1 Simplify Problems With Imaginary Roots Standards 5 & 6 Equation That Results In Imaginary Roots COMPLEX NUMBERS Complex Numbers In Context END SHOW Imaginary Roots Graph and Add Complex Numbers Simplify Expressions With Complex Conjugates A Problem Of Iterations Simplify Expressions With Complex Numbers PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 5: Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane. STANDARD 6: Students add, subtract, multiply, and divide complex numbers. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 ESTÁNDAR 5: Los estudiantes demuestran conocimiento de como los números reales y complejos están relacionados aritméticamente y gráficamente. En particular, ellos pueden graficar números como puntos en el plano. ESTÁNDAR 6: Los estudiantes suman, restan, multiplican, y dividen números complejos. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 Let’s remember the case where the INDEX is EVEN and the RADICAND NEGATIVE: Standards 5 & = (-1)(4)= (-1) 4 = i 4 = 2 i PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 IN GENERAL: Standards 5 & 6 - b = (-1)(b)= (-1) b = i b (-1) i = i = -1 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 Standards 5 & 6 a = b n a aa = b.... n b = a n EVEN ODD b > 0, positive b < 0, negative n b + n b – n b + None negative i n b n b – None positive one POWERS FACTORSROOTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 Standards 5 & 6 Simplify: – 63 = (–1)(9)(7) = –1 9 7 = i 3 7 = 3 i 7 –200x y 3 = (–1)(100) x 2 x y 2 = –1 100 x 2xy 2 = i 10 x 2xy = 10 i x 2xy 6 i 9 i = 54 i 2 = 54( ) 2 –1 = 54(–1) = –54 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 Standards 5 & 6 Simplify: –2 –8 –5 = (–1)(2) (–1)(4)(2) (–1)(5) = –1 2 –1 4 2 –1 5 i i i = i i –1 = i (–1)(4) 5 = – 4 i 5 4i4i 77 = 4 i i 76 1 = 4( i ) i 2 38 = 4(–1 ) i 38 = 4(1) i = 4 i PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 Standards 5 & 6 Solve: x +2x + 4 = 0 2 x +2x + 4 = 0 2 Using the quadratic equation: a= 1 b= 2 c= 4 x= -( ) ( ) - 4( )( ) 2( ) 2 +_ x= – 2 4 – ( 4 )( 4 ) 2 +_ – x= 2 +_ – 2 ( -1)(4)(3) x= 2 +_ – x= 2 +_ – 2 ( i )(2) 3 x= 2 +_ – 1 + i 3 x= – 1 – i 3 x= PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 Standards 5 & 6 Solve: x +2x + 4 = 0 2 – 1 + i 3 Checking the solution: ( ) +2( ) + 4 = – 2 i 3 + i 3 – i = – 1 + i 3 1 –2i 3 – 3 – 2 + 2i = 0 0 = 0 – 1 – i 3 ( ) +2( ) + 4 = i 3 + i 3 – 2 – 2 i = – 1 – i i 3 – 3 – 2 – 2i = 0 0 = 0 True PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 5 & 6 Solve: x +2x + 4 = 0 2 x +2x + 4 = 0 2 Using the quadratic equation: a= 1 b= 2 c= 4 x= -( ) ( ) - 4( )( ) 2( ) 2 +_ x= – 2 4 – ( 4 )( 4 ) 2 +_ – x= 2 +_ – 2 ( -1)(4)(3) x= 2 +_ – x= 2 +_ – 2 ( i )(2) 3 x= 2 +_ – 1 + i 3 x= – 1 – i 3 x= PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 Standards 5 & WHOLE NUMBERS NATURAL NUMBERS POSITIVE INTEGERS INTEGERS NEGATIVE INTEGERS THE NUMBER LINE NATURAL NUMBERS:1, 2, 3, 4, … WHOLE NUMBERS: 0, 1, 2, 3, 4, … POSITIVE INTEGERS: 1, 2, 3, 4, … NEGATIVE INTEGERS: -1, -2, -3, -4, … INTEGERS: …, -3, -2, -1, 0, 1, 2, 3, … REVIEW: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 REAL NUMBERS R= reals I I= irrationals Q Q= rationals