{"id":35173,"date":"2022-08-25T11:05:35","date_gmt":"2022-08-25T05:35:35","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35173"},"modified":"2022-08-25T11:13:01","modified_gmt":"2022-08-25T05:43:01","slug":"inter-2nd-year-maths-2a-complex-numbers-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2a-complex-numbers-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2A Complex Numbers Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2a-formulas\/\">Inter 2nd Year Maths 2A Formulas<\/a> PDF Chapter 1 Complex Numbers to solve questions creatively.<\/p>\n<h2>Intermediate 2nd Year Maths 2A Complex Numbers Formulas<\/h2>\n<p><strong>Definition of a complex number:<\/strong><br \/>\n\u2192 A number of the type z = x + yi, where x, y \u2208 R and i = \u221a- 1 i.e., i<sup>2<\/sup> = &#8211; 1 is called a complex number &#8216;x&#8217; is called real part of z, and &#8216;y&#8217; is called imaginary part of z. We write x = Re(z) and y = Im(z). A number z = x + yi is said to be purely real iff y = 0 (x \u2260 0) and purely imaginary iff x = 0 (y \u2260 0)<br \/>\nA complex number a + ib is an ordered pair of real numbers. It is denoted by (a, b), a \u2208 R, b \u2208 R.<\/p>\n<p>\u2192 Two complex numbers z<sub>1<\/sub> = (a, b), z<sub>2<\/sub> = (c, d) are said to be equal iff a = c and b = d.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Complex Numbers Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 If z<sub>1<\/sub> = (a, b), z<sub>2<\/sub> = (c, d) then<\/p>\n<ul>\n<li>z<sub>1<\/sub> + z<sub>1<\/sub> = (a + c, b + d)<\/li>\n<li>z<sub>1<\/sub> &#8211; z<sub>2<\/sub> = (a &#8211; c, b &#8211; d)<\/li>\n<li>z<sub>1<\/sub>. Z<sub>2<\/sub> = (ac &#8211; bd, ad + bc) and<\/li>\n<li>\\(\\frac{z_{1}}{z_{2}}=\\left(\\frac{a c+b d}{c^{2}+d^{2}}, \\frac{b c-a d}{c^{2}+d^{2}}\\right)\\)<\/li>\n<\/ul>\n<p><strong>Modulus and Amplitude:<\/strong><br \/>\n\u2192 The modulus of a complex number z = x + iy is defined as a non-negative real number r = \\(\\sqrt{x^{2}+y^{2}}\\). It is denoted by |z|.<\/p>\n<p>\u2192 Any real number \u03b8 satisfying the equation cos \u03b8 = \\(\\frac{x}{r}\\), sin \u03b8 = \\(\\frac{y}{r}\\) is called an amplitude or argument of z. The unique argument \u03b8 of z satisfying &#8211; \u03c0 &lt; \u03b8 \u2264 \u03c0 is called the principal argument of z and is denoted by Arg z.<\/p>\n<p>\u2192 The polar form or modulus amplitude form of the complex number<br \/>\nz = x + iy is r(cos \u03b8 + i sin \u03b8)<\/p>\n<p><strong>Conjugate of a complex number:<\/strong><br \/>\n\u2192 If z = x + iy i.e., x, y \u2208 R, then the complex number x &#8211; iy is called conjugate of z and is written as z\u0305. The sum and product of a complex number and its Conjugate is always purely real.<\/p>\n<p><strong>Some properties of modulus, amplitude and conjugate:<\/strong><\/p>\n<ul>\n<li>(z\u0305) = z<\/li>\n<li>z + z\u0305 = 2 Re (z) and z &#8211; z\u0305 = 2 Im (z)<\/li>\n<li>\\(\\overline{Z_{1} Z_{2}}\\) = \\(\\overline{Z_{1}}\\) \\(\\overline{Z_{2}}\\)<\/li>\n<li>\\(\\) and \\(\\) (z<sub>2<\/sub> \u2260 0) and |z<sub>1<\/sub>z<sub>2<\/sub>| = |z<sub>1<\/sub>| |z<sub>2<\/sub>|<\/li>\n<li>zz\u0305 = |z|<sup>2<\/sup><\/li>\n<li>|z| = |z\u0305|;<\/li>\n<li>|z<sub>1<\/sub> + z<sub>2<\/sub>| \u2264 |z<sub>1<\/sub>| + |z<sub>2<\/sub>|, |z<sub>1<\/sub> &#8211; z<sub>2<\/sub>| \u2264 |z<sub>1<\/sub>| + |z<sub>2<\/sub>|<\/li>\n<li>|z<sub>1<\/sub> &#8211; z<sub>2<\/sub>| &gt; | |z<sub>1<\/sub>| &#8211; |z<sub>2<\/sub>| |<br \/>\nIn (vii) and (viii) equality holds iff amp (z<sub>1<\/sub>) &#8211; amp (z<sub>2<\/sub>) is an integral multiple of 2\u03c0.