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CLEP General Mathematics: Complex Numbers

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1. 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
Affix
adding complex numbers
Absolute Value of a Complex Number
complex numbers

2. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
|z| = mod(z)
Polar Coordinates - r
cosh²y - sinh²y
The Complex Numbers

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

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4. Numbers on a numberline
sin iy
integers
z - z*
the vector (a -b)

5. Like pi
multiplying complex numbers
transcendental
How to multiply complex nubers(2+i)(2i-3)
i^4

6. 4th. Rule of Complex Arithmetic
zz*
Polar Coordinates - z
Complex Exponentiation
(a + bi) = (c + bi) = (a + c) + ( b + d)i

7. Starts at 1 - does not include 0
i^0
natural
z1 ^ (z2)
Imaginary number

8. A number that cannot be expressed as a fraction for any integer.
Complex Addition
transcendental
i^2 = -1
Irrational Number

9. A + bi
integers
subtracting complex numbers
standard form of complex numbers
Integers

10. Written as fractions - terminating + repeating decimals
How to multiply complex nubers(2+i)(2i-3)
i^2 = -1
(a + c) + ( b + d)i
rational

11. A subset within a field.
complex
Subfield
imaginary
-1

12. A complex number may be taken to the power of another complex number.
non-integers
|z| = mod(z)
four different numbers: i - -i - 1 - and -1.
Complex Exponentiation

13. When two complex numbers are divided.
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication
Complex Division
conjugate pairs

14. z1z2* / |z2|²
imaginary
v(-1)
z1 / z2
Liouville's Theorem -

15. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
z - z*
(cos? +isin?)n
De Moivre's Theorem
Complex Number

16. x / r
Polar Coordinates - cos?
Complex Division
Complex Numbers: Multiply
can't get out of the complex numbers by adding (or subtracting) or multiplying two

17. Real and imaginary numbers
Subfield
x-axis in the complex plane
i^2
complex numbers

18. The complex number z representing a+bi.
point of inflection
z1 ^ (z2)
Affix
Every complex number has the 'Standard Form': a + bi for some real a and b.

19. 1st. Rule of Complex Arithmetic
i^2 = -1
i^4
Roots of Unity
Complex Division

20. Not on the numberline
Complex Conjugate
Complex numbers are points in the plane
We say that c+di and c-di are complex conjugates.
non-integers

21. For real a and b - a + bi =
De Moivre's Theorem
0 if and only if a = b = 0
interchangeable
i²

22. When two complex numbers are added together.
Complex Addition
x-axis in the complex plane
cosh²y - sinh²y
Argand diagram

23. 1
complex
Complex Exponentiation
i^0
Imaginary number

24. I = imaginary unit - i² = -1 or i = v-1
Complex Number
Imaginary Numbers
i^2
'i'

25. 1
Roots of Unity
Polar Coordinates - cos?
cosh²y - sinh²y
point of inflection

26. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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27. I^2 =
(cos? +isin?)n
interchangeable
Real Numbers
-1

28. Derives z = a+bi
Complex Conjugate
i^2 = -1
Euler Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.

29. ½(e^(iz) + e^(-iz))
ln z
cos z
v(-1)
Subfield

30. I
natural
sin z
Polar Coordinates - z?¹
v(-1)

31. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 / z2
irrational

32. Divide moduli and subtract arguments
Polar Coordinates - Division
Complex Number Formula
i^0
conjugate pairs

33. 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
Polar Coordinates - r
We say that c+di and c-di are complex conjugates.
Complex Number Formula
cos z

34. A+bi
z - z*
Complex Number Formula
i^4
|z| = mod(z)

35. We can also think of the point z= a+ ib as
the vector (a -b)
'i'
i^3
Polar Coordinates - cos?

36. 2ib
Real and Imaginary Parts
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate
z - z*

37. E ^ (z2 ln z1)
z1 ^ (z2)
Complex Number Formula
Square Root
Euler's Formula

38. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - cos?
Polar Coordinates - sin?
Absolute Value of a Complex Number
How to find any Power

39. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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40. A² + b² - real and non negative
zz*
Polar Coordinates - Arg(z*)
We say that c+di and c-di are complex conjugates.
the vector (a -b)

41. Equivalent to an Imaginary Unit.
Imaginary number
zz*
i^4
integers

42. 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
x-axis in the complex plane
Complex Number Formula
Rules of Complex Arithmetic
i^4

43. xpressions such as ``the complex number z'' - and ``the point z'' are now
z1 ^ (z2)
i²
Rules of Complex Arithmetic
interchangeable

44. Cos n? + i sin n? (for all n integers)
subtracting complex numbers
(cos? +isin?)n
complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

45. I
conjugate pairs
a + bi for some real a and b.
imaginary
i^1

46. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
integers
Roots of Unity
Imaginary Unit
Complex Number

47. The product of an imaginary number and its conjugate is
The Complex Numbers
a real number: (a + bi)(a - bi) = a² + b²
the vector (a -b)
Imaginary Numbers

48. A plot of complex numbers as points.
Argand diagram
Square Root
real
radicals

49. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Numbers: Add & subtract
How to add and subtract complex numbers (2-3i)-(4+6i)
Liouville's Theorem -
e^(ln z)

50. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
z1 / z2
complex

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CLEP General Mathematics: Complex Numbers

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