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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
Square Root
cos z
We say that c+di and c-di are complex conjugates.
i^0

2. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
z1 / z2
|z| = mod(z)
Polar Coordinates - cos?

3. Has exactly n roots by the fundamental theorem of algebra
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos iy
Any polynomial O(xn) - (n > 0)
x-axis in the complex plane

4. Imaginary number

5. Derives z = a+bi
Imaginary Unit
i^1
Euler Formula
Polar Coordinates - z

6. I
Liouville's Theorem -
multiplying complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^1

7. Equivalent to an Imaginary Unit.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^1
Irrational Number
Imaginary number

8. 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.'
Roots of Unity
Complex Number
Any polynomial O(xn) - (n > 0)
subtracting complex numbers

9. z1z2* / |z2|²
Complex Number Formula
multiplying complex numbers
z1 / z2
i^3

10. A² + b² - real and non negative
zz*
conjugate pairs
Real and Imaginary Parts
|z-w|

11. 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.
radicals
Subfield
Complex numbers are points in the plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

12. The complex number z representing a+bi.
Affix
-1
conjugate pairs
(a + bi) = (c + bi) = (a + c) + ( b + d)i

13. 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
Field
Polar Coordinates - Multiplication by i
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane

14. 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.
Any polynomial O(xn) - (n > 0)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power
Complex numbers are points in the plane

15. Cos n? + i sin n? (for all n integers)
standard form of complex numbers
(cos? +isin?)n
'i'
(a + bi) = (c + bi) = (a + c) + ( b + d)i

16. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
natural
subtracting complex numbers
Complex Division

17. A plot of complex numbers as points.
Argand diagram
Complex Multiplication
i^3
v(-1)

18. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex numbers are points in the plane
Complex Numbers: Add & subtract
complex

19. 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 Multiplication
Real and Imaginary Parts
i^0
multiply the numerator and the denominator by the complex conjugate of the denominator.

20. When two complex numbers are subtracted from one another.
subtracting complex numbers
Complex Subtraction
Complex Addition
a + bi for some real a and b.

21. x / r
transcendental
Polar Coordinates - cos?
How to solve (2i+3)/(9-i)
How to add and subtract complex numbers (2-3i)-(4+6i)

22. Have radical
natural
sin iy
Polar Coordinates - z?¹
radicals

23. We can also think of the point z= a+ ib as
Imaginary number
Subfield
the vector (a -b)
the distance from z to the origin in the complex plane

24. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Every complex number has the 'Standard Form': a + bi for some real a and b.
point of inflection

25. 1
has a solution.
transcendental
Polar Coordinates - z
i^2

26. 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
We say that c+di and c-di are complex conjugates.
Real and Imaginary Parts
Polar Coordinates - Multiplication by i
Euler Formula

27. Written as fractions - terminating + repeating decimals
rational
the complex numbers
the distance from z to the origin in the complex plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

28. No i
Roots of Unity
cos iy
real
i^0

29. E ^ (z2 ln z1)
has a solution.
z1 ^ (z2)
i^2
a + bi for some real a and b.

30. 2a
a + bi for some real a and b.
z + z*
Polar Coordinates - Division
Any polynomial O(xn) - (n > 0)

31. 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
has a solution.
Imaginary Numbers
Rules of Complex Arithmetic
Complex Multiplication

32. Starts at 1 - does not include 0
i^3
Rules of Complex Arithmetic
natural
can't get out of the complex numbers by adding (or subtracting) or multiplying two

33. V(x² + y²) = |z|
z - z*
Polar Coordinates - r
adding complex numbers
Real and Imaginary Parts

34. All the powers of i can be written as
Complex Division
Square Root
conjugate pairs
four different numbers: i - -i - 1 - and -1.

35. I
point of inflection
v(-1)
natural
z - z*

36. V(zz*) = v(a² + b²)
|z| = mod(z)
Real and Imaginary Parts
i^4
Polar Coordinates - Division

37. Divide moduli and subtract arguments
Argand diagram
Polar Coordinates - Division
i^2 = -1
z1 / z2

38. We see in this way that the distance between two points z and w in the complex plane is
a real number: (a + bi)(a - bi) = a² + b²
|z-w|
Integers
i^4

39. ½(e^(iz) + e^(-iz))
Polar Coordinates - Multiplication by i
Real Numbers
Polar Coordinates - Arg(z*)
cos z

40. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
the distance from z to the origin in the complex plane
has a solution.
i^3
conjugate

41. 2nd. Rule of Complex Arithmetic

42. Every complex number has the 'Standard Form':
Polar Coordinates - Multiplication
a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)
Subfield

43. Not on the numberline
natural
non-integers
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Conjugate

44. R^2 = x
Polar Coordinates - cos?
Square Root
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0

45. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - Division
Rational Number
Irrational Number
How to multiply complex nubers(2+i)(2i-3)

46. ½(e^(-y) +e^(y)) = cosh y
'i'
conjugate pairs
integers
cos iy

47. 2ib
Complex Numbers: Multiply
z - z*
'i'
How to multiply complex nubers(2+i)(2i-3)

48. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - cos?

49. To simplify a complex fraction
ln z
Argand diagram
multiply the numerator and the denominator by the complex conjugate of the denominator.
We say that c+di and c-di are complex conjugates.

50. 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
i^0
How to find any Power
the complex numbers
subtracting complex numbers