70 necklaces 71 necklaces 72 necklaces 73 necklaces Don ’ t know

2. A jewelry maker has total revenue for necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces produces and sold. How many necklaces must be produced and sold in order to break even? • 70 necklaces • 71 necklaces • 72 necklaces • 73 necklaces • Don’t know

3. A jewelry maker has total revenue for necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the numberof necklacesmade and sold. How many necklaces must be produced and sold in order to break even? • x=number of necklaces • R(x)=number of $ earned for x necklaces • C(x)=number of $ spend to make x necklaces • x=? • R(x)=C(x) • 90.75x=24.50x+4770solve for x. • 66.25x=4770 • x=4770/66.25 • x=72 C

4. Complex Numbers

5. Time to dance!

6. Multiply each number by -1

7. Multiply each number by -1 again

8. Think: *√(-1)*√(-1) is the same as *(-1)

9. Multiply each number by √(-1)

10. Last time (Part of the truth) • Addition is shifting • Multiplication is stretching

11. This time (The whole truth) • Addition is shifting • Multiplication is stretching & rotating • You saw hints of this before: • Multiply by a negative causes a flip.

12. We need a new number line 2i i -i -2i

13. We need a new number line plane 2i i -i -2i 3-2i

14. Complex Numbers • 1 is a unit of rightward distance • -1 is a unit of leftward distance • i is a unit of forward distance • -i is a unit of backward distance • 1 and -1 cancel each other out, they can be combined. • i and –i cancel each other out, they can be combined. • 1 and i do not cancel each other out. You need them both.

15. Complex Numbers • 2+3i number that is 2 rights and 3 forwards. • 2-0.5i number that is rights and half a backward

16. Graphing becomes more complicated • We don’t have an x number line and a y number line anymore. We have an x number plane and a y number plane • Four Axes: • Re(x) • Im(x) • Re(y) • Im(y)

17. Arithmetic on complex numbers • 1 and i cannot be combined. They are on separate axes. • 1+i can’t be simplified, just like x+y can’t be simplified. • Treat i like a variable and you will be ok. • Remember that i2=-1 and √(-1)=i • This can be simplified

18. Examples

19. Examples • (2+3i)+(1-2i) =2+1+3i-2i =3+i • (2+3i)-(1-2i) =(2+3i)+(-1+2i) =1+5i

20. Examples

21. Examples You are not done until you have the real and imaginary parts completely separate

22. 80 -80 80i -80i None of the above Simplify:

23. Simplify: B

24. Powers of i i -1 1 -i

25. Powers of i Multiplying by i makes a quarter turn

26. Powers of i Each power of i is a quarter turn more • 1=i0,i4,i8,i12,etc… • i=i1,i5,i9,i13,etc… • -1=i2,i6,i10,i14,etc… • -i=i3,i7,i11,i15,etc…

27. Powers of i Number of turns is power/4 • 1=i0,i4,i8,i12,etc… • i=i1,i5,i9,i13,etc… • -1=i2,i6,i10,i14,etc… • -i=i3,i7,i11,i15,etc… TEST: Divide power by 4

28. Powers of i • 1=i0,i4,i8,i12,etc… • i=i1,i5,i9,i13,etc… • -1=i2,i6,i10,i14,etc… • -i=i3,i7,i11,i15,etc… TEST: Divide power by 4

29. Powers of i • 1=i0,i4,i8,i12,etc… • i=i1,i5,i9,i13,etc… • -1=i2,i6,i10,i14,etc… • -i=i3,i7,i11,i15,etc… 0,1,2,3,etc… TEST: Divide power by 4 .25,1.25,2.25,3.25, etc.. 0.5,1.5,2.5,3.5,etc… 0.75,1.75,2.75,3.75,etc..

30. Powers of i • 1=i0,i4,i8,i12,etc… • i=i1,i5,i9,i13,etc… • -1=i2,i6,i10,i14,etc… • -i=i3,i7,i11,i15,etc… Integer TEST: Divide power by 4 Decimal part 0.25 Decimal part 0.5 Decimal part 0.75

31. Example • i14 • 14/4=3.5 • Starting at 1: • Integer part 3  three full turns (still at 1) • Decimal part 0.5  half turn (wind up at -1) • End result: -1 • i14=-1

32. Simplify: i2,789 • i • -1 • -i • 1 • None of these

33. Simplify: i2,789 2789/4=697.25 Integer part is 697  697 full turns Decimal part is 0.25  one quarter turn End result: one quarter turn A) i