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The Negative Double
The negative double is a vital convention for players who open 5-card majors.  Without this convention, there is a danger that the partnership will not find available 8-card major suit fits.

The most common situation which creates a need for the negative double is when partner opens 1♣ or 1 and your RHO overcalls 1♠.  As responder, you may have wanted to bid 1, but your opponent’s bid prevents that call.  A 2 bid requires a 5-card suit and 10+ points.  If you have 4 hearts or even 5 hearts without the required 10+ points, you will be stuck for a constructive bid.  This is the situation the negative double helps resolve.
Consider this auction:

East (partner) South   West (you)
1♣/1                1        Double

If you and your partner are playing negative doubles, your double says: “Partner, I wanted to bid hearts, but my hand does not allow me to bid them at the two level.”
The bid promises at least 4 hearts.  The remainder of your hand is still undefined.  You may have support for partner’s minor suit, the unbid minor, or perhaps 5 hearts but lacking the required 10 points.  
Consider the following auctions:  Imagine you are sitting North.

North        East     South       West
1 ♣/1     1 ♠       double*

North        East     South         West
1 ♣/1    1       double**

* I have four hearts and at least six points.
** I have four spades and at least six points.

North        East     South       West       *I have 5 hearts and at least
1 ♣/1     1 ♠       2 *                            10 points

North        East     South         West       ** I have 5 spades and at
 1 ♣/1    1       1♠**                              least 6 points.  

Consider Example 1 and 2 below.

         Example 1                                             
  7    K     6    Q
  6    J      5    8
      5     3     6
      3     2

        Example 2
  K    A    K    Q
  2    Q    5    8
       J     3    6
       4    2
You are South and your partner has opened the bidding 1, your RHO overcalls 1♠.  With each of these hands, you intended to bid 1, but your opponent’s bid prevents that call.  A negative double allows you to show partner your four-card heart suit.  With example 1, the auction may look like this:

North (partner) East   South (you)   West
 1                    1♠         Double        ???

West will bid whatever his hand calls for.  She may raise East’s spades or pass (or even, perhaps, bid a new suit of her own).  Regardless, of what West does, North will know that you have 4 hearts and at least 6 points.

With example two, the auction will be the same, but note the strength of your hand.  In this case, West will most likely pass.  After North bids you will place the partnership in the appropriate game.  

The Negative Double at the two level.

The negative double can occasionally be used at the two level.  Consider the following hand:

   K    7     K     J
   Q    6     5    8
   8            3    6
   3           2

The auction is as follows:

North (partner) East   South (you)   West
  1                    2        Double        pass
Your negative double tells partner that you have four spades and 8+ points.  Think carefully about the auction and you will see how helpful the negative double convention is in this situation.

Consider it from North’s perspective.  If North has four spades he can bid spades at the two level.  Most important suppose North does not have spades and must rebid his heart suit.  North can make his rebid at the two level.   
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