 Hello and welcome to the session. In this session we will learn about complex numbers and urban plane. Now we know about real numbers where all the equations of first degree can be solved over the real number system. For example, 2x plus 3 is equal to 0 implies x is equal to minus 3 back 2 and similarly x square is equal to 4 implies x is equal to plus minus 2. So these equations can be solved over the real number system. But we cannot solve x square plus 5 is equal to 0 as this implies x square is equal to minus 5. So we cannot solve this equation over the set of real numbers as there is no real number whose square is a negative real number. Therefore the symbol iota is introduced minus 1 and iota square is equal to minus 1 and this symbol is also used as imaginary unit. So we have iota is equal to square root of minus 1. Now let us discuss the integral parts of iota. Now 2000 is equal to 1. iota squared is equal to minus 1 and iota cubed is equal to iota square into iota which is equal to, now iota squared is minus 1. So, minus 1 into iota will be equal to minus iota is equal to into iota square which is equal to minus 1 into minus 1 which is equal to 1. Now, let 2 into iota and let 3 to the power 4 on dividing let beta be the remainder. So, we have dividend which is n is equal to the divisor which is 4 into quotient which is n plus remainder. So, this is less than equal to r is less than 4 and n will be equal to iota raise to power now n is equal to 4n plus r so it will be equal to iota raise to power 4n plus r which is equal to iota raise to power 4n into iota raise to power r which is equal to iota now we know that iota raise to power 4 is equal to 1 and 1 raise to power m will be equal to 1. So, this is equal to iota raise to power r therefore the value of iota raise to power n for n is greater than is equal to iota raise to power r where r is the remainder on dividing because the negative integral parts of iota now iota raise to power minus 1 can be written as 4 into iota raise to power minus 1 as iota raise to power 4 is 1 now this is equal to iota raise to power 4 minus 1 that is iota raise to power 3 and iota raise to power 3 is equal to minus i this we have proved earlier can be written as iota raise to power 4 into iota raise to power minus 2 which is equal to iota which is equal to iota square and iota square is equal to minus 1 now iota raise to power minus 3 can be written as iota raise to power 4 into iota raise to power minus 3 which is equal to iota raise to power 1 minus 3 which is equal to iota raise to power 1 which is equal to iota iota raise to power minus 4 is equal to 1 over iota raise to power 4 now iota raise to power 4 is 1 so this this is equal to 1 by 1 which is equal to 1. Now for n is greater than 4, iota raise to power minus n will be equal to 1 over iota raise to power n. Now, so this is equal to 1 over iota raise to power r is the remainder when n is divided by 4. We can observe that iota raise to power n power 1 out of the 4 values i.e. minus iota minus 1 iota and now it is discussed complex numbers. Now we have already discussed the concept of iota. Now every number of the form is b iota where and b are real numbers equal to square root of minus 1 is known as complex number is denoted by now in this complex number a is called the real part of the complex number b is called the imaginary part of the complex number i.e. it can be written as the real part of z and b can be written as the imaginary part of z where z is equal to a plus b iota. Now let us see another example z is a complex number which is equal to 5 plus 6 iota then we have to form a plus b iota. So the real part of the complex number z will be equal to the imaginary part of the complex number z is equal to b which has some points about complex numbers. Therefore we can say it belongs to a set of complex numbers. Now for a complex number z which is equal to a plus b iota equal to 0 then real number and for a complex number z which is equal to a plus b iota equal to 0 then the complex number z will be equal to b iota imaginary number of z is equal to 0 to the imaginary number if the real part of z which is a is equal to 0 the complex number z is equal to 0 the real part of the complex number is equal to 0 and the imaginary part of the complex number is equal to 0. Now the complex number z1 is equal to a plus b iota and z2 is equal to c plus b iota c that is the real part of z1 is equal to the real part of z2 b is equal to d that is the imaginary part of z1 is equal to the imaginary part of z2. Now let us discuss an example. Now let the complex number z1 is equal to 5 plus y iota and the complex number z2 is equal to the real part of the complex number z1 is equal to the real part of the complex number z2 that is 5 is equal to x the imaginary part of the complex number z1 is equal to the imaginary part of the complex number z2 that is y is equal to 6. Now let us discuss the definition of complex numbers in terms of audit pairs. Now any complex number z which is equal to a plus b I have denoted by the audit pair audit pair is the real part of the complex number of the audit pair is the real coefficient as an example. Now the complex number z1 is equal to 39 can be denoted by the audit pair 3 minus 2. The audit pair 3 minus 2 is associated with the complex number 3 minus 2 iota. The real numbers is a way of naming a complex number. Now let us discuss the argon plane. Now every complex number in a wide plane the complex number can be represented by it can be represented by the unique point. Now the point x0 the complex number x plus 0 iota that is the real number x as the imaginary part of the complex number is 0 as a point on the axis as such the x of the real axis the point 0 on the y axis represents the complex number 0 plus y iota that is the imaginary number therefore the y axis are represented as points on this x. Now here the positive real numbers are represented by the points on the positive and the negative x axis is represented by the original code. And the imaginary numbers with the imaginary part of the complex number greater than 0 are represented on the positive y axis that is they are represented by the points on the positive y axis. And the imaginary numbers with the imaginary part of the complex number less than 0 are represented by the points on the negative y axis. The complex numbers the complex example in which we will represent this complex number on the argon plane. Now the complex number 3 minus 2 iota is represented as 3 minus 2 therefore this point is representing the complex number 3 minus 2 iota on the argon plane. We have learnt about argon plane so this completes our session hope you all have enjoyed the session.