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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. z1z2* / |z2|²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 / z2
i²
Imaginary number

2. Has exactly n roots by the fundamental theorem of algebra
the complex numbers
Any polynomial O(xn) - (n > 0)
e^(ln z)
Polar Coordinates - sin?

3. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Polar Coordinates - sin?
conjugate
Euler's Formula

4. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - z
Roots of Unity
'i'
Complex Subtraction

5. Derives z = a+bi
Euler Formula
adding complex numbers
sin iy
Irrational Number

6. 3rd. Rule of Complex Arithmetic
real
Polar Coordinates - sin?
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - cos?

7. A number that can be expressed as a fraction p/q where q is not equal to 0.
complex
Complex Numbers: Multiply
How to multiply complex nubers(2+i)(2i-3)
Rational Number

8. When two complex numbers are added together.
Imaginary number
real
i^3
Complex Addition

9. Every complex number has the 'Standard Form':
(cos? +isin?)n
cosh²y - sinh²y
a + bi for some real a and b.
Polar Coordinates - z

10. Have radical
Affix
radicals
e^(ln z)
Complex Addition

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

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12. 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
the complex numbers
cos z
Euler Formula
Real and Imaginary Parts

13. Cos n? + i sin n? (for all n integers)
standard form of complex numbers
De Moivre's Theorem
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n

14. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
complex
Euler Formula
Polar Coordinates - z?¹
multiplying complex numbers

15. 1
Polar Coordinates - z
(a + c) + ( b + d)i
-1
i^0

16. 1
subtracting complex numbers
ln z
i²
x-axis in the complex plane

17. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
0 if and only if a = b = 0
zz*
Liouville's Theorem -

18. Like pi
has a solution.
x-axis in the complex plane
transcendental
Absolute Value of a Complex Number

19. A number that cannot be expressed as a fraction for any integer.
Complex Addition
Liouville's Theorem -
Irrational Number
natural

20. (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
Complex Number Formula
Irrational Number
How to solve (2i+3)/(9-i)
sin iy

21. When two complex numbers are subtracted from one another.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Subtraction
How to find any Power

22. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.

23. 1
Euler's Formula
real
cos iy
i^2

24. V(x² + y²) = |z|
Polar Coordinates - r
rational
How to find any Power
Irrational Number

25. (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)
Complex Division
(a + c) + ( b + d)i
Liouville's Theorem -

26. 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.
How to find any Power
interchangeable
conjugate
sin z

27. The square root of -1.
Imaginary Unit
cos z
Complex Numbers: Multiply
i²

28. V(zz*) = v(a² + b²)
Absolute Value of a Complex Number
Square Root
|z| = mod(z)
How to multiply complex nubers(2+i)(2i-3)

29. E ^ (z2 ln z1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 ^ (z2)
Rules of Complex Arithmetic
cos z

30. Written as fractions - terminating + repeating decimals
z - z*
(cos? +isin?)n
rational
natural

31. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Division
sin z
natural
complex

32. 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
subtracting complex numbers
-1
Complex numbers are points in the plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

33. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
cos iy
Polar Coordinates - Arg(z*)
Real and Imaginary Parts

34. When two complex numbers are divided.
irrational
Complex Division
Complex Number
subtracting complex numbers

35. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Square Root
cos z
Rules of Complex Arithmetic

36. 2ib
Any polynomial O(xn) - (n > 0)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental
z - z*

37. Imaginary number

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38. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Affix
cosh²y - sinh²y
z1 / z2

39. 2nd. Rule of Complex Arithmetic

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40. Equivalent to an Imaginary Unit.
Roots of Unity
complex numbers
z1 / z2
Imaginary number

41. Rotates anticlockwise by p/2
Complex Numbers: Add & subtract
Irrational Number
The Complex Numbers
Polar Coordinates - Multiplication by i

42. 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
integers
Rules of Complex Arithmetic
adding complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

43. E^(ln r) e^(i?) e^(2pin)
ln z
Complex Numbers: Add & subtract
Polar Coordinates - Division
e^(ln z)

44. (a + bi)(c + bi) =
zz*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.
four different numbers: i - -i - 1 - and -1.

45. y / r
Real and Imaginary Parts
Complex Division
Polar Coordinates - sin?
cos iy

46. I
v(-1)
Subfield
sin iy
Polar Coordinates - Multiplication

47. 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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48. Multiply moduli and add arguments
De Moivre's Theorem
Polar Coordinates - Multiplication
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

49. All the powers of i can be written as
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.
sin z
e^(ln z)

50. Root negative - has letter i
0 if and only if a = b = 0
imaginary
multiplying complex numbers
z1 / z2