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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. Cos n? + i sin n? (for all n integers)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rational Number
Real and Imaginary Parts
(cos? +isin?)n

2. No i
conjugate pairs
real
Complex Number
z - z*

3. xpressions such as ``the complex number z'' - and ``the point z'' are now
z + z*
Complex Subtraction
interchangeable
real

4. Multiply moduli and add arguments
Complex Number Formula
standard form of complex numbers
Polar Coordinates - Multiplication
conjugate

5. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Any polynomial O(xn) - (n > 0)
four different numbers: i - -i - 1 - and -1.
i^2

6. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Field
z + z*
Polar Coordinates - r

7. The field of all rational and irrational numbers.
i^2 = -1
Real and Imaginary Parts
Real Numbers
transcendental

8. (a + bi)(c + bi) =
Liouville's Theorem -
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
Imaginary number

9. 3rd. Rule of Complex Arithmetic
The Complex Numbers
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)

10. 5th. Rule of Complex Arithmetic
Real Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the distance from z to the origin in the complex plane
|z-w|

11. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
transcendental
|z| = mod(z)
Integers
Imaginary number

12. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
standard form of complex numbers
ln z
How to add and subtract complex numbers (2-3i)-(4+6i)
i^4

13. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Arg(z*)
|z-w|
zz*
a + bi for some real a and b.

14. 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
multiplying complex numbers
i^3
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts

15. For real a and b - a + bi =
0 if and only if a = b = 0
multiply the numerator and the denominator by the complex conjugate of the denominator.
non-integers
the complex numbers

16. 1
Liouville's Theorem -
Polar Coordinates - sin?
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

17. The reals are just the
four different numbers: i - -i - 1 - and -1.
interchangeable
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

18. When two complex numbers are divided.
point of inflection
Irrational Number
i²
Complex Division

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. Where the curvature of the graph changes
point of inflection
Complex Conjugate
Polar Coordinates - cos?
a + bi for some real a and b.

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

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22. (e^(iz) - e^(-iz)) / 2i
Argand diagram
sin z
(cos? +isin?)n
(a + bi) = (c + bi) = (a + c) + ( b + d)i

23. V(zz*) = v(a² + b²)
Complex Number
|z| = mod(z)
|z-w|
We say that c+di and c-di are complex conjugates.

24. E ^ (z2 ln z1)
cos z
z1 ^ (z2)
Complex Number Formula
For real a and b - a + bi = 0 if and only if a = b = 0

25. Any number not rational
non-integers
irrational
i^3
How to multiply complex nubers(2+i)(2i-3)

26. 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
transcendental
Absolute Value of a Complex Number
Complex Numbers: Add & subtract
Real and Imaginary Parts

27. R^2 = x
Field
Square Root
-1
For real a and b - a + bi = 0 if and only if a = b = 0

28. Real and imaginary numbers
0 if and only if a = b = 0
complex numbers
De Moivre's Theorem
Polar Coordinates - z

29. (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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
i^1
irrational

30. A + bi
multiplying complex numbers
standard form of complex numbers
Complex Conjugate
i²

31. E^(ln r) e^(i?) e^(2pin)
Complex Exponentiation
e^(ln z)
i^2
ln z

32. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Liouville's Theorem -
Subfield
Rules of Complex Arithmetic

33. Derives z = a+bi
i^1
Irrational Number
Euler Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

34. When two complex numbers are subtracted from one another.
Integers
De Moivre's Theorem
Complex Subtraction
Absolute Value of a Complex Number

35. Every complex number has the 'Standard Form':
Rules of Complex Arithmetic
Liouville's Theorem -
a + bi for some real a and b.
standard form of complex numbers

36. 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
Polar Coordinates - Arg(z*)
irrational
Subfield
Rules of Complex Arithmetic

37. A complex number may be taken to the power of another complex number.
the distance from z to the origin in the complex plane
Euler Formula
Complex Exponentiation
i^2

38. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
e^(ln z)
Rules of Complex Arithmetic
|z| = mod(z)

39. A complex number and its conjugate
Field
conjugate
conjugate pairs
rational

40. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Complex Number
Polar Coordinates - Multiplication
complex numbers

41. z1z2* / |z2|²
z1 / z2
Rules of Complex Arithmetic
natural
Absolute Value of a Complex Number

42. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers
i^2
(a + bi) = (c + bi) = (a + c) + ( b + d)i

43. 2a
i²
z + z*
Complex Subtraction
conjugate pairs

44. The square root of -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary Unit
(cos? +isin?)n
Euler Formula

45. 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.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex numbers are points in the plane
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. ½(e^(-y) +e^(y)) = cosh y
z + z*
x-axis in the complex plane
cos iy
Affix

47. A² + b² - real and non negative
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number
zz*
integers

48. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
standard form of complex numbers
Roots of Unity
conjugate
Absolute Value of a Complex Number

49. 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.'
conjugate pairs
Polar Coordinates - Multiplication
Any polynomial O(xn) - (n > 0)
Complex Number

50. Imaginary number

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CLEP General Mathematics: Complex Numbers

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