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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. 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
Any polynomial O(xn) - (n > 0)
has a solution.
adding complex numbers
Complex Subtraction

2. 4th. Rule of Complex Arithmetic
v(-1)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²

3. Not on the numberline
z - z*
cos iy
sin iy
non-integers

4. A number that can be expressed as a fraction p/q where q is not equal to 0.
irrational
Rational Number
Subfield
non-integers

5. 2ib
Affix
z - z*
(cos? +isin?)n
Real Numbers

6. 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.
interchangeable
i^2 = -1
How to find any Power
Polar Coordinates - z

7. ½(e^(-y) +e^(y)) = cosh y
Subfield
cos iy
i^0
has a solution.

8. 1
i^0
Complex Number Formula
adding complex numbers
Polar Coordinates - cos?

9. Rotates anticlockwise by p/2
Polar Coordinates - sin?
Polar Coordinates - Multiplication by i
standard form of complex numbers
Complex Numbers: Add & subtract

10. All the powers of i can be written as
0 if and only if a = b = 0
a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.
Liouville's Theorem -

11. (e^(-y) - e^(y)) / 2i = i sinh y
The Complex Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
For real a and b - a + bi = 0 if and only if a = b = 0
sin iy

12. The modulus of the complex number z= a + ib now can be interpreted as
adding complex numbers
Absolute Value of a Complex Number
non-integers
the distance from z to the origin in the complex plane

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

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14. The field of all rational and irrational numbers.
Complex Addition
Real Numbers
Rational Number
zz*

15. When two complex numbers are divided.
Complex Multiplication
transcendental
real
Complex Division

16. 3
Integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^3
|z| = mod(z)

17. ? = -tan?
Real and Imaginary Parts
cos iy
Polar Coordinates - Arg(z*)
(cos? +isin?)n

18. The reals are just the
x-axis in the complex plane
a real number: (a + bi)(a - bi) = a² + b²
Imaginary number
Polar Coordinates - z

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

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20. 1st. Rule of Complex Arithmetic
integers
z + z*
i^3
i^2 = -1

21. Any number not rational
standard form of complex numbers
Complex Addition
sin z
irrational

22. All numbers
complex
Complex Number Formula
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

23. 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.
Complex numbers are points in the plane
the vector (a -b)
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

24. 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.'
sin iy
We say that c+di and c-di are complex conjugates.
0 if and only if a = b = 0
Complex Number

25. A complex number may be taken to the power of another complex number.
De Moivre's Theorem
Complex Exponentiation
point of inflection
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

26. 5th. Rule of Complex Arithmetic
i²
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)

27. 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
conjugate
(a + c) + ( b + d)i
e^(ln z)

28. ½(e^(iz) + e^(-iz))
For real a and b - a + bi = 0 if and only if a = b = 0
cos z
De Moivre's Theorem
imaginary

29. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
sin z
i^4
Complex numbers are points in the plane

30. 1
zz*
e^(ln z)
i^2
standard form of complex numbers

31. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
radicals
Complex Conjugate
i²

32. 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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33. 1
How to solve (2i+3)/(9-i)
rational
Imaginary Numbers
cosh²y - sinh²y

34. x + iy = r(cos? + isin?) = re^(i?)
transcendental
The Complex Numbers
Complex Addition
Polar Coordinates - z

35. A + bi
standard form of complex numbers
'i'
-1
irrational

36. I^2 =
Real Numbers
-1
i^2
Irrational Number

37. z1z2* / |z2|²
sin iy
z1 ^ (z2)
complex numbers
z1 / z2

38. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Complex Numbers: Add & subtract
0 if and only if a = b = 0
How to find any Power

39. No i
real
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
a real number: (a + bi)(a - bi) = a² + b²

40. (e^(iz) - e^(-iz)) / 2i
Imaginary Numbers
cosh²y - sinh²y
the distance from z to the origin in the complex plane
sin z

41. The square root of -1.
Complex Exponentiation
Complex Addition
We say that c+di and c-di are complex conjugates.
Imaginary Unit

42. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
complex
Affix
subtracting complex numbers

43. The product of an imaginary number and its conjugate is
Square Root
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
Complex numbers are points in the plane

44. Starts at 1 - does not include 0
Euler's Formula
natural
Polar Coordinates - z
point of inflection

45. Cos n? + i sin n? (for all n integers)
Argand diagram
i^4
cos z
(cos? +isin?)n

46. A complex number and its conjugate
multiplying complex numbers
cos iy
cos z
conjugate pairs

47. Where the curvature of the graph changes
Liouville's Theorem -
Complex Multiplication
point of inflection
rational

48. Written as fractions - terminating + repeating decimals
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
Affix
rational

49. Given (4-2i) the complex conjugate would be (4+2i)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - cos?
Complex Conjugate
complex numbers

50. 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
Complex Numbers: Add & subtract
Polar Coordinates - z?¹
Complex Division
four different numbers: i - -i - 1 - and -1.