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## Chapter 39

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**Chapter 39**Complex Numbers By Chtan FYHS-Kulai**consist of all positive and negative integers, all rational**numbers and irrational numbers. By Chtan FYHS-Kulai**Rational numbers are of the form p/q, where p, q are**integers. Irrational numbers are By Chtan FYHS-Kulai**Complex number notation**By Chtan FYHS-Kulai**Algebra form**Trigonometric form Notation index form By Chtan FYHS-Kulai**Complex numbers**are defined as numbers of the form : or By Chtan FYHS-Kulai**is represented by**are real numbers. A complex number consists of 2 parts. Real part and imaginary part. By Chtan FYHS-Kulai**Notes:**• When b=0, the complex number is Real • When a=0, the complex number is imaginary • The complex number is zero iff a=b=0 By Chtan FYHS-Kulai**VENN DIAGRAMRepresentation**• All numbers belong to the Complex number field, C. The Real numbers, R, and the imaginary numbers, i, are subsets of C asillustrated below. Complex Numbers a + bi Real Numbers a + 0i Imaginary Numbers 0 + bi**Conjugate complex number**By Chtan FYHS-Kulai**Conjugate complex numbers**The complex numbers and are called conjugate numbers. By Chtan FYHS-Kulai**is conjugate of .**By Chtan FYHS-Kulai**e.g. 1**Solve the quadratic equation Soln: By Chtan FYHS-Kulai**e.g. 2**Factorise . Soln: By Chtan FYHS-Kulai**Representation of complex number in an Argand diagram**By Chtan FYHS-Kulai**(Im)**y P(a,b) x (Re) 0 P’(-a,-b) Argand diagram By Chtan FYHS-Kulai**e.g. 3**If P, Q represent the complex numbers 2+i, 4-3i in the Argand diagram, what complex number is represented by the mid-point of PQ? By Chtan FYHS-Kulai**y**(Im) Soln: P(2,1) x (Re) 0 Q(4,-3) Mid-point of PQ is (3,-1) is the complex number. By Chtan FYHS-Kulai**Do pg.272 Ex 20a**By Chtan FYHS-Kulai**Equality of complex numbers**By Chtan FYHS-Kulai**The complex numbers**and are said to be equal if, and only if, a=c and b=d. By Chtan FYHS-Kulai**e.g. 4**Find the values of x and y if (x+2y)+i(x-y)=1+4i. Soln: x+2y=1; x-y=4 2y+y=1-4; 3y=-3, y=-1 x-(-1)=4, x=4-1=3 By Chtan FYHS-Kulai**Addition of complex numbers**By Chtan FYHS-Kulai**If**then By Chtan FYHS-Kulai**Subtraction of complex numbers**By Chtan FYHS-Kulai**If**then By Chtan FYHS-Kulai**Do pg.274 Ex 20b**By Chtan FYHS-Kulai**Multiplication of complex numbers**By Chtan FYHS-Kulai**e.g. 5**If , find the values of (i) (ii) Soln: (i) (ii) By Chtan FYHS-Kulai**If**then By Chtan FYHS-Kulai**Division of complex numbers**By Chtan FYHS-Kulai**If**then By Chtan FYHS-Kulai**e.g. 6**Express in the form . Soln: By Chtan FYHS-Kulai**e.g. 7**By Chtan FYHS-Kulai**e.g. 8**If z=1+2i is a solution of the equation where a, b are real, find the values of a and b and verify that z=1-2i is also a solution of the equation. By Chtan FYHS-Kulai**The cube roots of unity**By Chtan FYHS-Kulai**If is a cube root of 1,**By Chtan FYHS-Kulai**Notice that the complex roots have the property that one is**the square of the other, By Chtan FYHS-Kulai**let**So the cube roots of unity can be expressed as By Chtan FYHS-Kulai**If we take**then or vice versa. By Chtan FYHS-Kulai**(1) As is a solution of**(2) As is a solution of By Chtan FYHS-Kulai**(3)**(4) (5) … etc By Chtan FYHS-Kulai**e.g. 9**Solve the equation . By Chtan FYHS-Kulai**e.g. 10**By Chtan FYHS-Kulai**Soln:**By Chtan FYHS-Kulai**Do pg.277 Ex 20c**By Chtan FYHS-Kulai**The (r,θ) form of a complex number**By Chtan FYHS-Kulai