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CLEP General Mathematics: Complex Numbers

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1. 3rd. Rule of Complex Arithmetic
integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Absolute Value of a Complex Number
For real a and b - a + bi = 0 if and only if a = b = 0

2. y / r
Euler's Formula
Polar Coordinates - sin?
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.

3. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
subtracting complex numbers
the complex numbers
complex numbers
Integers

4. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
natural
sin z
How to multiply complex nubers(2+i)(2i-3)
Integers

5. We see in this way that the distance between two points z and w in the complex plane is
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers
|z-w|
e^(ln z)

6. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
z1 / z2
conjugate
complex
For real a and b - a + bi = 0 if and only if a = b = 0

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
The Complex Numbers
Field
non-integers
point of inflection

8. V(x² + y²) = |z|
Polar Coordinates - r
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiplying complex numbers
|z-w|

9. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex numbers are points in the plane
Complex Number Formula
real
De Moivre's Theorem

10. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
(a + c) + ( b + d)i
Absolute Value of a Complex Number
Rules of Complex Arithmetic
i^3

11. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
z - z*
Rules of Complex Arithmetic
i²
|z| = mod(z)

12. A + bi
standard form of complex numbers
has a solution.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - cos?

13. A+bi
Complex Number Formula
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z

14. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
complex
the distance from z to the origin in the complex plane
Argand diagram
How to multiply complex nubers(2+i)(2i-3)

15. 1
i²
standard form of complex numbers
cos iy
(cos? +isin?)n

16. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Numbers: Add & subtract
multiplying complex numbers
a + bi for some real a and b.
Complex Number Formula

17. The modulus of the complex number z= a + ib now can be interpreted as
conjugate
ln z
the distance from z to the origin in the complex plane
|z| = mod(z)

18. The complex number z representing a+bi.
Affix
Complex numbers are points in the plane
Complex Division
Argand diagram

19. (a + bi)(c + bi) =
Real and Imaginary Parts
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number

20. ½(e^(iz) + e^(-iz))
Polar Coordinates - sin?
standard form of complex numbers
Complex Number
cos z

21. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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22. (e^(iz) - e^(-iz)) / 2i
i^1
sin z
Real Numbers
Roots of Unity

23. Cos n? + i sin n? (for all n integers)
zz*
z - z*
(cos? +isin?)n
How to multiply complex nubers(2+i)(2i-3)

24. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Subfield
How to multiply complex nubers(2+i)(2i-3)
sin z
adding complex numbers

25. E ^ (z2 ln z1)
How to solve (2i+3)/(9-i)
Roots of Unity
i^2 = -1
z1 ^ (z2)

26. 2a
For real a and b - a + bi = 0 if and only if a = b = 0
z + z*
Imaginary Unit
Rules of Complex Arithmetic

27. Multiply moduli and add arguments
i^2
Polar Coordinates - cos?
Polar Coordinates - Division
Polar Coordinates - Multiplication

28. 1
i^4
cos iy
the distance from z to the origin in the complex plane
Polar Coordinates - Division

29. (e^(-y) - e^(y)) / 2i = i sinh y
has a solution.
Complex Conjugate
Real Numbers
sin iy

30. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
the vector (a -b)
Field
How to solve (2i+3)/(9-i)
z + z*

31. 2ib
z - z*
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)
i²

32. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Liouville's Theorem -
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

33. I
subtracting complex numbers
De Moivre's Theorem
i^1
standard form of complex numbers

34. 2nd. Rule of Complex Arithmetic

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35. No i
Complex Numbers: Multiply
real
Complex Number
irrational

36. Root negative - has letter i
transcendental
the distance from z to the origin in the complex plane
e^(ln z)
imaginary

37. 1
Field
i^2
integers
(cos? +isin?)n

38. V(zz*) = v(a² + b²)
has a solution.
Complex Number
|z| = mod(z)
irrational

39. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
(a + c) + ( b + d)i
Argand diagram
How to add and subtract complex numbers (2-3i)-(4+6i)

40. 5th. Rule of Complex Arithmetic
Real and Imaginary Parts
irrational
i^3
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

41. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts

42. I^2 =
Real and Imaginary Parts
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex numbers

43. Divide moduli and subtract arguments
z1 / z2
Polar Coordinates - Division
four different numbers: i - -i - 1 - and -1.
Complex Addition

44. Real and imaginary numbers
Complex Division
the vector (a -b)
i^2
complex numbers

45. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
-1
|z-w|
multiply the numerator and the denominator by the complex conjugate of the denominator.
We say that c+di and c-di are complex conjugates.

46. Not on the numberline
the vector (a -b)
non-integers
i^2 = -1
Polar Coordinates - z?¹

47. 1
i^0
x-axis in the complex plane
z1 / z2
i^2

48. Starts at 1 - does not include 0
0 if and only if a = b = 0
'i'
Polar Coordinates - Multiplication
natural

49. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
point of inflection
has a solution.
zz*
ln z

50. When two complex numbers are multipiled together.
How to add and subtract complex numbers (2-3i)-(4+6i)
Real and Imaginary Parts
Complex Multiplication
natural

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CLEP General Mathematics: Complex Numbers

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