In the mathematics book of 6th grade, there is a chapter about fractions and grouping operators like (), {}, [] and fourth – the horizontal bar ________ which is placed over a math expression like the example given below:
$$x=2\cdot\left[1/2+\left\{4-\left(3\times\overline{2+3}\right)\right\}\right]$$
So far, my understanding was that the bar is just a grouping operator as brackets, but has the highest priority to solve first of all. But I was challenged by a math teacher: he said that not only solve it first, but also the signs will be converted to their opposite signs (like plus to negative and multiplication to division). This means that to solve the given example above, it will take
$$3 \times (-2 – 3) \to 3 \times -5 \to -15$$
And I was thinking that it should be resulted as $3 \times 5 \to 15$.
I could not find neither any reference to prove my idea nor to verify his idea.
Can any one here please come up with some references about how to solve the expression with bar over them?
Best Answer
Compiling various comments into an answer, by request.
The bar is formally called a vinculum. (In TeX mathematical markup, it's called
overline; I think most people would just call it a "bar".) It's used for many things, but nowadays only rarely as a grouping symbol, probably because we now have the facility to typeset ever-larger parentheses to satisfy our grouping needs. (I'm surprised a modern-day textbook even has exercises with such notation.) It's worth noting that the square root symbols was initially just "$\sqrt{}$", and a vinculum grouped the terms to which the square root applied. Since then, the vinculum has become a part of the square root symbol itself.To the extent that it is used as a grouping symbol, the vinculum, in and of itself, doesn't indicate "highest priority" over other groups. In the problem at hand, you might be inclined to find value under the vinculum first, but only because it's the "deepest" grouping.
I'm not sure what your instructor means about the signs. If you were to follow deepest-to-shallowest grouping order, and standard order of operations, you'd have
$$2[1/2+\{4−(3\times\overline{2+3})\}]\quad\to\quad 2[1/2+{4−(3\times 5)}]\quad\to\quad 2[1/2+\{4−15\}]\quad\to\quad\cdots$$ (which seems to be your interpretation). Perhaps the instructor was talking about distributing the "$-$" from the subtraction, but that seems a lot more complicated than just adding $2+3$ and moving on.
I'll point out (for those who may not know) that, a bit later in mathematics, one encounters "complex numbers" that have the form $a+bi$ (where $i$ is the so-called "imaginary unit", but that's not important here). In that context, the vinculum is used to represent a notion called "conjugation", which changes one sign:
$$\overline{a+bi} = a-b i$$
I suppose that there's a (very remote) possibility that your instructor is mis-interpreting the vinculum as the conjugation operation, and then also mis-applying that operation to change the sign of everything under the bar. [According to a follow-up comment, this is exactly what happened!]