<\/li>\n<li>amp (z<sub>1<\/sub> z<sub>2<\/sub>) = amp (z<sub>1<\/sub>) + amp (z<sub>2<\/sub>) + n\u03c0, for some n \u2208 {- 1, 0, 1}<\/li>\n<li>amp \\(\\left(\\frac{Z_{1}}{Z_{2}}\\right)\\) = amp (z<sub>1<\/sub>) &#8211; amp (z<sub>2<\/sub>) + n\u03c0, for some n \u2208 {- 1, 0, 1}<\/li>\n<li>\\(\\frac{1}{{cis} \\alpha}\\) = cis(- \u03b1)<\/li>\n<li>cis \u03b1 cis \u03b2 = cis (\u03b1 + \u03b2)<\/li>\n<li>\\(\\frac{{cis} \\alpha}{{cis} \\beta}\\) = cis (\u03b1 &#8211; \u03b2)<\/li>\n<\/ul>\n<p><strong>De-Moivre&#8217;s theorem:<\/strong><\/p>\n<ul>\n<li>If n is any integer, then (cos \u03b8 + i sin \u03b8)<sup>n<\/sup> = cos n\u03b8 + i sin n\u03b8<\/li>\n<li>If n = \\(\\frac{p}{q}\\), where p and q are integers having no common factor and q &gt; 1, then cos n\u03b8 + i sin n\u03b8 is one of the q values of (cos \u03b8 + i sin \u03b8)<sup>n<\/sup><\/li>\n<li>If z<sub>0<\/sub> = r<sub>0<\/sub> cis \u03b8<sub>0<\/sub> \u2260 0, then the n<sup>th<\/sup> roots of z<sub>0<\/sub> are \u03b1<sub>k<\/sub> = r<sub>0<\/sub><sup>1\/n<\/sup> cis \\(\\left(\\frac{2 k \\pi+\\theta_{0}}{n}\\right)\\),<br \/>\nk = 0, 1, 2, 3 &#8230; (n-1)<\/li>\n<\/ul>\n<p><strong>Cube roots of unity:<\/strong><\/p>\n<ul>\n<li>The cube roots of unity are 1, \u03c9 = \\(\\frac{-1+\\sqrt{3 i}}{2}\\) and \u03c9<sup>2<\/sup> = \\(\\frac{-1-\\sqrt{3 i}}{2}\\)<\/li>\n<li>1 + \u03c9 + \u03c9<sup>2<\/sup> = 0 and w<sup>3<\/sup> = 1; 1 + \u03c9 = &#8211; \u03c9 <sup>2<\/sup>, 1 + \u03c9<sup>2<\/sup> = &#8211; \u03c9, \u03c9 + \u03c9<sup>2<\/sup> = &#8211; 1<\/li>\n<li>Either of the two non-real cube roots of unity is square of the other.<\/li>\n<li>For either of the two non &#8211; real cube roots a.and of unity \u03b1 + \u03b2 = &#8211; 1, \u03b1\u03b2 = &#8211; 1, \u03b1<sup>2<\/sup> = \u03b2, \u03b2<sup>2<\/sup> = \u03b1 and \u03b1<sup>3<\/sup> = \u03b2<sup>3<\/sup> = 1<\/li>\n<li>(-1)<sup>1\/3<\/sup> = &#8211; 1, &#8211; \u03c9 &#8211; \u03c9<sup>2<\/sup><\/li>\n<li>The n<sup>th<\/sup> roots of unity are cis\\(\\left(\\frac{2 k \\pi}{n}\\right)\\), k = 0, 1, 2, &#8230;.. (n &#8211; 1)<\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Complex Numbers Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p><strong>Formulae:<\/strong><\/p>\n<p>\u2192 Modulus of Z = \\(\\sqrt{x^{2}+y^{2}}\\)<\/p>\n<p>\u2192 If \\(\\sqrt{a+i b}\\) = (x + iy), then x = \\(\\sqrt{\\frac{\\sqrt{a^{2}+b^{2}}+a}{2}}\\) and y = \\(\\sqrt{\\frac{\\sqrt{a^{2}+b^{2}}-a}{2}}\\)<\/p>\n<p>\u2192 Conjugate of a + ib = a &#8211; ib<\/p>\n<p>\u2192 Conjugate of a &#8211; ib = a + ib<\/p>\n<p>\u2192 Any number of the form x + iy where x, y \u2208 R and i<sup>2<\/sup> = -1 is called a Complex Number.<\/p>\n<p>\u2192 In the complex number x + iy, x is called the real part and y is called the imaginary part of the complex number.<\/p>\n<p>\u2192 A complex number is said to be purely imaginary if its real part is zero and is said to be purely real if its imaginary part is zero.<br \/>\n(a) Two complex numbers are said to be equal if their real parts are equal and their imaginary parts are equal.