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if we are told to find the difference between 3 and 5,
then we usually subtract 3 from 5 ,5-3=2 and thus, we say that the difference is 2. but why can't we subtract 5 from 3 ,3-5= -2 to get the difference -2?? which result is right? is the difference ( 2) or (-2)?
Also tell me how can we calculate the difference if we are told to calculate difference between two numbers,-5 and 2 on the number line.
Traditionally, the “difference" between two numbers refers to the distance on a number line between the points corresponding to each of the two numbers, a.k.a. the absolute value. Analogously, if you asked “What is the distance from Toronto to Vancouver?” or "What is the distance from Vancouver to Toronto?", you would expect the same answer: the [positive] distance separating the two cities, regardless of the direction of travel.
On the other hand, if asked “What is the result when you subtract 3 from 5?”, you should give a different answer (2) than if you were asked “What is the result if you subtract 5 from 3?” (-2).
As for calculating on the number line:
The term you're looking to use is "absolute value."
Imagine the number line as a single axis of a graph plane; a dimension, rather. Number lines are commonly represented in whole numbers, each number representing a scaling increment of that line. Counting the total, positive amount increments between two numbers on the same number line will give you a hands-on approach at discovering absolute value. Absolute value looks like this:
$|-4| = 4$
$|9| = 9$
Positive and negative numbers come out as positive after determining absolute value.
Like usual, a number line will have negative numbers (numbers less than 0) to the left of 0, and positive numbers (numbers greater than 0) to the right of 0. Imagine a placeholder on your starting number which we'll say is $-5$. When subtracting a positive number from any number in this case, you will move your placeholder left. Subtracting $3$ from $-5$ will move your starting placeholder left two increments to $-8$.
Conversely, subtracting a negative number from any number becomes addition. If our placeholder starts on $3$, this will move it to the right towards more positive numbers. When subtracting $-5$ from $3$, we tell $3$ to no longer be burdened by losing $5$ from its value, and so the $-5$ is taken away.
I hope this paints a decent picture. I can edit this later with some Photoshopped number lines and other visuals if need be.
As stated before, finding the difference between two numbers, is just to subtract them. So if you want to find the difference, you take the bigger one minus the smaller one.
But if you want to find the $distance$ between two number, you use the absolute value. For example, if you want to find the angle between two angles, lets say the angle between $90°$ and $30°$, we know it is $60°$ because the bigger one, $90°$, minus the smaller one, $30°$, is $90°-30° = 60°$. But what if you dont know which one is bigger? For example if you want to express the "difference" in the general case, using variables, you have to account for the fact that you don't know which is bigger, so you don't know if you should write $θ-ϕ$ or $ϕ-θ$. Then, to express that you want to find the difference between $ϕ$ and $θ$, you can write $|θ-ϕ|$, because when you take the absolute value of the whole expression, the order doesn't matter.
And indeed, $|90°-30°| = 60° = |30°-90°| = |-60°| = 60°$.
In short, to find the difference, subtract the bigger one from the smaller one, to find $distance$ between the numbers on the number line, take the absolute value of one minus the other, because when you take absolute value of a subtraction, order doesn't matter.
As I understand it, the difference between two given numbers is a result of subtraction, so it depends on the order of data:
the difference between $5$ and $3$ is $5-3 = 2$, while
the difference between $3$ and $5$ is $3-5 = -2$
However the distance on the number line is a modulus (an absolute value) of the difference, hence independent on the order:
$dist(3, 5) = |3-5| = |-2| = 2 = |2| = |5-3| = dist(5, 3).$
The difference between the plus and minus sign is that the plus sign in the difference implies a growth and the minus sign implies a decrease. That assumes that you are calculating a difference between two numbers that show the time variation of a magnitude.
If just the difference between two numbers want to be observed, then you should get the positive value and define the difference between two numbers as the absolute value of their difference, obtaining always the positive value. In the -5 and 2 example the difference would be $|-5-2|=|2-(-5)|=7$.
The difference between two numbers on a number line is the distance between them and it varies whether you are going from left to right or right to left. The commonly used way is to go from right to left, which gives us a positive number.
For calculating difference between -5 and 2, plot both of them on the number line. The distance between them is |-5|+|2| = 7. If you go from right to left, the answer is 7 and it is -7 if we go left to right.
Hope it helps.
In my opinion the word “difference” can have different meanings depending on context.
We (sometimes) call $x-y$ a difference even though it can be negative. But if you ask for the difference between $x$ and $y,$ it’s not clear which to subtract from which. So in that context “difference” is usually distance as others have said.
But if I said we had sales of $3000$ dollars last month and $2000$ this month, and I ask for the difference, I’m probably talking about how the sales amount changed. And describing the difference with a negative number tells me that the sales decreased rather than increased, which I think is rather important.
So I think usually the absolute value is desired, but not always.
