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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 number that cannot be expressed as a fraction for any integer.
Irrational Number
z + z*
imaginary
zz*

2. The product of an imaginary number and its conjugate is
sin z
z - z*
a real number: (a + bi)(a - bi) = a² + b²
z1 / z2

3. 4th. Rule of Complex Arithmetic
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
-1
Polar Coordinates - Division

4. A + bi
Complex numbers are points in the plane
Rules of Complex Arithmetic
radicals
standard form of complex numbers

5. Real and imaginary numbers
complex numbers
Complex Numbers: Multiply
has a solution.
Real Numbers

6. We see in this way that the distance between two points z and w in the complex plane is
the complex numbers
-1
|z-w|
i^0

7. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
transcendental
Polar Coordinates - z?¹
Absolute Value of a Complex Number

8. The modulus of the complex number z= a + ib now can be interpreted as
zz*
the distance from z to the origin in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
'i'

9. ? = -tan?
adding complex numbers
Polar Coordinates - Arg(z*)
Irrational Number
Polar Coordinates - cos?

10. 2a
z + z*
Complex Conjugate
Complex Subtraction
'i'

11. 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
four different numbers: i - -i - 1 - and -1.
Liouville's Theorem -
adding complex numbers
(a + c) + ( b + d)i

12. Have radical
point of inflection
radicals
irrational
cos iy

13. 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.'
Complex Number
Polar Coordinates - Multiplication by i
Complex Addition
Argand diagram

14. 1
i^2 = -1
|z| = mod(z)
Field
i^4

15. Derives z = a+bi
Polar Coordinates - cos?
We say that c+di and c-di are complex conjugates.
Euler Formula
the complex numbers

16. When two complex numbers are added together.
Complex Addition
Subfield
adding complex numbers
conjugate

17. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
conjugate pairs
z1 ^ (z2)
Integers

18. 3rd. Rule of Complex Arithmetic
i^2 = -1
cosh²y - sinh²y
e^(ln z)
For real a and b - a + bi = 0 if and only if a = b = 0

19. Given (4-2i) the complex conjugate would be (4+2i)
zz*
Complex Numbers: Add & subtract
Complex Addition
Complex Conjugate

20. 1st. Rule of Complex Arithmetic
i^2 = -1
conjugate
Euler Formula
'i'

21. When two complex numbers are subtracted from one another.
Complex Subtraction
How to multiply complex nubers(2+i)(2i-3)
has a solution.
Complex Numbers: Multiply

22. Rotates anticlockwise by p/2
'i'
Polar Coordinates - Multiplication by i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
The Complex Numbers

23. Where the curvature of the graph changes
Complex Numbers: Multiply
Polar Coordinates - sin?
z - z*
point of inflection

24. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
Subfield
zz*

25. When two complex numbers are divided.
i^0
Real and Imaginary Parts
the distance from z to the origin in the complex plane
Complex Division

26. Root negative - has letter i
De Moivre's Theorem
imaginary
adding complex numbers
Euler Formula

27. (a + bi) = (c + bi) =
conjugate
Complex Numbers: Add & subtract
a + bi for some real a and b.
(a + c) + ( b + d)i

28. 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: Multiply
transcendental
Complex numbers are points in the plane
imaginary

29. (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
How to solve (2i+3)/(9-i)
subtracting complex numbers
non-integers
i²

30. Imaginary number

31. (a + bi)(c + bi) =
the complex numbers
Imaginary Numbers
Square Root
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

32. A² + b² - real and non negative
Argand diagram
subtracting complex numbers
zz*
Euler Formula

33. 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
How to solve (2i+3)/(9-i)
|z| = mod(z)
multiplying complex numbers

34. 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.
Polar Coordinates - Multiplication by i
i²
Complex Numbers: Multiply
z - z*

35. No i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix
real
Complex Subtraction

36. xpressions such as ``the complex number z'' - and ``the point z'' are now
i^1
Polar Coordinates - Multiplication by i
Every complex number has the 'Standard Form': a + bi for some real a and b.
interchangeable

37. R?¹(cos? - isin?)
complex numbers
Polar Coordinates - z?¹
Subfield
cos z

38. All numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
complex
Polar Coordinates - sin?
Affix

39. 2nd. Rule of Complex Arithmetic

40. For real a and b - a + bi =
multiplying complex numbers
-1
0 if and only if a = b = 0
The Complex Numbers

41. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts
i^1
(cos? +isin?)n

42. Like pi
How to add and subtract complex numbers (2-3i)-(4+6i)
transcendental
imaginary
De Moivre's Theorem

43. V(zz*) = v(a² + b²)
transcendental
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z| = mod(z)
conjugate pairs

44. Has exactly n roots by the fundamental theorem of algebra
real
z1 / z2
Any polynomial O(xn) - (n > 0)
conjugate

45. 2ib
Polar Coordinates - z?¹
i^3
z - z*
four different numbers: i - -i - 1 - and -1.

46. ½(e^(iz) + e^(-iz))
a real number: (a + bi)(a - bi) = a² + b²
cos z
How to find any Power
multiplying complex numbers

47. x + iy = r(cos? + isin?) = re^(i?)
0 if and only if a = b = 0
z1 / z2
Polar Coordinates - z
Polar Coordinates - Multiplication

48. 1
natural
irrational
cosh²y - sinh²y
i^1

49. Not on the numberline
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Unit
non-integers
Real and Imaginary Parts

50. A+bi
cos z
e^(ln z)
cosh²y - sinh²y
Complex Number Formula