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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. ½(e^(-y) +e^(y)) = cosh y
cos iy
The Complex Numbers
Polar Coordinates - z?¹
transcendental

2. For real a and b - a + bi =
real
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0
Imaginary Unit

3. A plot of complex numbers as points.
Argand diagram
i^2
Polar Coordinates - sin?
i^2 = -1

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

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5. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Multiplication
z + z*
multiplying complex numbers
|z| = mod(z)

6. Equivalent to an Imaginary Unit.
Imaginary number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real Numbers
The Complex Numbers

7. The field of all rational and irrational numbers.
Real Numbers
z1 ^ (z2)
i^2 = -1
Complex Exponentiation

8. The square root of -1.
0 if and only if a = b = 0
transcendental
Imaginary Unit
integers

9. 5th. Rule of Complex Arithmetic
i^4
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Real Numbers
Complex Subtraction

10. (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
real
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Arg(z*)
i^0

11. Numbers on a numberline
integers
i^2 = -1
How to find any Power
Square Root

12. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
Complex Numbers: Add & subtract
Complex Division
Euler Formula
the complex numbers

13. I = imaginary unit - i² = -1 or i = v-1
cosh²y - sinh²y
complex
Imaginary Numbers
Roots of Unity

14. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
x-axis in the complex plane
Imaginary number

15. 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
cos z
adding complex numbers
Complex Conjugate
Complex Subtraction

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

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17. E^(ln r) e^(i?) e^(2pin)
zz*
Polar Coordinates - Multiplication by i
sin z
e^(ln z)

18. When two complex numbers are subtracted from one another.
Complex Subtraction
Complex Exponentiation
i^3
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

19. R^2 = x
Square Root
radicals
Integers
z - z*

20. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Multiplication by i
|z-w|
i^3
Polar Coordinates - z

21. Written as fractions - terminating + repeating decimals
standard form of complex numbers
rational
i^2
Rules of Complex Arithmetic

22. 1
0 if and only if a = b = 0
ln z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4

23. 1st. Rule of Complex Arithmetic
(a + c) + ( b + d)i
Real Numbers
Polar Coordinates - Multiplication by i
i^2 = -1

24. To simplify the square root of a negative number
the complex numbers
i^3
z + z*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

25. 3
a real number: (a + bi)(a - bi) = a² + b²
Complex numbers are points in the plane
i^3
Complex Division

26. Root negative - has letter i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number
imaginary
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. 1
i^0
Complex Multiplication
cos z
conjugate pairs

28. V(x² + y²) = |z|
Square Root
How to find any Power
non-integers
Polar Coordinates - r

29. 4th. Rule of Complex Arithmetic
Euler Formula
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin z

30. 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
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²
subtracting complex numbers
Complex Numbers: Add & subtract

31. We can also think of the point z= a+ ib as
standard form of complex numbers
radicals
ln z
the vector (a -b)

32. Derives z = a+bi
0 if and only if a = b = 0
Euler Formula
Real and Imaginary Parts
the complex numbers

33. 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.'
e^(ln z)
Complex Number
radicals
i^0

34. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
z1 / z2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Roots of Unity

35. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
i^0
The Complex Numbers
Polar Coordinates - cos?
cos iy

36. Like pi
Complex Subtraction
-1
imaginary
transcendental

37. 2ib
z - z*
Complex Conjugate
radicals
Polar Coordinates - Arg(z*)

38. 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
Polar Coordinates - Multiplication by i
Roots of Unity
z - z*
Real and Imaginary Parts

39. Any number not rational
complex numbers
irrational
real
De Moivre's Theorem

40. A number that can be expressed as a fraction p/q where q is not equal to 0.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
zz*
Roots of Unity
Rational Number

41. 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.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply

42. y / r
conjugate
Euler Formula
zz*
Polar Coordinates - sin?

43. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - Division
0 if and only if a = b = 0
Complex Division
sin iy

44. 1
How to find any Power
integers
adding complex numbers
i²

45. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers
Polar Coordinates - z?¹
ln z

46. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Real and Imaginary Parts
conjugate
For real a and b - a + bi = 0 if and only if a = b = 0
ln z

47. I
v(-1)
z1 ^ (z2)
adding complex numbers
Real and Imaginary Parts

48. To simplify a complex fraction
Imaginary Numbers
De Moivre's Theorem
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex

49. x / r
-1
complex numbers
Polar Coordinates - cos?
z1 / z2

50. Not on the numberline
Imaginary number
non-integers
e^(ln z)
Complex Conjugate

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CLEP General Mathematics: Complex Numbers

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