = x$, solve for $x$

He was getting $-4.3$, where the correct answer is of course $-3.7$.

It turns out the reason he got the answer wrong is because he was doing the subtraction directly, and when he subtracted $.2$ from $.5$ he got $.3$. I showed him that he could multiply both sides by $-1$ and solve for $-x$, negate the answer and thus get the right result. Being the curious kid that he is, he asked why the two results weren't the same, and I couldn't give him an answer.

Why doesn't the direct subtraction work?

Being the curious kid that he is he asked why the two results weren't the same, and I couldn't give him an answer.

That's because the two results are the same, and he is implicitly using a slightly different and context-dependent notation to express his answer.

The arithmetic is correct, but $-4$ is not a decimal digit in the usual scheme of things.

A correct answer of $(-4).3$ was found, with an intended meaning of $-4 +0.3$. That notation is non-standard, and writing it as $-4.3$ gives the wrong answer when read as a standard decimal.

Although it's clear what an expression like $(-4).3$ should mean here, to represent that result in the standard system with digits 0-9, the minus sign can only apply to all digits in the number at once. The conversion to standard notation is $("-4").3 = -(3.7) = -3.7 $

He's trying to solve this problem:

$$\begin{equation} \frac{ \begin{array}[b]{r} 4.5 \\ - 8.2 \end{array} }{ } \end{equation}$$

The $.5-.2$ part doesn't cause any heartburn, but when he gets to the $4-8$ part, he'll need to borrow from the tens column. But there's nothing to borrow from!

When performing the subtraction algorithm, the minuend has to be greater than or equal to the subtrahend.

It seems the minus in \begin{align*} 4.5-8.2 \end{align*} looks so dominant. We are tempted to think

• at first we subtract $8$ from $4$ giving $4-8=-4$

• and then we have to subtract $0.5-0.2$ resulting in $-4-0.3=-4.3$

But the correct way is

• at first we subtract $8$ from $4$ giving $4-8=-4$

• and then we have to add $0.5-0.2$ resulting in $-4+0.3=-3.7$

We have to add the values since \begin{align*} 4.5-8.2&=(4\color{blue}{+}0.5)-(8\color{blue}{+}0.2)\\ &=(4-8)\color{blue}{+}(0.5-0.2)\\ &=-4\color {blue}{+}0.3\\ &=-3.7 \end{align*}