Express in the Modulus-amplitude form (i) -i (ii) √(3)-i (iii) (1+i)(-1+i √(3)) (iv) ((1-i)(1+√(

Express in the Modulus-amplitude form (i) -i (ii) √(3)-i...

Key Concepts

Complex Numbers

Complex numbers are expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. They are foundational in many areas of mathematics because they extend the real numbers and allow for the solution of equations that have no real solutions.

Modulus (Absolute Value)

The modulus of a complex number is its distance from the origin in the complex plane. For a complex number a + bi, the modulus is calculated as ?(a² + b²) and represents the magnitude of the number.

Argument (Amplitude)

The argument of a complex number is the angle that the line connecting the origin to the point representing the number makes with the positive real axis. This angle, measured in radians, indicates the direction of the complex number in the plane.

Polar Representation

The polar or modulus-amplitude form of a complex number expresses it in terms of its modulus and argument, typically written as r(cos? + i sin?) or equivalently as re^(i?). This form simplifies operations like multiplication and division, which become straightforward when the numbers are represented in polar form.

Euler's Formula

Euler's formula states that e^(i?) = cos? + i sin?. This fundamental formula connects exponential functions with trigonometric functions, and it is instrumental in converting a complex number from rectangular form to polar form.

Transcript

00:01 So in the given question we are told to express the following complex numbers in the modulus amplitude form.

00:08 In the modulus amplitude form.

00:13 So what is this form? so this is the form of writing a complex number of the type x plus iy in r times cost theta plus i.

00:29 Sine theta terms right so if you write if we write the complex number x plus i y in this form then we can call this we can call this form the modulus amplitude form right over here we can see that we can write x plus i y is equal to r cost theta plus i times r sine and we compare the x and y terms x is equal to r cost theta and y is equal to r sine theta and we can find the value of r by taking square root of x squared plus y square right so these are the things we need to know about the modulus amplitude form so in the question we have several complex numbers and we are told to find they are told to write the these complex numbers in the modulus amplitude form.

01:28 So the first complex number that is given in the question is minus i.

01:35 So we can write this complex number as 0 plus minus 1 times i or i times minus 1.

01:44 So over here x is equal to 0, y is equal to minus 1.

01:50 So r can be found as square root of 0 square plus minus 1.

01:55 Square which is equal to 1 so r is equal to 1 right now we need to find theta also right so theta can be found as tan inverse of y by x tan inverse of y by x so we should write this in a box also since this is a property of the modulus amplitude form so what we can find from this is theta is equal to tan inverse of y which is minus 1 divided by x which is 0.

02:32 So minus 1 by 0 is not defined and this value of this value of the tan inverse comes when theta is equal to minus pi by 2 right when theta is equal to minus pi by 2, sign gives us the value of minus 1 and cost gives us the value of so when we take tan of minus pi by 2 we get minus 1 by 0 as the fraction so then we can write theta as minus pi by 2 and now we can write the modulus amplitude form as x would be equal to 0 so r is equal to 1 right so 1 times cos theta which is cos by minus pi by 2 plus i sign of minus pi by 2.

03:31 So this is the required modulus amplitude form and we can simplify this and write it as cos pi by 2.

03:39 Pos of minus theta is cost theta.

03:43 So cost of pi by 2 and sine of minus minus sine theta is equal to minus sine theta.

03:49 So cost pi by 2 minus i sine by 2 is the required modulus amplitude form of the first complex number given in the question.

03:59 Now let's move on to the second number, second complex number which is given as root 3 minus i.

04:09 So over here we can write this complex number as root 3 by 2 minus 1 by 2 i.

04:17 Right? and now what we can do with this is we can take root 3 by 2.

04:24 We should know the value of cost pi by 6 as root 3 by 2 and sine pi by 6 is equal to 1 by 2.

04:38 So over here what we should do is we can write this as 2 times cos pi by 6 minus i sine pi by 6.

04:51 Right so this would be the required modulus amplitude form of the given complex number root 3 minus i right so when we take this what we should know is we can write this as we can take the angle minus pi by 6 instead of pi by 6 and write this as cos pi by 6 plus i sign minus 5 by 6.

05:25 So we can write it like this and then this would be the required modus amplitude form of the given complex number root 3 minus 7 right so next up we have the complex number which is given as 1 plus i times 1 plus i times minus minus 1 plus i root 3.

05:58 So in this complex number what we can do is to first write 1 plus i in the modulus amplitude form...

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