{"id":638,"date":"2018-04-22T20:43:37","date_gmt":"2018-04-22T15:13:37","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=638"},"modified":"2023-08-14T12:38:39","modified_gmt":"2023-08-14T07:08:39","slug":"polar-form-of-complex-numbers","status":"publish","type":"post","link":"http:\/\/ibelitetutor.com\/blog\/polar-form-of-complex-numbers\/","title":{"rendered":"Polar Form of Complex Numbers"},"content":{"rendered":"

Polar Form of complex numbers<\/span><\/h2>\n

In the previous post, Our IB Maths Tutors<\/a><\/strong> discussed the basics of complex numbers. This is the second and final post on complex numbers. Here we shall discuss, polar form of complex numbers
\nWe consider complex numbers like vectors and every vector must have some magnitude and a certain direction. If we write a complex number in form of a point in the Cartesian\u00a0plane\/ordered a pair like x+iy=(x,y) <\/strong>then the distance of this point from the origin (0,0) is equal to the magnitude of our complex number while the angle it’s making with x-axis will show it’s direction.\"<img<\/a><\/p>\n

In the above right triangle, using Pythagoras\u00a0theorem<\/p>\n

\"{r^2}\u00a0so the magnitude of the complex number\/vector<\/p>\n

\"\\left|<\/p>\n

\"\\left|\u00a0 \u00a0 \u00a0and in the same triangle,\"\\tan\u00a0 \u00a0\"Sin\\theta\u00a0 \u00a0 \u00a0<\/span>\"Cos\\theta<\/p>\n

<\/p>\n

and\u00a0 \u00a0 \u00a0 \u00a0\u00a0\"\\theta\u00a0 \u00a0 here\u00a0\u00a0\"\\theta\u00a0represents the direction of complex number\/vector. This is called the argument of a complex number<\/p>\n

\u00a0 \u00a0\"z<\/p>\n

\u00a0 \u00a0 \u00a0\" <\/p>\n

\u00a0 \u00a0 \u00a0\"<\/p>\n

<\/p>\n

<\/p>\n

\"z\u00a0this is called the polar form of complex numbers z. Magnitude\u00a0\u00a0\"\\left|\u00a0and argument\u00a0\u00a0\"\\theta\u00a0 can be calculated using above-mentioned formulas. So this way we find the polar form of the complex number<\/p>\n

VERY IMPORTANT NOTE:-<\/strong><\/span> As we have already discussed, a complex number is like a point that can lie in any quadrant of the Cartesian plane. We should always consider the quadrant of the point while calculating argument and finding the\u00a0 polar form of complex numbers<\/p>\n

Example:- Find the polar form of complex numbers given below<\/p>\n

\u00a0 \u00a0<\/strong><\/p>\n

Solution<\/em><\/strong><\/p>\n

(a)<\/strong>\u00a0Let\u2019s find out r<\/p>\n

 <\/p>\n

We can see, ordered-pair\u00a0of the given complex number is (-1,\"\\sqrt). It should be lying in the second quadrant so the argument must satisfy<\/p>\n

Tan is negative in both second and fourth quadrants but we shall only consider second quadrant value\u00a0because our complex number is lying there.<\/p>\n

so<\/p>\n

therefore the polar form complex number is-<\/p>\n

<\/p>\n

If we are after all the possible value of the argument then:<\/p>\n

<\/p>\n

Exponential Form of a Complex Number-<\/h3>\n

After rectangular form and polar form of complex numbers, this is the third form of a complex number. To find it, we take help from Euler’s Theorem-<\/p>\n

<\/p>\n

as we know Z=x+iy is also equal to\u00a0<\/p>\n

so\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/p>\n

This is known as the exponential form of a complex number<\/p>\n

De Moivre\u2019s Theorem-<\/h3>\n

As we know that\u00a0\u00a0<\/p>\n

so\u00a0 \u00a0 <\/p>\n

<\/p>\n

this relationship is called De Moivre\u2019s Theorem and this is one of the most important concepts of Mathematics.<\/p>\n

This can also be written like this-<\/p>\n

\"(Cos{\\theta=\u00a0\"Cos({\\theta<\/p>\n

<\/p>\n

Example2:\u00a0\u00a0<\/i><\/b>\u00a0Evaluate\u00a0\u00a0<\/span><\/b><\/p>\n

Answer- First of all we convert the given complex number 3+4i into its polar form by using the above-mentioned method<\/p>\n

<\/p>\n

we raise both sides to a power\u00a0of 5<\/p>\n

<\/p>\n

The same trick\u00a0can be used to find the Square root, cube root or nth root of a complex number<\/p>\n

\"ib<\/p>\n

Whatsapp at +919911262206 or fill the form<\/h5>\n\n
\n

<\/p>

    <\/ul><\/div>\n
    \n
    \n<\/fieldset>\n

    \n<\/p>

    <\/div>\n<\/form>\n<\/div>\n\n

     <\/p>\n

     <\/p>\n

     <\/p>\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    Polar Form of complex numbers In the previous post, Our IB Maths Tutors discussed the basics of complex numbers. This is the second and final […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-638","post","type-post","status-publish","format-standard","hentry","category-ib-mathematics-tutors"],"_links":{"self":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/638","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/comments?post=638"}],"version-history":[{"count":0,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/638\/revisions"}],"wp:attachment":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/media?parent=638"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/categories?post=638"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/tags?post=638"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}