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?
 A: 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*}
A: how about draw the number line? That helped me when I was little. When 4.5-8.2, you first -4.5 to get to the 0 point, and you need another part keeping going to reach the total length 8.2. so that will be 8.2-4.5=3.7 away from 0 on the negative side. Hope it will help.
A: 
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 $
A: 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.
A: Performing subtraction one has to do two calculations. At first to determine the sign of the difference by comparing the sizes of the minuend and the subtrahend. Example: the sign of
$$3-5$$
is negative as $5$ is greater than $3$.  So by now we have
$$3-5={-}$$
And know you calculate how much $5$ is greater than $3$, namely $5-3=2$. So finally 
$$3-5=-2.$$
As for $4.5-8.2$ we have firstly a negative sign and, as $8.2-4.5=3.7$, we arrive in $-3.7$.
A: He subtracted the decimal part twice, once to get the 0.3 and then again to go from -4 to -4.3 -- he should have added the 0.3 to -4 to get the final answer.
You can subtract wherever you want, but you must only do it once. Otherwise, you're adding rather than subtracting.
A: Expanding on @John's answer https://math.stackexchange.com/a/1948174/169560. You can borrow from "nothing" you just need to remember to give it back:
\begin{equation}
\frac{
    \begin{array}[b]{r}
       4.5 \\
      - 8.2
    \end{array}
}{}=
\frac{
    \begin{array}[b]{r}
       \ ^{-1}14.5 \\
      - 8.2
    \end{array}
}{}=
\frac{
    \begin{array}[b]{r}
       \ ^{-1}14.5 \\
      - 8.2
    \end{array}
}{-10+6.3}=-3.7
\end{equation}
This way, if the minuend is smaller than the subtrahend, you will end up with something like $-100 \ldots 000 + YY \ldots YY.yy \ldots yy$ where $x$, $Y$ and $y$ stand for any digits, and the number of zeroes is the same as the number of $Y$s. It should be easy to do this last step, as you just need to pad to a potence of 10 - and you train questions like $100 - 22 = ?$ (which some people answer wrongly as $88$) as a bonus.
A: The standard subtraction algorithm only works when the larger-magnitude number is on top (in this case, the $8.2$). The most straightforward calculation is to note that the result here will be obviously negative (because the number being subtracted is of larger magnitude), and then perform $8.2 - 4.5$ for the magnitude of the result.
Failing to pay attention to which magnitude is larger results in exactly this kind of subtraction error. 
A: Well, too bad nobody uses tape decks these days any more since this is a case of "tape deck counter arithmetic" (which is, as mentioned in other answers, 10s complement arithmetic).  999, 998, 997 ... are dual representations of -1, -2, -3.  Your son arrived at 3 for the last digit which is the dual of -7.  And in tape counterese, his result was 996.3 which is the dual of -3.7: if you forward your tape from a counter setting of 996.3 by 3.7, it will be zero again.
Whether the last digit is really 3 or 7 is something which you can only decide once you actually know whether the result as a whole should be positive or negative, and you don't know that yet when starting from the right.
A: I think that John above was onto something: "There was nothing to borrow from".  The real problem is that the subtraction was never completed.   If it were, the result would be an infinitely long string of leading nines.  
In early calculating machines, even soroban and addiator, subtraction was handled by tens complement addition.   
