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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

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Answer 50 questions in 15 minutes.
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1. 2nd. Rule of Complex Arithmetic

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2. I
v(-1)
Roots of Unity
non-integers
Complex Number

3. V(zz*) = v(a² + b²)
zz*
a real number: (a + bi)(a - bi) = a² + b²
|z| = mod(z)
Imaginary number

4. (a + bi) = (c + bi) =
i²
a + bi for some real a and b.
Complex numbers are points in the plane
(a + c) + ( b + d)i

5. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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6. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
|z-w|
conjugate
Absolute Value of a Complex Number
i^2

7. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Euler Formula
conjugate
We say that c+di and c-di are complex conjugates.
i^2 = -1

8. ½(e^(iz) + e^(-iz))
i^0
Complex numbers are points in the plane
cos z
z1 / z2

9. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - sin?
irrational

10. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
four different numbers: i - -i - 1 - and -1.
i²
a real number: (a + bi)(a - bi) = a² + b²
Roots of Unity

11. For real a and b - a + bi =
Rules of Complex Arithmetic
0 if and only if a = b = 0
a + bi for some real a and b.
adding complex numbers

12. Derives z = a+bi
Polar Coordinates - r
sin iy
Euler Formula
standard form of complex numbers

13. 1
Polar Coordinates - sin?
i^4
How to find any Power
Every complex number has the 'Standard Form': a + bi for some real a and b.

14. Any number not rational
z1 / z2
point of inflection
irrational
conjugate

15. (e^(-y) - e^(y)) / 2i = i sinh y
i^2
cos iy
sin iy
interchangeable

16. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
i^4
z1 ^ (z2)
Every complex number has the 'Standard Form': a + bi for some real a and b.

17. Numbers on a numberline
integers
z + z*
complex
Rational Number

18. 3
(a + c) + ( b + d)i
irrational
Polar Coordinates - Multiplication by i
i^3

19. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
Irrational Number
Real and Imaginary Parts
Complex Numbers: Multiply
Complex Numbers: Add & subtract

20. Written as fractions - terminating + repeating decimals
0 if and only if a = b = 0
rational
Complex numbers are points in the plane
Liouville's Theorem -

21. 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.
non-integers
Complex numbers are points in the plane
Complex Number Formula
How to add and subtract complex numbers (2-3i)-(4+6i)

22. When two complex numbers are added together.
four different numbers: i - -i - 1 - and -1.
Argand diagram
Complex Addition
Polar Coordinates - z?¹

23. Given (4-2i) the complex conjugate would be (4+2i)
The Complex Numbers
Complex Subtraction
standard form of complex numbers
Complex Conjugate

24. When two complex numbers are multipiled together.
Polar Coordinates - z
Complex Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)

25. 3rd. Rule of Complex Arithmetic
Absolute Value of a Complex Number
For real a and b - a + bi = 0 if and only if a = b = 0
Rules of Complex Arithmetic
i^3

26. The complex number z representing a+bi.
Affix
i²
subtracting complex numbers
a real number: (a + bi)(a - bi) = a² + b²

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

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28. 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
We say that c+di and c-di are complex conjugates.
Complex Addition
z + z*
Polar Coordinates - sin?

29. I
i^1
subtracting complex numbers
i^2 = -1
z + z*

30. Like pi
the complex numbers
transcendental
0 if and only if a = b = 0
Field

31. We can also think of the point z= a+ ib as
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)
ln z
Irrational Number

32. No i
real
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
the vector (a -b)

33. In this amazing number field every algebraic equation in z with complex coefficients
Complex Addition
Roots of Unity
a real number: (a + bi)(a - bi) = a² + b²
has a solution.

34. A complex number may be taken to the power of another complex number.
i^1
Complex Exponentiation
Euler's Formula
Integers

35. V(x² + y²) = |z|
i^1
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r

36. 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
has a solution.
Complex Numbers: Add & subtract
Complex Conjugate
Integers

37. Has exactly n roots by the fundamental theorem of algebra
the complex numbers
cos z
conjugate pairs
Any polynomial O(xn) - (n > 0)

38. A+bi
Complex Number Formula
point of inflection
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 ^ (z2)

39. 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
Imaginary Numbers
Rules of Complex Arithmetic
interchangeable
i^1

40. z1z2* / |z2|²
z1 / z2
i^2 = -1
Complex Conjugate
Complex Number Formula

41. Rotates anticlockwise by p/2
Square Root
Polar Coordinates - Multiplication by i
z + z*
Imaginary Numbers

42. 1
Polar Coordinates - Multiplication by i
How to solve (2i+3)/(9-i)
i²
standard form of complex numbers

43. ½(e^(-y) +e^(y)) = cosh y
cos iy
ln z
|z| = mod(z)
conjugate pairs

44. y / r
rational
complex
Polar Coordinates - sin?
Rules of Complex Arithmetic

45. 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
adding complex numbers
Complex Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)
Rational Number

46. Not on the numberline
a + bi for some real a and b.
non-integers
complex
Complex Exponentiation

47. I^2 =
sin z
'i'
i^4
-1

48. To simplify a complex fraction
non-integers
Euler's Formula
complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.

49. xpressions such as ``the complex number z'' - and ``the point z'' are now
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
interchangeable
Argand diagram
cos iy

50. x / r
Polar Coordinates - cos?
cos iy
four different numbers: i - -i - 1 - and -1.
Complex Multiplication

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CLEP General Mathematics: Complex Numbers

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