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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. A plot of complex numbers as points.
cos z
Irrational Number
Argand diagram
(cos? +isin?)n

2. Cos n? + i sin n? (for all n integers)
0 if and only if a = b = 0
(cos? +isin?)n
Euler's Formula
z + z*

3. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
i^1
Complex Addition
The Complex Numbers
|z| = mod(z)

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

5. All the powers of i can be written as
irrational
We say that c+di and c-di are complex conjugates.
Every complex number has the 'Standard Form': a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.

6. Numbers on a numberline
Imaginary Numbers
i^2
integers
Real and Imaginary Parts

7. Where the curvature of the graph changes
0 if and only if a = b = 0
integers
point of inflection
z + z*

8. 2a
Liouville's Theorem -
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z + z*
Argand diagram

9. 2nd. Rule of Complex Arithmetic

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10. I
v(-1)
e^(ln z)
transcendental
Field

11. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
De Moivre's Theorem
Field
Imaginary Numbers

12. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
radicals
e^(ln z)
Complex Subtraction

13. A + bi
Complex Number
Polar Coordinates - Arg(z*)
standard form of complex numbers
non-integers

14. When two complex numbers are divided.
Complex Division
Polar Coordinates - sin?
i^1
-1

15. y / r
conjugate
(cos? +isin?)n
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - sin?

16. When two complex numbers are multipiled together.
the distance from z to the origin in the complex plane
Complex Multiplication
adding complex numbers
i^1

17. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - Multiplication by i
non-integers
The Complex Numbers
Roots of Unity

18. A+bi
Polar Coordinates - cos?
Imaginary Numbers
i^0
Complex Number Formula

19. R^2 = x
non-integers
Square Root
z1 ^ (z2)
Absolute Value of a Complex Number

20. I = imaginary unit - i² = -1 or i = v-1
Field
z - z*
|z-w|
Imaginary Numbers

21. 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
e^(ln z)
Absolute Value of a Complex Number
(a + c) + ( b + d)i

22. 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 Number Formula
Subfield
Complex Numbers: Add & subtract
Complex Numbers: Multiply

23. A subset within a field.
Subfield
transcendental
rational
Polar Coordinates - z

24. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
Field
Polar Coordinates - sin?

25. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers
Complex Conjugate

26. 4th. Rule of Complex Arithmetic
conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Multiplication
point of inflection

27. ½(e^(iz) + e^(-iz))
sin z
cos z
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Subtraction

28. 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
standard form of complex numbers
How to find any Power
subtracting complex numbers
Complex Division

29. A complex number may be taken to the power of another complex number.
sin z
Integers
(a + c) + ( b + d)i
Complex Exponentiation

30. (e^(-y) - e^(y)) / 2i = i sinh y
i^0
z1 / z2
sin iy
Complex Number Formula

31. To simplify the square root of a negative number
Complex Subtraction
Complex numbers are points in the plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)

32. To simplify a complex fraction
imaginary
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Addition
For real a and b - a + bi = 0 if and only if a = b = 0

33. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
How to multiply complex nubers(2+i)(2i-3)
Square Root
Integers

34. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - sin?
'i'
the distance from z to the origin in the complex plane
i^2

35. 3rd. Rule of Complex Arithmetic
i²
For real a and b - a + bi = 0 if and only if a = b = 0
-1
transcendental

36. ? = -tan?
Polar Coordinates - Arg(z*)
v(-1)
z1 / z2
Any polynomial O(xn) - (n > 0)

37. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Subtraction
How to add and subtract complex numbers (2-3i)-(4+6i)
i^2
How to multiply complex nubers(2+i)(2i-3)

38. No i
the distance from z to the origin in the complex plane
real
We say that c+di and c-di are complex conjugates.
Imaginary Numbers

39. V(x² + y²) = |z|
Polar Coordinates - r
a + bi for some real a and b.
De Moivre's Theorem
z1 / z2

40. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Number
Polar Coordinates - cos?
Irrational Number
interchangeable

41. The complex number z representing a+bi.
cos iy
irrational
Affix
Argand diagram

42. 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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43. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
the complex numbers
Complex Numbers: Multiply
De Moivre's Theorem
Field

44. The product of an imaginary number and its conjugate is
z - z*
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n
Complex Conjugate

45. A complex number and its conjugate
-1
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0
Irrational Number

46. (a + bi)(c + bi) =
v(-1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
cos iy

47. 3
The Complex Numbers
0 if and only if a = b = 0
i^3
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

48. I^2 =
-1
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Unit
irrational

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

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50. I
point of inflection
i^0
i^1
i²

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CLEP General Mathematics: Complex Numbers

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