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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. 1
i^0
conjugate
Affix
a + bi for some real a and b.

2. E ^ (z2 ln z1)
Polar Coordinates - z
Polar Coordinates - Division
Polar Coordinates - r
z1 ^ (z2)

3. y / r
Polar Coordinates - sin?
Real Numbers
Affix
Polar Coordinates - z?¹

4. Starts at 1 - does not include 0
Complex Numbers: Add & subtract
subtracting complex numbers
Rational Number
natural

5. Like pi
De Moivre's Theorem
z1 / z2
cosh²y - sinh²y
transcendental

6. V(x² + y²) = |z|
Polar Coordinates - r
Polar Coordinates - Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
De Moivre's Theorem

7. 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.
Euler Formula
i^0
Complex numbers are points in the plane
x-axis in the complex plane

8. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
z - z*
How to add and subtract complex numbers (2-3i)-(4+6i)
Roots of Unity
|z| = mod(z)

9. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
standard form of complex numbers
Integers
Field
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

10. 2nd. Rule of Complex Arithmetic

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11. The field of all rational and irrational numbers.
Complex Subtraction
Polar Coordinates - sin?
Real Numbers
point of inflection

12. 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
Field
'i'
Absolute Value of a Complex Number

13. 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.
Complex Division
How to add and subtract complex numbers (2-3i)-(4+6i)
How to find any Power
How to multiply complex nubers(2+i)(2i-3)

14. 2ib
four different numbers: i - -i - 1 - and -1.
|z| = mod(z)
How to multiply complex nubers(2+i)(2i-3)
z - z*

15. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Subtraction
Euler's Formula
Complex Numbers: Add & subtract

16. 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
point of inflection
imaginary
|z-w|
Rules of Complex Arithmetic

17. When two complex numbers are added together.
Complex Addition
Complex Numbers: Multiply
v(-1)
|z-w|

18. All the powers of i can be written as
0 if and only if a = b = 0
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication
has a solution.

19. Any number not rational
Roots of Unity
Polar Coordinates - Multiplication
irrational
interchangeable

20. 1
Rules of Complex Arithmetic
i²
Euler Formula
i^2

21. 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.
x-axis in the complex plane
Polar Coordinates - z?¹
Complex Numbers: Multiply
sin z

22. When two complex numbers are multipiled together.
|z-w|
four different numbers: i - -i - 1 - and -1.
cosh²y - sinh²y
Complex Multiplication

23. A+bi
i^2
Every complex number has the 'Standard Form': a + bi for some real a and b.
z - z*
Complex Number Formula

24. A number that can be expressed as a fraction p/q where q is not equal to 0.
interchangeable
four different numbers: i - -i - 1 - and -1.
sin z
Rational Number

25. E^(ln r) e^(i?) e^(2pin)
Imaginary Numbers
Real and Imaginary Parts
'i'
e^(ln z)

26. The modulus of the complex number z= a + ib now can be interpreted as
the vector (a -b)
multiply the numerator and the denominator by the complex conjugate of the denominator.
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication

27. Real and imaginary numbers
Liouville's Theorem -
Complex numbers are points in the plane
Complex Multiplication
complex numbers

28. When two complex numbers are divided.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
Complex Number Formula
Polar Coordinates - Arg(z*)

29. Root negative - has letter i
sin iy
|z-w|
standard form of complex numbers
imaginary

30. x + iy = r(cos? + isin?) = re^(i?)
cosh²y - sinh²y
a + bi for some real a and b.
Polar Coordinates - z
How to multiply complex nubers(2+i)(2i-3)

31. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
non-integers
Field
cos z
Rules of Complex Arithmetic

32. ? = -tan?
adding complex numbers
cosh²y - sinh²y
i^1
Polar Coordinates - Arg(z*)

33. I
conjugate
Roots of Unity
i^1
zz*

34. The product of an imaginary number and its conjugate is
complex
multiplying complex numbers
a real number: (a + bi)(a - bi) = a² + b²
z + z*

35. Multiply moduli and add arguments
0 if and only if a = b = 0
Complex Multiplication
Polar Coordinates - Multiplication
Polar Coordinates - cos?

36. (e^(-y) - e^(y)) / 2i = i sinh y
Real and Imaginary Parts
Every complex number has the 'Standard Form': a + bi for some real a and b.
complex numbers
sin iy

37. For real a and b - a + bi =
0 if and only if a = b = 0
non-integers
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Division

38. 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
Polar Coordinates - Multiplication
adding complex numbers
Rational Number
the complex numbers

39. ½(e^(-y) +e^(y)) = cosh y
Irrational Number
cosh²y - sinh²y
cos iy
point of inflection

40. Have radical
transcendental
radicals
z1 ^ (z2)
four different numbers: i - -i - 1 - and -1.

41. 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
interchangeable
cos z
i^2 = -1

42. A² + b² - real and non negative
i^4
Polar Coordinates - cos?
four different numbers: i - -i - 1 - and -1.
zz*

43. ½(e^(iz) + e^(-iz))
a real number: (a + bi)(a - bi) = a² + b²
standard form of complex numbers
cos z
Absolute Value of a Complex Number

44. (a + bi)(c + bi) =
Complex Division
De Moivre's Theorem
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex

45. V(zz*) = v(a² + b²)
|z| = mod(z)
Field
The Complex Numbers
real

46. Derives z = a+bi
Euler Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
conjugate

47. Written as fractions - terminating + repeating decimals
rational
(a + c) + ( b + d)i
v(-1)
Euler Formula

48. Equivalent to an Imaginary Unit.
point of inflection
(cos? +isin?)n
Imaginary number
Polar Coordinates - Multiplication by i

49. 1
(a + c) + ( b + d)i
natural
i^4
e^(ln z)

50. 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
complex
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
We say that c+di and c-di are complex conjugates.
i²