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LU Bible

Math 101 – Linear Algebra By: Abdulsalam Shalashtein. LU Bible. Why is it called a complex number ? Unlike the name suggests the chapter is very easy it’s called a complex number, because it’s made up of a real part and an imaginary one . Why is it denoted by Z ?

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LU Bible

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  1. Math 101 – Linear Algebra By: Abdulsalam Shalashtein LU Bible

  2. Why is it called a complex number ? • Unlike the name suggests the chapter is very easy it’s called a complex number, because it’s made up of a real part and an imaginary one. • Why is it denoted by Z ? - If you look at a mirror you’ll see you hieght and width ( Y-axis & X-axis ) but you won’t see your back ( Z-axis), however your back is there somewhere so we call it imaginary Chapter 1 – Complex Numbers

  3. Numbers … Complex Numbers Real Numbers Quotient Numbers Integers Natural Numbers

  4. Imaginary numbers are special because when squared the result is a negative number . Proof : i2 = (0 + i)2 = (0 + i)(0 + i) = (0×0 - 1×1) + (0×1 + 1×0)i = -1 + 0i = -1 The i

  5. To add two complex numbers we add each element separately: (a+bi) + (c+di) = (a+c) + (b+d)i Example: (3 + 2i) + (1 + 7i) = (4 + 9i) Adding two complex numbers

  6. To multiply complex numbers: Each part of the first complex number gets multiplied byeach part of the second complex number Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" Multiplying Complex Numbers Firsts: a·c Outers: a·di Inners: bi·c Lasts: bi·di (a+bi)(c+di) = ac + adi + bci + bdi2

  7. A conjugate is achieved when you switch the sign of the imaginary number Multiplying by the conjugate : (a + bi)(a - bi) = a2 + b2 Example: (4 - 5i)(4 + 5i) = 42 + 52 Conjugate _ Z = a - ib Z = a + ib

  8. = 0  z = 0 ’ = ’ ’ = ’ If Z’ then = = Z R(z) = R() and I(z) = - I() Properties of conjugate

  9. The modulus of a complex number Z is the distance between the center O (0,0) and the point of affix Z. represented as or r Modulus of a complex number Z |Z| or “r” = r

  10. An argument is the angle between and How to Calculate ? - It’s a triangle ! So you can calculate it in three ways ( take Z = a + ib ) : 1- = 2-= 3-= Argument

  11. Forms of a complex number

  12. The trigonometric form z=r() can be written as [r,] zz’ = rr’ ( = ( = = remarks

  13. Zn = rn[cos(n)+i sin(n)] Proven by Induction Demoivre’s formula

  14. An nth root of Z is every complex number that would satisfy un = Z. While an nth root of unity is every nth root of z=1 nth root of z

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