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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. I^2 =
-1
(cos? +isin?)n
z1 / z2
Complex Number Formula

2. V(x² + y²) = |z|
Rules of Complex Arithmetic
Polar Coordinates - r
real
radicals

3. (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
point of inflection
z - z*
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Multiplication by i

4. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
The Complex Numbers
We say that c+di and c-di are complex conjugates.
Irrational Number

5. Root negative - has letter i
Complex Subtraction
x-axis in the complex plane
Euler Formula
imaginary

6. I
'i'
irrational
i^1
adding complex numbers

7. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
cos iy
Imaginary Numbers
conjugate
ln z

8. (e^(-y) - e^(y)) / 2i = i sinh y
Rational Number
z1 / z2
Absolute Value of a Complex Number
sin iy

9. y / r
z - z*
Subfield
Polar Coordinates - Multiplication
Polar Coordinates - sin?

10. 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
rational
Complex Numbers: Add & subtract
subtracting complex numbers
Affix

11. 1st. Rule of Complex Arithmetic
Polar Coordinates - z
i^2 = -1
'i'
point of inflection

12. Rotates anticlockwise by p/2
Euler's Formula
i^3
Polar Coordinates - Multiplication by i
For real a and b - a + bi = 0 if and only if a = b = 0

13. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
cos iy
v(-1)
Imaginary Numbers
Roots of Unity

14. I
Polar Coordinates - Multiplication
the complex numbers
v(-1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

15. A complex number may be taken to the power of another complex number.
cos z
Complex Exponentiation
Polar Coordinates - r
conjugate pairs

16. A² + b² - real and non negative
Polar Coordinates - Multiplication by i
zz*
'i'
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. A number that cannot be expressed as a fraction for any integer.
Irrational Number
i^0
Polar Coordinates - cos?
z1 ^ (z2)

18. 2nd. Rule of Complex Arithmetic

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19. 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
cos iy
Complex Numbers: Multiply
We say that c+di and c-di are complex conjugates.
i^4

20. 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
Complex Division
Real and Imaginary Parts
Polar Coordinates - z
Polar Coordinates - sin?

21. Written as fractions - terminating + repeating decimals
rational
ln z
complex numbers
Absolute Value of a Complex Number

22. Multiply moduli and add arguments
i²
x-axis in the complex plane
Polar Coordinates - Multiplication
can't get out of the complex numbers by adding (or subtracting) or multiplying two

23. 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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24. (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 Division
conjugate pairs
How to solve (2i+3)/(9-i)
Complex Addition

25. A plot of complex numbers as points.
i^1
Field
Argand diagram
z1 ^ (z2)

26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Imaginary number
has a solution.
complex numbers
Field

27. When two complex numbers are added together.
z - z*
Complex Addition
Polar Coordinates - Arg(z*)
integers

28. All the powers of i can be written as
the distance from z to the origin in the complex plane
rational
adding complex numbers
four different numbers: i - -i - 1 - and -1.

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

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30. 1
i^0
v(-1)
Euler Formula
i^2

31. ? = -tan?
i^2
natural
i^4
Polar Coordinates - Arg(z*)

32. The modulus of the complex number z= a + ib now can be interpreted as
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
Absolute Value of a Complex Number
the distance from z to the origin in the complex plane

33. The complex number z representing a+bi.
0 if and only if a = b = 0
z - z*
Imaginary number
Affix

34. The square root of -1.
real
i²
Imaginary Unit
multiplying complex numbers

35. 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.
subtracting complex numbers
Complex numbers are points in the plane
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication by i

36. Given (4-2i) the complex conjugate would be (4+2i)
i²
Polar Coordinates - Arg(z*)
Complex Conjugate
radicals

37. 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
interchangeable
the complex numbers
subtracting complex numbers
|z-w|

38. Starts at 1 - does not include 0
natural
has a solution.
Complex Subtraction
0 if and only if a = b = 0

39. Where the curvature of the graph changes
rational
point of inflection
Subfield
complex

40. Like pi
transcendental
-1
four different numbers: i - -i - 1 - and -1.
The Complex Numbers

41. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Square Root
Complex Multiplication
v(-1)

42. When two complex numbers are multipiled together.
Complex Multiplication
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
Polar Coordinates - cos?

43. A number that can be expressed as a fraction p/q where q is not equal to 0.
complex
Rational Number
i^0
Every complex number has the 'Standard Form': a + bi for some real a and b.

44. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate
i²
Complex Subtraction

45. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number
Argand diagram
Polar Coordinates - r

46. When two complex numbers are subtracted from one another.
Complex Subtraction
|z| = mod(z)
Polar Coordinates - z?¹
multiplying complex numbers

47. 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
De Moivre's Theorem
v(-1)
Complex numbers are points in the plane
the complex numbers

48. ½(e^(iz) + e^(-iz))
cos z
|z-w|
Polar Coordinates - z
Absolute Value of a Complex Number

49. 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.'
adding complex numbers
Liouville's Theorem -
Complex Number
Rules of Complex Arithmetic

50. 3
Real and Imaginary Parts
i^3
natural
Complex Number