 Hello and welcome to the session. In this session we shall discuss that in complex numbers, the midpoint of a segment is the average of the numbers at its end points. We will understand this with the help of an example. Let us consider two complex numbers say z is equal to minus of 4 plus iota and w is equal to 2 plus 5 iota. Let us plot these complex numbers in complex plane. Here is the complex plane where x represents real axis and iota y represents imaginary axis. Now we can represent this complex number by the point p with coordinates minus 4, 1 and this complex number by the point q with coordinates 2, 5. Now we shall plot these points in the complex plane. Point p has coordinates minus 4, 1. So we mark this point as p and it represents the complex number minus 4 plus iota. Similarly we can mark point q with the coordinates 2, 5 here and it represents the complex number given by 2 plus 5 iota. Now we join origin o with point p as well as point q and we get o p and o q and now we have to find the midpoint between these two complex points that is point p and point q. For this let us complete the parallelogram o p, r q and draw its diagonals p q and r o. Now we see that p q is one of these diagonals and it is the line segment joining two complex points p and q. Now we shall find the midpoint of p q and we know that diagonals of a parallelogram bisect each other. So let point m be the midpoint of both the diagonals. Thus we say that m is midpoint of p q. Now from the plane we can see that point m is given by the coordinates minus 1, 3. So midpoint of p q is given by m with the coordinates minus 1, 3. Also as a brightly we know that if x 1, y 1 and x 2, y 2 are any two points in a plane then its midpoint is given by x 1 plus x 2 the whole upon 2, y 1 plus y 2 the whole upon 2. So here also p and q are two points in a plane which coordinates x 1, y 1 as minus 4, 1 and x 2, y 2 as 2, 5 then its midpoint m is given by x 1 plus x 2 the whole upon 2 that is minus 4 plus 2 whole upon 2, y 1 plus y 2 the whole upon 2 that is 1 plus 5 the whole upon 2 and this is equal to minus 2 upon 2 6 upon 2 that is minus 1, 3 which is same as we had found on the plane. Thus we say that midpoint of a segment is average of the numbers at its end points. So midpoint is given by average of x coordinate average of y coordinate here midpoint m with coordinates minus 1, 3 represents complex number that is minus 1 plus 3 iota. We can also find the complex number given by midpoint directly by taking average of the two complex numbers. We were given two complex numbers z and w. Now to find the midpoint of these two complex numbers in the complex plane we can directly take the average of these two complex numbers that is we have z plus w whole upon 2 which gives the average of the two complex numbers. So we get minus of 4 plus iota the whole plus 2 plus 5 iota the whole whole upon 2 now grouping real and imaginary parts in the numerator we get minus 4 plus 2 the whole plus iota plus 5 iota the whole whole upon 2 and this is equal to minus 2 plus 6 iota whole upon 2 which can be written as minus 2 upon 2 plus 6 iota upon 2 which is further equal to minus 1 plus 3 iota. Thus we have got the average of two given complex number as the complex number minus 1 plus 3 iota so the midpoint is given by the complex number minus 1 plus 3 iota or on plane it is represented by the coordinates minus 1 3. Thus in this session we have learnt that in complex numbers the midpoint of a segment is the average of the numbers at its end points this completes our session hope you enjoyed this session