Z Z= integers W W= Wholes N N= naturals Standards 5 & 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 Standards 5 & 6 COMPLEX NUMBERS a + b i then REAL NUMBERS then IMAGINARY NUMBERS then PURE IMAGINARY then NON-PURE IMAGINARY If b = 0 RATIONALES IRRATIONALS If a = 0 4i4i - 7 i i = 7 1 = 5 = -8 1 = 5 1 = i PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 Standards 5 & 6 ( i ) + 6 i SIMPLIFY: = i + 6 i = 10 + (5 + 6) i = i (1 + i ) + (3 i – 5) = (1 + i ) + (–5 + 3 i ) = 1 + i –5 + 3 i = 1 – 5 +1 i + 3 i = – 4 + (1 + 3) i = – i Grouping like terms Distributive property Grouping like terms Distributive property PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 Standards 5 & 6 SIMPLIFY: (4 – 7 i ) – (9 + i ) Grouping like terms Distributive property = 4 – 7 i –9 – i = 4 – 9 –7 i – 1 i = –5 + (–7 – 1) i = –5 + (–8) i = –5 –8 i PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 4i a b Imaginary Real i Standards 5 & 6 Graph: (-5, -7) i i (3, 2) 4i4i (0, 4) i 3 + 2i Complex numbers comply with vector’s properties. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 4i4i a b Imaginary Real i Standards 5 & 6 – 5 – 7 i Now let’s add two of them graphically: ( ) + ( ) = - 5 – 3 i - 5 – 3 i (-5,-3) Draw parallel vectors. Draw a vector from the origin over the diagonal of the resulting parallelogram. (-5, -7) - 5 – 7 i 4i4i (0, 4) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 Standards 5 & 6 Find x and y so the equation is true: i = 6x + 3y i For two complex numbers to be equal the real part is equal in both of them, and the imaginary part is equal in both of them as well. 4 = 6x 4 i = 3y i 6 3 i x = y = 4 3 Check: i = 6( ) + 3( ) i i = + i i = i true PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 Standards 5 & 6 SIMPLIFY: (5 – 3 i )(2 + 4 i ) F O I L (5)(2) + (5)(4 i ) + (-3 i )(2) + (-3 i )(4 i ) i – 6 i – 12 i i –12(-1) i i PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21 Standards 5 & 6 SIMPLIFY: (5 – 3 i )(5 + 3 i ) = 25 – (3 i ) 2 = 25 – 9 i 2 Difference of squares COMPLEX CONJUGATES = 25 – 9(-1) = = 34 The product of COMPLEX CONJUGATES is always a REAL NUMBER. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 Standards 5 & 6 1 – 3 i i 2 4 i –3 5 4 i – 3 i i 5 2 – i i i 6 – 5 i Complete the complex conjugates for the following: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 Standards 5 & – 5 i i 36 – 25 i 4(6 + 5 i ) 2 36 – 25 i i 2 36 – 25 (-1) i i i SIMPLIFY: Multiply both numerator and denominator by the COMPLEX CONJUGATE of the denominator. Difference of squares. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

24 Standards 5 & 6 3i3i 9 i – 1 SIMPLIFY: 3i3i 9 i – 1 i i Multiply both numerator and denominator by i to obtain i at the denominator and eliminate the imaginary part there. 2 3i3i 9 i – i 2 2 3(-1) 9 (-1) – i – i 3 + i 1 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

25 Standards 5 & 6 Iterate three times the following function with the initial value of x = 2 i : f(x) = x f(2i) = ( ) i2i = 4 i = 4(-1) + 1 = = -3 f(-3) = ( ) = = 10 f(10) = ( ) = = 101 First Iteration: Second Iteration: Third Iteration: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved