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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. (a + bi)(c + bi) =
Subfield
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)
Rational Number

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
How to multiply complex nubers(2+i)(2i-3)
i^2 = -1
Absolute Value of a Complex Number
z + z*

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

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4. 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
the complex numbers
Absolute Value of a Complex Number
Square Root
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

5. The square root of -1.
adding complex numbers
Imaginary Unit
point of inflection
Imaginary Numbers

6. I
x-axis in the complex plane
i^0
Irrational Number
v(-1)

7. 2ib
z - z*
cos iy
conjugate
x-axis in the complex plane

8. A plot of complex numbers as points.
i^0
cosh²y - sinh²y
Argand diagram
Affix

9. Rotates anticlockwise by p/2
(cos? +isin?)n
rational
sin z
Polar Coordinates - Multiplication by i

10. Given (4-2i) the complex conjugate would be (4+2i)
e^(ln z)
Complex Conjugate
i^4
rational

11. The reals are just the
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z + z*
Every complex number has the 'Standard Form': a + bi for some real a and b.

12. 4th. Rule of Complex Arithmetic
How to solve (2i+3)/(9-i)
ln z
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i

13. y / r
Polar Coordinates - sin?
zz*
multiplying complex numbers
Rules of Complex Arithmetic

14. 3rd. Rule of Complex Arithmetic
Imaginary Unit
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - sin?
i^2 = -1

15. Every complex number has the 'Standard Form':
a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)
De Moivre's Theorem

16. R^2 = x
Absolute Value of a Complex Number
Polar Coordinates - Arg(z*)
Square Root
0 if and only if a = b = 0

17. To simplify the square root of a negative number
Argand diagram
Roots of Unity
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z + z*

18. 2nd. Rule of Complex Arithmetic

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19. V(x² + y²) = |z|
Polar Coordinates - Multiplication by i
Polar Coordinates - r
z1 ^ (z2)
non-integers

20. 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
Rules of Complex Arithmetic
Polar Coordinates - Multiplication by i
Complex Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)

21. 5th. Rule of Complex Arithmetic
Roots of Unity
ln z
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

22. 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
Liouville's Theorem -
z1 / z2
Complex Numbers: Add & subtract
i^1

23. No i
Complex Multiplication
cos iy
real
integers

24. A complex number and its conjugate
conjugate pairs
Argand diagram
cos z
Complex Multiplication

25. When two complex numbers are subtracted from one another.
the vector (a -b)
natural
Complex Subtraction
Real and Imaginary Parts

26. R?¹(cos? - isin?)
Polar Coordinates - z?¹
(cos? +isin?)n
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler Formula

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

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28. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
-1

29. 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
|z-w|
Real and Imaginary Parts
zz*
standard form of complex numbers

30. A subset within a field.
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary Unit
Subfield
point of inflection

31. E ^ (z2 ln z1)
four different numbers: i - -i - 1 - and -1.
0 if and only if a = b = 0
z1 ^ (z2)
'i'

32. z1z2* / |z2|²
z1 / z2
Subfield
standard form of complex numbers
Polar Coordinates - Multiplication

33. To simplify a complex fraction
four different numbers: i - -i - 1 - and -1.
zz*
Polar Coordinates - r
multiply the numerator and the denominator by the complex conjugate of the denominator.

34. A number that cannot be expressed as a fraction for any integer.
rational
Euler's Formula
conjugate
Irrational Number

35. Cos n? + i sin n? (for all n integers)
real
Imaginary number
(cos? +isin?)n
Roots of Unity

36. 1
cos z
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^4
z - z*

37. We can also think of the point z= a+ ib as
the vector (a -b)
Polar Coordinates - sin?
cos z
complex

38. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Every complex number has the 'Standard Form': a + bi for some real a and b.
rational
complex numbers

39. Any number not rational
Euler Formula
Field
Complex Multiplication
irrational

40. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - r
the distance from z to the origin in the complex plane
-1
rational

41. Real and imaginary numbers
complex numbers
Imaginary number
Any polynomial O(xn) - (n > 0)
Imaginary Unit

42. Equivalent to an Imaginary Unit.
Euler Formula
Imaginary number
conjugate pairs
Any polynomial O(xn) - (n > 0)

43. The complex number z representing a+bi.
Affix
cosh²y - sinh²y
Polar Coordinates - r
For real a and b - a + bi = 0 if and only if a = b = 0

44. (a + bi) = (c + bi) =
real
(a + c) + ( b + d)i
Polar Coordinates - Division
z1 / z2

45. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
z1 ^ (z2)
Polar Coordinates - r
Imaginary Unit

46. 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.'
Complex Number
We say that c+di and c-di are complex conjugates.
non-integers
Field

47. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - z?¹
interchangeable
i²
standard form of complex numbers

48. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary number
the distance from z to the origin in the complex plane
i^0

49. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
e^(ln z)
'i'
conjugate
-1

50. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - z?¹
cos iy
Polar Coordinates - cos?
standard form of complex numbers

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CLEP General Mathematics: Complex Numbers

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