This is a very simple question and I apologise, but I am so confused. I know that when removing the brackets with negative sign I have to change the signs.
Consider: $$10 - (3+5) = 2$$
If I want to remove the brackets, I know it would become:
$$10 - 3 - 5 = 2.$$
But why so?
Thank you and sorry for this very basic question.
On
It may be useful to think about this in the context of a number line. In your equation above
$$ 10-(3+5) = 2 $$
you might think of it as starting at position $10$ on the number line, and then moving left ("negatively") by a number of steps equal to $3+5$. That is equivalent to moving left first by $3$ steps, and then moving left again by $5$ steps, which we might represent by $(10-3)-5$, which is just $10-3-5$, which is just $2$.
Does that help?
On
The following are some of many fundamental properties of arithmetic:
Distributivity of multiplication over addition: $a\cdot (b+c) = a\cdot b + a\cdot c$
Multiplication by negative one transforms into additive inverse: $(-1)\cdot a = (-a)$
Subtraction is adding by additive inverse: $a-b = a+(-b)$
We have then:
$10-(3+5)=10+(-1(3+5)) = 10+((-1)(3)+(-1)(5))=10+(-1)(3)+(-1)(5)$
$=10+(-3)+(-5)=10-3-5$
On
The brackets mean: "do me first".
$$\begin{align}10 -(3+5) & = 10-(8) \\ &= 2\end{align}$$
So if we remove the brackets, we should change their contents so that the whole result equals the same thing. Without the brackets, the leftmost subtraction/addition is performed first.
$$\begin{align}10-3-5 &= 7-5 \\ &= 2\end{align}$$
So in general $-(a+b) ~=~ (-a)+(-b) ~=~ -a-b$
To better explain the reason, I think that writing the sign of every number can shed some light: $$10-(3+5)=10-[(+3)+(+5)]=10+(-3)+(-5)=10-3-5$$
Intuitively, if you have $10$ marbles on a table and you choose $3$ of them, then choose another $5$, and remove all the $8$ chosen marbles, is the same than remove $3$ and then remove $5$ more marbles.