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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. 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
cos z
Integers
Imaginary Unit
Complex Numbers: Add & subtract

2. Not on the numberline
non-integers
a + bi for some real a and b.
Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.

3. When two complex numbers are multipiled together.
Liouville's Theorem -
Complex Multiplication
Complex Exponentiation
Polar Coordinates - Multiplication

4. When two complex numbers are divided.
Complex Division
Polar Coordinates - z
Complex Number
(a + c) + ( b + d)i

5. 1
complex numbers
i²
Real Numbers
Euler's Formula

6. Cos n? + i sin n? (for all n integers)
complex numbers
Complex Numbers: Multiply
(cos? +isin?)n
Polar Coordinates - Multiplication

7. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
(cos? +isin?)n
standard form of complex numbers
Affix

8. ½(e^(iz) + e^(-iz))
e^(ln z)
the distance from z to the origin in the complex plane
cos z
subtracting complex numbers

9. A+bi
standard form of complex numbers
Complex Number Formula
Complex Numbers: Multiply
sin z

10. 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)
Rational Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

11. A complex number may be taken to the power of another complex number.
Complex Exponentiation
How to solve (2i+3)/(9-i)
sin z
i^2

12. V(zz*) = v(a² + b²)
adding complex numbers
integers
Complex Number
|z| = mod(z)

13. Root negative - has letter i
Imaginary Unit
imaginary
Complex Numbers: Multiply
Complex Subtraction

14. 3
i^0
i²
i^3
Complex Conjugate

15. z1z2* / |z2|²
We say that c+di and c-di are complex conjugates.
How to find any Power
Liouville's Theorem -
z1 / z2

16. Imaginary number

17. The complex number z representing a+bi.
The Complex Numbers
Polar Coordinates - Arg(z*)
Affix
imaginary

18. Numbers on a numberline
Imaginary Unit
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Argand diagram
integers

19. x + iy = r(cos? + isin?) = re^(i?)
natural
Complex numbers are points in the plane
Polar Coordinates - z
the distance from z to the origin in the complex plane

20. A² + b² - real and non negative
Any polynomial O(xn) - (n > 0)
zz*
z1 / z2
(a + c) + ( b + d)i

21. All the powers of i can be written as
i^4
four different numbers: i - -i - 1 - and -1.
i^3
i^2

22. 4th. Rule of Complex Arithmetic
cos z
Complex Subtraction
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the distance from z to the origin in the complex plane

23. To simplify a complex fraction
Rational Number
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
Argand diagram

24. 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
radicals
i^2
subtracting complex numbers
Square Root

25. y / r
Affix
z1 / z2
rational
Polar Coordinates - sin?

26. ? = -tan?
i^1
Complex Number
sin iy
Polar Coordinates - Arg(z*)

27. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Addition
Integers
i^0
Any polynomial O(xn) - (n > 0)

28. Divide moduli and subtract arguments
Polar Coordinates - Division
Complex Number
complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

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

30. Any number not rational
transcendental
irrational
sin iy
real

31. A complex number and its conjugate
conjugate pairs
z1 ^ (z2)
Complex Division
The Complex Numbers

32. Like pi
Argand diagram
z1 / z2
transcendental
irrational

33. Starts at 1 - does not include 0
0 if and only if a = b = 0
natural
(cos? +isin?)n
Polar Coordinates - sin?

34. Derives z = a+bi
i^2 = -1
Euler Formula
irrational
Field

35. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
Euler's Formula
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z
Imaginary Numbers

36. 1
z + z*
multiplying complex numbers
i^2
transcendental

37. The modulus of the complex number z= a + ib now can be interpreted as
v(-1)
Integers
the distance from z to the origin in the complex plane
sin z

38. I
Real Numbers
v(-1)
cosh²y - sinh²y
i^2

39. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
cos iy
multiplying complex numbers
non-integers

40. The reals are just the
Rules of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
x-axis in the complex plane
The Complex Numbers

41. 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
Subfield
natural
adding complex numbers
subtracting complex numbers

42. 2ib
Polar Coordinates - cos?
z - z*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
point of inflection

43. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
-1
How to multiply complex nubers(2+i)(2i-3)
the complex numbers
conjugate

44. I^2 =
Euler's Formula
zz*
i²
-1

45. The field of all rational and irrational numbers.
Polar Coordinates - r
Polar Coordinates - Division
z - z*
Real Numbers

46. V(x² + y²) = |z|
integers
cos z
Polar Coordinates - r
sin z

47. When two complex numbers are added together.
Affix
Complex Addition
Liouville's Theorem -
a real number: (a + bi)(a - bi) = a² + b²

48. 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.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power
Complex Numbers: Multiply
point of inflection

49. 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
(cos? +isin?)n
the complex numbers
i²
i^2 = -1

50. I
i^0
For real a and b - a + bi = 0 if and only if a = b = 0
i^1
Complex Subtraction