COMPLEX NUMBERS
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COMPLEX NUMBERS. Consider the quadratic Equation X 2 + 1 = 0 What is its solution ?. X 2 = - 1 or x =. But this number is not known to us. What is i ?. unreal. i is an imaginary number. complex. Or a complex number. imaginary. Or an unreal number. The terms are inter-changeable. - PowerPoint PPT Presentation
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COMPLEX NUMBERS
COMPLEX NUMBERS
COMPLEX NUMBERSConsider the quadratic EquationX2 + 1 = 0
What is its solution ?But this number is not known to us.
COMPLEX NUMBERSCOMPLEX NUMBERS
imaginarycomplexunrealWhat is i ?i is an imaginary numberOr a complex numberOr an unreal numberThe terms are inter-changeableCOMPLEX NUMBERSCOMPLEX NUMBERSIf b = 0, the number a + bi = a is a real number. Example: 5= 5+ i 0
If a = 0, the number a + bi is called an imaginary numberExample: -2i= 0+ (-2)i
COMPLEX NUMBERSCOMPLEX NUMBERSIRRATIONAL NUMBERSCOMPLEX NUMBERSReal numbers and imaginary numbers are subsets of the set of complex numbers.
Complex Numbers Imaginary NumbersReal NumbersRationalIrrationalCOMPLEX NUMBERSCOMPLEX NUMBERSPractice Time!!!!
Simplify Evaluate 3i x -4i COMPLEX NUMBERSAddition of Complex Numbers Let z1=a+ib and z2=c+id be any two complex numbers. Then the sum of those two complex numbers is defined as : z1 +z2 = (a+bi) + (c+di) = (a+c) + (b+d)i
Addition of two complex numbers can be done geometrically by constructing a parallelogram COMPLEX NUMBERSPractice Time!!!!
Simplify
(2+3i ) + (4 -3i)
(-3+4i) + (-2- i10)
COMPLEX NUMBERSProperties of addition
COMPLEX NUMBERSProperties of addition
COMPLEX NUMBERSDifference of two complex numbersGiven any two complex numbers z1 and z2, the difference z1 - z2 is defined as follows : z1 - z2 = z1 +(-z2)
Simplify (3i+2i) ( -2 + i3)COMPLEX NUMBERSMultiplication of two complex numbers Multiplying complex numbers is similar to multiplying polynomials and combining like terms. Let a+ib and c+id be any two complex numbers. Then the product of those two complex numbers is defined as follows: (a+ib) (c+id) = (ac bd) + i(ad + bc)
COMPLEX NUMBERSPROPERTIES OF MULTIPLICATION
COMPLEX NUMBERSPractice Time!!!!Simplify
(2+3i)(4-3i)
(-4+2i)(7-12i)COMPLEX NUMBERS
COMPLEX NUMBERSThe following identifies are true for complex numbers
COMPLEX NUMBERS
POWERS OF iIn general, for any integer k, i4k = 1, i4k+1 = i, i4k+2 = -1.COMPLEX NUMBERS
COMPLEX NUMBERSUSEFUL RESULTS
COMPLEX NUMBERSUSEFUL RESULTS
COMPLEX NUMBERS USEFUL RESULTS
COMPLEX NUMBERSCOMPLEX NUMBERSIZICOMPLEX NUMBERSxy1231232 + 3iWe can represent complex numbers as a point.COMPLEX NUMBERSLet the point P represent the non zero complex number z = x + iy. Let the directed line segment OP be the length r and be the angle which OP makes with the positive direction of x-axis
Polar representation of a Complex numberCOMPLEX NUMBERSWe may note that the point P is uniquely determined by the ordered pair of real numbers (r, ), called the polar coordinates of the point P.We consider the origin as the pole and the positive direction of the x-axis as the initial line.
COMPLEX NUMBERSWe have , x = r cos , y = r sin and therefore , z = r(cos + i sin ). The latter is said to be the polar form of the complex number.Here is the modules of z and is called the argument of z which is denoted by arg z.
COMPLEX NUMBERSCOMPLEX NUMBERS
COMPLEX NUMBERS
COMPLEX NUMBERS
COMPLEX NUMBERS
COMPLEX NUMBERS
COMPLEX NUMBERSENJOY COMPLEX NUMBERS & LIFE
COMPLEX NUMBERS