<br \/>\n(b) In the set of complex numbers, there is no meaning to the phrases one complex is greater than or less than another i.e. If two complex numbers are not equal, we say they are unequal.<br \/>\n(c) a+ ib &gt; c + id is meaningful only when b = 0, d = 0.<\/p>\n<p>\u2192 Two complex numbers are conjugate if their sum and product are both real. They are of the form a + ib, a &#8211; ib.<\/p>\n<p>\u2192 cis\u03b8<sub>1<\/sub> cis\u03b8<sub>2<\/sub> = cis(\u03b8<sub>1<\/sub> + \u03b8<sub>2<\/sub>), \\(\\frac{{cis} \\theta_{1}}{{cis} \\theta_{2}}\\) = cis(\u03b8<sub>1<\/sub> &#8211; \u03b8<sub>2<\/sub>), \\(\\frac{1}{\\cos \\theta+i \\sin \\theta}\\) = cos\u03b8 &#8211; isin\u03b8<\/p>\n<p>\u2192 \\(\\frac{a_{1}+i b_{1}}{a_{2}+i b_{2}}=\\frac{\\left(a_{1} a_{2}+b_{1} b_{2}\\right)+i\\left(a_{2} b_{1}-a_{1} b_{2}\\right)}{a_{2}^{2}+b_{2}^{2}}\\)<\/p>\n<p>\u2192 \\(\\frac{1+i}{1-i}\\) = i, \\(\\frac{1-i}{1+i}\\) = i<\/p>\n<p>\u2192 \\(\\sqrt{x^{2}+y^{2}}\\) is called the modulus of the complex number x + iy and is denoted by r or |x + iy|<\/p>\n<p>\u2192 Any value of 0 obtained from the equations cos \u03b8 = \\(\\frac{x}{r}\\), sin \u03b8 = \\(\\frac{y}{r}\\) is called an amplitude of the complex number.<\/p>\n<p>\u2192 The amplitude lying between -\u03c0 and \u03c0 is called the principal amplitude of the complex number. Rule for choosing the principal amplitude.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Complex Numbers Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 If \u03b8 is the principal amplitude, then -\u03c0 &lt; 0 &lt; \u03c0<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-20209\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/08\/Inter-2nd-Year-Maths-2A-Complex-Numbers-Formulas-1.png\" alt=\"Inter 2nd Year Maths 2A Complex Numbers Formulas 1\" width=\"160\" height=\"127\" \/><\/p>\n<p>\u2192If \u03b1 is the principle amplitude of a complex number, general amplitude = 2n\u03c0 + \u03b1 where n \u2208 Z.<\/p>\n<ul>\n<li>Amp (Z<sub>1<\/sub> Z<sub>2<\/sub>) = Amp Z<sub>1<\/sub> + AmpZ<sub>2<\/sub><\/li>\n<li>Amp z + Amp z\u0305 = 2\u03c0 (when z is a negative real number) = 0 (otherwise)<\/li>\n<\/ul>\n<p>\u2192 r(cos \u03b8 + i sin \u03b8) is the modulus amplitude form of x + iy.<\/p>\n<p>\u2192 If the amplitude of a complex number is \\(\\frac{\\pi}{2}\\), its real part is zero.<\/p>\n<p>\u2192 If the amplitude of a complex number is \\(\\frac{\\pi}{4}\\), its real part is equal to its imaginary part.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 1 Complex Numbers to solve questions creatively. Intermediate 2nd Year Maths 2A Complex Numbers Formulas Definition of a complex number: \u2192 A number of the type z = x + yi, where x, y \u2208 R and i = \u221a- 1 i.e., i2 = &#8211; &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[14],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35173"}],"collection":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/comments?post=35173"}],"version-history":[{"count":3,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35173\/revisions"}],"predecessor-version":[{"id":35197,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35173\/revisions\/35197"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/media?parent=35173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/categories?post=35173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/tags?post=35173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}