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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. 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
sin z
cos z
(cos? +isin?)n

2. (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
i^2 = -1
sin z
How to multiply complex nubers(2+i)(2i-3)
-1

3. 3rd. Rule of Complex Arithmetic
interchangeable
zz*
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0

4. A + bi
Imaginary Numbers
a real number: (a + bi)(a - bi) = a² + b²
standard form of complex numbers
v(-1)

5. All numbers
i^3
complex
Complex Numbers: Multiply
Polar Coordinates - z?¹

6. E ^ (z2 ln z1)
z1 ^ (z2)
has a solution.
Complex Number
We say that c+di and c-di are complex conjugates.

7. All the powers of i can be written as
imaginary
Polar Coordinates - Multiplication
four different numbers: i - -i - 1 - and -1.
Integers

8. ? = -tan?
(a + c) + ( b + d)i
Irrational Number
Polar Coordinates - Arg(z*)
Imaginary Unit

9. When two complex numbers are subtracted from one another.
Complex Subtraction
cos z
Complex Numbers: Multiply
Polar Coordinates - r

10. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Imaginary Numbers
|z| = mod(z)
For real a and b - a + bi = 0 if and only if a = b = 0
Field

11. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex numbers are points in the plane
real
irrational

12. We can also think of the point z= a+ ib as
De Moivre's Theorem
Polar Coordinates - Multiplication by i
How to solve (2i+3)/(9-i)
the vector (a -b)

13. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Complex numbers are points in the plane
Complex Addition
the vector (a -b)

14. Every complex number has the 'Standard Form':
z - z*
Polar Coordinates - Division
a + bi for some real a and b.
Polar Coordinates - Arg(z*)

15. No i
real
rational
Affix
0 if and only if a = b = 0

16. Equivalent to an Imaginary Unit.
|z-w|
Imaginary number
complex numbers
(cos? +isin?)n

17. A complex number may be taken to the power of another complex number.
Real and Imaginary Parts
Complex Exponentiation
(a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)

18. xpressions such as ``the complex number z'' - and ``the point z'' are now
Every complex number has the 'Standard Form': a + bi for some real a and b.
interchangeable
sin z
Complex Conjugate

19. 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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20. Rotates anticlockwise by p/2
interchangeable
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0
v(-1)

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

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22. Real and imaginary numbers
integers
Complex numbers are points in the plane
complex numbers
sin z

23. 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
Real Numbers
sin z
z1 / z2
Real and Imaginary Parts

24. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Rules of Complex Arithmetic
non-integers
Polar Coordinates - cos?

25. 3
the distance from z to the origin in the complex plane
complex
Polar Coordinates - Multiplication by i
i^3

26. The reals are just the
multiply the numerator and the denominator by the complex conjugate of the denominator.
x-axis in the complex plane
radicals
i²

27. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)
adding complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

28. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
a real number: (a + bi)(a - bi) = a² + b²
i^1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number

29. ½(e^(-y) +e^(y)) = cosh y
i^2 = -1
cos iy
z + z*
has a solution.

30. When two complex numbers are multipiled together.
Complex Multiplication
z1 / z2
cos iy
i^2 = -1

31. The square root of -1.
Polar Coordinates - z
Complex Exponentiation
point of inflection
Imaginary Unit

32. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
Absolute Value of a Complex Number
ln z
Argand diagram
How to find any Power

33. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
i^0
the distance from z to the origin in the complex plane
e^(ln z)

34. Imaginary number

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35. The product of an imaginary number and its conjugate is
Euler Formula
non-integers
cos z
a real number: (a + bi)(a - bi) = a² + b²

36. The field of all rational and irrational numbers.
i^3
rational
Real Numbers
We say that c+di and c-di are complex conjugates.

37. (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
Roots of Unity
z + z*
has a solution.
How to solve (2i+3)/(9-i)

38. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Complex Numbers: Multiply
How to add and subtract complex numbers (2-3i)-(4+6i)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - cos?

39. A complex number and its conjugate
We say that c+di and c-di are complex conjugates.
conjugate pairs
(a + c) + ( b + d)i
i^1

40. 2a
z + z*
cosh²y - sinh²y
i^4
multiply the numerator and the denominator by the complex conjugate of the denominator.

41. Any number not rational
irrational
cos z
i^2 = -1
Every complex number has the 'Standard Form': a + bi for some real a and b.

42. When two complex numbers are divided.
Polar Coordinates - z?¹
Complex Division
the vector (a -b)
sin z

43. A number that cannot be expressed as a fraction for any integer.
ln z
e^(ln z)
natural
Irrational Number

44. I
v(-1)
Any polynomial O(xn) - (n > 0)
Affix
z + z*

45. Has exactly n roots by the fundamental theorem of algebra
Imaginary Unit
z + z*
radicals
Any polynomial O(xn) - (n > 0)

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

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47. x / r
Polar Coordinates - cos?
Complex numbers are points in the plane
Complex Addition
Any polynomial O(xn) - (n > 0)

48. To simplify the square root of a negative number
cosh²y - sinh²y
Liouville's Theorem -
sin z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

49. ½(e^(iz) + e^(-iz))
interchangeable
i^1
cos z
non-integers

50. A+bi
Polar Coordinates - Multiplication by i
Argand diagram
Complex Conjugate
Complex Number Formula

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CLEP General Mathematics: Complex Numbers

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