Value of pieces: beginner question

I am sorry if my question sounds stupid but I must ask: why is a rook (5) worthier than a bishop or a knight (3) in terms of exchange ? I mean all of these pieces seem to be on the same level to me, they just have different movement possibilities and utility.

If one had to be worthier I think it would be the knight because of its ability to jump over any other piece on its way and its particular "L" movement which can set up terrible forks as well as making it the only piece able to threaten the queen without putting itself in danger.

So why is a rook worthier according to this points system ?

Thank you.

PS: I am sorry for any mistake, english is not my native language.

Well, just look at how many max. squares an R may attack or occupy. Let's say it's on A1. It can move (obviously, if unobstructed) from A1 to A8. Or from A1 to H1. That's 15 squares. 

Its all about the number of squares a piece can cover (attack, defend, threaten), as e4nf3 points out. Put any piece on the board and count the number of squares it can reach, this will give you a relative strength of the piece concerned.

The relative values of the pieces express the average worth of the pieces if you examine millions of games.

In some positions a knight simply is winning over a rook, making it's relative value infinite.

In other positions a bishop will crush a knight.

And so on.

But the averages are a great guideline when it's impossible to use anything but a best guess to evaulate a position precisely because they tend to hold.

Well why not compare the two, using pros and cons?

Knight:

Pros:

It can jump pieces

It is useful for forking pieces when facing an opponent who isn't careful

It is great in closed positions

Cons

It is a short range piece

In open positions, bishops can dominate it

it takes several moves to move from one square to another (a specific square, so like from g1 to f6)

Rooks:

Pros:

Long range piece

Great for attacking weak pawns

Useful for setting up attacks

Useful defensive piece

When used with the other rook, you can totally dominate the position

Cons

Can't jump pieces

not very helpful at the beginning of the game or until the position is open

So, if we compare the two, we see that knights are better when the position is closed and there are no open/semi open files, while a rook is better when the position is open due to a long range effect. This rule is not always true, it depends on the position, but it can be a very general rule. 

A rook can also play a very useful role in an endgame: he can set up an invisible fence on an entire rank or file which keeps the enemy king stuck far away from the action. Then it's almost like you have an extra piece.

Another powerful role a rook can play in an endgame is both protect a pawn where it stands and after it moves.

King and rook can checkmate a sole king. That's impossible with bishop or knight.

If you compare the two sliding pieces rook and bishop, you'll notice that the bishop is restricted to the squares of one colour. Hence the rook is better.

The knight has the big disadvantage that he has to abandon all squares he attacks/protects when forced to move.

Thank you all for your answers, there's so much useful advice in there altogether. And sorry for the late thanks.

Unless it's a pawn or knight endgame, and in some other situations... then rook and knight pawns are much more valuable than center pawns heh.

I think the given number values were calculated by the sum of number of squares the piece can cover when standing on every single square of the chess boards, then dividing it by 64 (the number of squares on the chess board), forming an average. Those averages are then simplified to lowest terms.

Interesting if that's how they started.  I wonder what these numbers would be?

Also interesting is that over many years the values humans had settled on were very close to reality (at least when computers crunched the numbers for many thousands of positions).

Rook is a powerful piece because it can checkmate an enemy king by himself. Bishop and knight can't do this by themselves.

Besides not being able to cover as many squares as a rook, a bishop is also weaker because it is limited to only one color.  A rook is also more powerful in the endgame because once it lines up on the same file as a pawn, the pawn cannot escape its attack by moving.  The knight is weak on an open board because it has very short range and few squares it can move to can be easily covered.  A knight at the edge of the board can be stalemated be either a rook or a bishop, taking it out of action.

One other thing is that a knight can attack squares of either colour, which a bishop cannot do.

If that was the case a bishop would be worth much more than a knight.

But only the Pope can truly pontificate.

Well, I'll give it a go.

Since a rook covers 14 squares no matter which square it is on, value of rook equals (14*64)/64, which equals 14.

To calculate the value of a bishop, the chess board will have to be divided into four eccentric squares. A bishop located in each square that makes up the sides of the outermost eccentric square (A) controls 7 squares. In square B, it can control 9. C, 11 squares. D (the centre squares), 13 squares. So the value of the bishop equals (28*7+20*9+12*11+4*13)/64, or (196+180+132+52)/64, which equals 8.75.

A knight involves more individual calculation. Going up the files, a knight on the squares in the a-file control 2+3+4+4+4+4+3+2, or 26 squares. For the b-file, those numbers are 3+4+6+6+6+6+4+3, or 38 squares. For c- to f-files, the numbers are 4+6+8+8+8+8+6+4, or 52 squares. The g- and h-files have the same figures as the a- and b-files. Those numbers bring the value of the knight to (2*26+2*38+4*52)/64, which equals 5.25.

A queen is very easy now that I have the values for the rook and the bishop - I just add the two together. 14+8.75=22.75

Queen: 22.75

Rook: 14

Bishop: 8.75

Knight: 5.25

The value of the pawn equals (7*12)/48 (because the pawns cannot stand on the first rank, and the pawn is no longer a pawn on the eighth rank), which equals 1.75.

So if the value of the pieces are divided by the value of the pawn, you get

Queen: 13

Rook: 8

Bishop: 5

Knight: 3

Pawn: 1

However, this is based purely on an empty board, if I were to place other pieces - or even a position - on the board, and calculate individual pieces and combinations of them, it would provide a much more accurate number to rate the chess pieces' abilities.

Can anyone check my calculations for any errors? Thanks!

And if I went further by placing more obstacles on the board, the value of the bishop would go down considerably, while the knights' value would not change as much, for example.