Adding numbers in AngularJS

No, it shouldn't. The controller function is executed once, when angular sees the ng-controller in the DOM. It's not executed every time you type something in the input boxes. So the code computing the result is only executed once, before the user has had a chance to type anything in the inputs.

You don't need the keyup, though (and that wouldn't work if you pasted the value with your mouse, BTW). All you need is

$scope.add = function(){
     return (Number($scope.a || 0)+ Number($scope.b || 0));
}

and in the view:

<span>{{ add() }}</span>

When doing it that way, angular will reevaluate the add() expression after each event that modifies the model, and update the DOM with the result of the expression.

If you want to compute the result only when a or b changes, you should be using ng-change, not ng-keyup.

Note that if your inputs are supposed to contain numbers only, you should use an input of type number, not text. In that case, $scope.a and $scope.b will be of type number, not string.

scope variable (a) and (b) are being updated since they are two-way binded.

variable a1 and b1 won't get the update since they only get updated when you initiate your controller "calc" and they are not two-way binded.

To update your result scope variable you can use ng-change to set variable a1 and b1 with updates from $scope.a and $scope.b

<div ng-app="cal">
<div  ng-controller="calc">


<input type="text" ng-model="a" ng-change="updateResult()"/>
<input type="text" ng-model="b" ng-change="updateResult()"/>

<span>{{result}}</span>
</div>

/**
 * Created by rajat on 12-Nov-16.
 */
var app=angular.module("cal", []);


app.controller("calc",function calc($scope) {
    $scope.updateResult = function () {
      var a1 = Number($scope.a || 0);
      var b1 = Number($scope.b || 0);
      $scope.result=a1+b1;
    }
});